Each of the three tensor product operations $\otimes^{!}$, $\otimes^{*}$, and
$\mathbin{\overset{\leftarrow}{\vphantom{o}\smash{\otimes}}}$ defines an
associative tensor/monoidal structure on the category of topological vector
spaces; the former two tensor products are also commutative. In particular,
given a topological associative algebra ${\mathfrak{R}}$ and a $k$-vector
space $P$, there is a natural isomorphism of (nontopological) vector spaces
${\mathfrak{R}}\otimes^{^}\,({\mathfrak{R}}\otimes^{^}\,P)\simeq({\mathfrak{R}}\mathbin{\overset{\leftarrow}{\vphantom{o}\smash{\otimes}}}{\mathfrak{R}})\otimes^{^}\,P$;
hence the monad multiplication map
$\phi_{P}\colon{\mathfrak{R}}\otimes^{^}\,({\mathfrak{R}}\otimes^{^}\,P)\longrightarrow{\mathfrak{R}}\otimes^{^}\,P$
from Section 2.3 defined whenever open right ideals form a base of
neighborhoods of zero in ${\mathfrak{R}}$ [52, Section D.5.2].

For any topological vector space $V$, denote by $\bigwedge^{2,*}(V)$ the
completion of the nontopological exterior square $\bigwedge^{2}(V)$ with
respect to the topology in which a vector subspace $T\subset\bigwedge^{2}(V)$
is open if and only if there exists an open subspace $V^{\prime}\subset V$
such that $\bigwedge^{2}(V^{\prime})\subset T$ and for any vector $v\in V$
there exists an open subspace $V^{\prime\prime}\subset V$ such that $v\wedge
V^{\prime\prime}\subset T$. For any topological vector space $U$, a skew-
commutative bilinear map $V\times V\longrightarrow U$ is continuous if and
only if it can be (uniquely) extended to a continuous linear map
$\bigwedge^{2,*}(V)\longrightarrow U$. The topological vector space
$\bigwedge^{2,*}(V)$ can be also viewed as a closed vector subspace in
$V\otimes^{*}V$.

A _contramodule_ ${\mathfrak{P}}$ over a topological Lie algebra
${\mathfrak{g}}$ over a field $k$ is a $k$-vector space endowed with a
contraaction map
$\pi\colon{\mathfrak{g}}\otimes^{^}\,{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$
satisfying the following _contra-Jacobi equation_. The composition

$\textstyle\bigwedge^{2,*}({\mathfrak{g}})\otimes^{^}\,{\mathfrak{P}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathfrak{g}}\otimes^{^}\,{\mathfrak{P}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathfrak{P}}$

of the map induced by the Lie bracket
$\bigwedge^{2,*}({\mathfrak{g}})\longrightarrow{\mathfrak{g}}$ of
${\mathfrak{g}}$ with the contraaction map should be equal to the composition
of the map induced by the natural maps of topological vector spaces
$\bigwedge^{2}(V)\longrightarrow V\otimes^{*}V\longrightarrow
V\mathbin{\overset{\leftarrow}{\vphantom{o}\smash{\otimes}}}V$ considered in
the case $V={\mathfrak{g}}$ with the iterated contraaction map

$\textstyle\bigwedge^{2,*}(V)\otimes^{^}\,{\mathfrak{P}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}({\mathfrak{g}}\otimes^{*}{\mathfrak{g}})\otimes^{^}\,{\mathfrak{P}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}({\mathfrak{g}}\mathbin{\overset{\leftarrow}{\vphantom{o}\smash{\otimes}}}{\mathfrak{g}})\otimes^{^}\,{\mathfrak{P}}\\\
\text{$\mskip 1.5mu$}\simeq\text{$\mskip
1.5mu$}{\mathfrak{g}}\otimes^{^}\,({\mathfrak{g}}\otimes^{^}\,{\mathfrak{P}})\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathfrak{g}}\otimes^{^}\,{\mathfrak{P}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathfrak{P}}.$

This definition can be found in [52, Section D.2.6].

A _discrete module_ ${\mathcal{M}}$ over a topological Lie algebra
${\mathfrak{g}}$ is a ${\mathfrak{g}}$-module for which the action map
${\mathfrak{g}}\times{\mathcal{M}}\longrightarrow{\mathcal{M}}$ is continuous
in the given topology of ${\mathfrak{g}}$ and the discrete topology of
${\mathcal{M}}$. In other words, this means that the annihilator of any
element of ${\mathcal{M}}$ is an open subalgebra in ${\mathfrak{g}}$. So one
can say that discrete ${\mathfrak{g}}$-modules are a good category to be
assigned to ${\mathfrak{g}}$ when open subalgebras form a base of
neighborhoods of zero in ${\mathfrak{g}}$ (cf. [9, Sections 1.4 and 2.4]);
otherwise, a discrete ${\mathfrak{g}}$-module is the same thing as a discrete
module over the completion of ${\mathfrak{g}}$ in the new topology with a base
consisting of of the open subalgebras in the original one. For any discrete
${\mathfrak{g}}$-module ${\mathcal{M}}$ and any (nontopological) $k$-vector
space $E$, the vector space
${\mathfrak{P}}=\operatorname{Hom}_{k}({\mathcal{M}},E)$ has a natural
${\mathfrak{g}}$-contramodule structure provided by the contraaction map
$\pi\colon{\mathfrak{g}}\otimes^{^}\,\operatorname{Hom}_{k}({\mathcal{M}},E)$
defined symbolically by the formula

$\pi(x\otimes^{^}\,f)(m)=-f(xm),$

where $x\in{\mathfrak{g}}$, $m\in{\mathcal{M}}$,
$f\in\operatorname{Hom}_{k}({\mathcal{M}},E)$ and the expression in the right-
hand side makes sense due to the definition of the completed tensor product
${\mathfrak{g}}\otimes^{^}\,P$ and the discreteness condition on the
${\mathfrak{g}}$-module ${\mathcal{M}}$ [52, Section D.2.6].

The category ${\mathfrak{g}}{\operatorname{\mathsf{--discr}}}$ of discrete
${\mathfrak{g}}$-modules is abelian and the embedding/forgetful functors
${\mathfrak{g}}{\operatorname{\mathsf{--discr}}}\longrightarrow{\mathfrak{g}}{\operatorname{\mathsf{--mod}}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ from it to the categories of arbitrary
${\mathfrak{g}}$-modules and $k$-vector spaces are exact. Both infinite direct
sums and infinite products exist in
${\mathfrak{g}}{\operatorname{\mathsf{--discr}}}$; the infinite direct sums
are also preserved by the forgetful functors. It follows that filtered
inductive limits are exact in
${\mathfrak{g}}{\operatorname{\mathsf{--discr}}}$. In other words, the
category ${\mathfrak{g}}{\operatorname{\mathsf{--discr}}}$ satisfies the
axioms Ab5 and Ab3*, but not in general Ab4*. It also admits a set of
generators, so it has enough injective objects.

Any ${\mathfrak{g}}$-contamodule ${\mathfrak{P}}$ has an underlying structure
of a module over the Lie algebra ${\mathfrak{g}}$ viewed as an abstract
(nontopological) Lie algebra; it is provided by the composition of maps
${\mathfrak{g}}\otimes{\mathfrak{P}}\longrightarrow{\mathfrak{g}}\otimes^{^}\,{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$.
The category ${\mathfrak{g}}{\operatorname{\mathsf{--contra}}}$ is abelian and
the forgetful functors
${\mathfrak{g}}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathfrak{g}}{\operatorname{\mathsf{--mod}}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ are exact. Infinite products exist in the
category ${\mathfrak{g}}{\operatorname{\mathsf{--contra}}}$ and are preserved
by the forgetful functor. The theorem below, when it is applicable, allows one
to say more (cf. Sections 1.7 and 2.1).

The enveloping algebra $U({\mathfrak{g}})$ of a topological Lie algebra
${\mathfrak{g}}$ can be endowed with a natural topology in two opposite ways.
Let us denote by $U_{l}^{^}\,({\mathfrak{g}})$ the completion of
$U({\mathfrak{g}})$ in the topology where the left ideals in
$U({\mathfrak{g}})$ generated by open subspaces in ${\mathfrak{g}}$ form a
base of neighborhoods of zero, and by $U_{r}^{^}\,({\mathfrak{g}})$ the
completion of $U({\mathfrak{g}})$ in the similar topology with a base formed
by the right ideals generated by open subspaces in ${\mathfrak{g}}$. Using the
assumption of continuity of the bracket in ${\mathfrak{g}}$, one can easily
check that the multiplication in $U({\mathfrak{g}})$ can be extended to
continuous multiplications in $U_{l}^{^}\,({\mathfrak{g}})$ and
$U_{r}^{^}\,({\mathfrak{g}})$. This construction was considered in [7, Section
3.8.17], [9, Section 2.4], and [52, Section D.5.1].

######  Theorem.

(a) For any topological Lie algebra ${\mathfrak{g}}$, the category of discrete
${\mathfrak{g}}$-modules is naturally isomorphic to the category of discrete
left $U_{l}^{^}\,({\mathfrak{g}})$-modules,
${\mathfrak{g}}{\operatorname{\mathsf{--discr}}}\simeq
U_{l}^{^}\,({\mathfrak{g}}){\operatorname{\mathsf{--discr}}}$. The datum of a
discrete ${\mathfrak{g}}$-module structure on a vector space ${\mathcal{M}}$
is equivalent to the datum of a discrete left
$U_{l}^{^}\,({\mathfrak{g}})$-module structure on ${\mathcal{M}}$.

(b) For any topological Lie algebra ${\mathfrak{g}}$ admitting a countable
base of neighborhoods of zero consisting of open Lie subalgebras in
${\mathfrak{g}}$, the category of ${\mathfrak{g}}$-contramodules is naturally
isomorphic to the category of left
$U_{r}^{^}\,({\mathfrak{g}})$-contramodules,
${\mathfrak{g}}{\operatorname{\mathsf{--contra}}}\simeq
U_{r}^{^}\,({\mathfrak{g}}){\operatorname{\mathsf{--contra}}}$. The datum of a
${\mathfrak{g}}$-contramodule structure on a vector space ${\mathfrak{P}}$ is
equivalent to the datum of a left $U_{r}^{^}\,({\mathfrak{g}})$-contramodule
structure on ${\mathfrak{P}}$.

###### Proof.

Part (a): any ${\mathfrak{g}}$-module can be viewed as an
$U({\mathfrak{g}})$-module and vice versa; it is obvious from the definitions
that a ${\mathfrak{g}}$-module is discrete if and only if it its
$U({\mathfrak{g}})$-module structure extends to a structure of discrete left
module over $U_{l}^{^}\,({\mathfrak{g}})$. The proof of part (b) is more
complicated; see [52, Section D.5.3]. ∎

######  Remark.

General topology and topological algebra are known to be treacherous ground,
and caution is advisable when working with topological vector spaces with
linear topologies, as many assertions which appear to be natural at first
glance turn out to be problematic at a closer look. In particular, the
exposition in the paper [9], while correcting several mistakes or unfortunate
definitions found in the previous book [7], may be still too optimistic on a
few points.

For example, it is _not_ clear how to prove that the quotient space $V/U$ of a
topological vector space $V$ by a closed vector subspace $U$ is _complete_ in
the quotient topology, or even if it is, that the induced map of topological
tensor products $V\otimes^{!}W\longrightarrow(V/U)\otimes^{!}W$ or complete
tensor products $V\otimes^{^}\,P\longrightarrow(V/U)\otimes^{^}\,P$ is always
surjective. In both cases, the question is how to show that a map between
projective limits of vector spaces is surjective. (Cf. [11, Exercice
IV.4.10.b.$\alpha$], where a related counterexample in the setting of
topological vector spaces with nonlinear topologies compatible with the
topology of the basic field of real numbers is considered.)

The problem does not arise for topological vector spaces with countable bases
of neighborhoods of zero, as countable projective limits are better behaved,
and indeed, any closed subspace $U$ that has a countable base of neighborhoods
of zero is a topological direct summand in $V$. However, the $*$-tensor
product operation leads outside of the class of topological vector spaces with
countable topologies.

In particular, we formulate our system of contra-Jacobi equations as being
indexed by the complete tensor product
$\bigwedge^{2,*}({\mathfrak{g}})\otimes^{^}\,{\mathfrak{P}}$, while the
somewhat simpler alternative of having it indexed by the complete tensor
product $({\mathfrak{g}}\otimes^{*}{\mathfrak{g}})\otimes^{^}\,{\mathfrak{P}}$
would work just as well when the characteristic of the field $k$ is different
from $2$. Indeed, the natural map
$({\mathfrak{g}}\otimes^{*}{\mathfrak{g}})\otimes^{^}\,{\mathfrak{P}}\longrightarrow\bigwedge^{2,*}({\mathfrak{g}})\otimes^{^}\,{\mathfrak{P}}$
is surjective in this case, the topological vector space
$\bigwedge^{2,*}({\mathfrak{g}})$ being a direct summand in
${\mathfrak{g}}\otimes^{*}{\mathfrak{g}}$. Then the contra-Jacobi equation
could be written in the familiar form of the difference between the two
compositions
$({\mathfrak{g}}\otimes^{*}{\mathfrak{g}})\otimes^{^}\,{\mathfrak{P}}\rightrightarrows{\mathfrak{g}}\otimes^{^}\,({\mathfrak{g}}\otimes^{^}\,{\mathfrak{P}})\longrightarrow{\mathfrak{g}}\otimes^{^}\,{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$
being equal to the composition
$({\mathfrak{g}}\otimes^{*}{\mathfrak{g}})\otimes^{^}\,{\mathfrak{P}}\longrightarrow{\mathfrak{g}}\otimes^{^}\,{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$.
The desire to incorporate the characteristic $2$ case leads to the somewhat
more complicated definition above.

###  2.5. Contramodules over corings

The following scheme of categorical buildup is discussed in the introduction
to the book [52]. Let ${\mathsf{K}}$ be a category endowed with an
(associative, noncommutative) monoidal (tensor) category structure,
${\mathsf{M}}$ be a left module category over it, ${\mathsf{N}}$ be a right
module category, and ${\mathsf{V}}$ be a category for which there is a pairing
between the module categories ${\mathsf{N}}$ and ${\mathsf{M}}$ over
${\mathsf{K}}$ taking values in ${\mathsf{V}}$.

This means that, in addition to the multiplication functor
$\otimes\colon{\mathsf{K}}\times{\mathsf{K}}\longrightarrow{\mathsf{K}}$,
there are also action functors

$\otimes\colon{\mathsf{N}}\times{\mathsf{K}}\longrightarrow{\mathsf{N}}\quad\text{and}\quad\otimes\colon{\mathsf{K}}\times{\mathsf{M}}\longrightarrow{\mathsf{M}}$

and a pairing functor
$\otimes\colon{\mathsf{N}}\times{\mathsf{M}}\longrightarrow{\mathsf{V}}$.
Furthermore, there are associativity constraints for the ternary
multiplications

${\mathsf{K}}\times{\mathsf{K}}\times{\mathsf{K}}\longrightarrow{\mathsf{K}},\quad{\mathsf{N}}\times{\mathsf{K}}\times{\mathsf{K}}\longrightarrow{\mathsf{K}},\quad{\mathsf{K}}\times{\mathsf{K}}\times{\mathsf{M}}\longrightarrow{\mathsf{M}},\quad{\mathsf{N}}\times{\mathsf{K}}\times{\mathsf{M}}\longrightarrow{\mathsf{V}}$

satisfying the pentagonal diagram equations for products of four factors.

Let $A$ be an associative ring object in ${\mathsf{K}}$. Then one can consider
the category ${}_{A}{\mathsf{K}}_{A}$ of $A$-$A$-bimodule objects in
${\mathsf{K}}$, the category ${}_{A}{\mathsf{M}}$ of left $A$-module objects
in ${\mathsf{M}}$, and the category ${\mathsf{N}}_{A}$ of right $A$-module
objects in ${\mathsf{N}}$. When the categories ${\mathsf{K}}$, ${\mathsf{M}}$,
${\mathsf{N}}$, and ${\mathsf{V}}$ are abelian (or additive categories with
cokernels, or, at least, admit coequalizers), there are the functors
$\otimes_{A}$ of tensor product over $A$, making ${}_{A}{\mathsf{K}}_{A}$ a
tensor category, ${}_{A}{\mathsf{M}}$ a left module category over it,
${\mathsf{N}}_{A}$ a right module category, and providing a pairing
${\mathsf{N}}_{A}\times{}_{A}{\mathsf{M}}\longrightarrow{\mathsf{V}}$. The new
tensor structures $\otimes_{A}$ are associative whenever the original tensor
product functors $\otimes$ were right exact (preserved coequalizers).

Inverting the arrows in all the four categories, one comes to considering the
situation of a _co_ ring object $C\in{\mathsf{K}}$. Then there is the category
${}_{C}{\mathsf{K}}_{C}$ of $C$-$C$-bicomodule objects in ${\mathsf{K}}$, the
category ${}_{C}{\mathsf{M}}$ of left $C$-comodule objects in ${\mathsf{M}}$,
and the category ${\mathsf{N}}_{C}$ of right $C$-comodule objects in
${\mathsf{N}}$. When the categories ${\mathsf{K}}$, ${\mathsf{M}}$,
${\mathsf{N}}$, and ${\mathsf{V}}$ are abelian (or, at least, admit
equalizers), there are the functors $\mathbin{\text{\smaller$\square$}}_{C}$
of _cotensor product_ over $C$, making ${}_{C}{\mathsf{K}}_{C}$ a tensor
category, ${}_{C}{\mathsf{M}}$ a left module category over it,
${\mathsf{N}}_{C}$ a right module category, and providing a pairing
${\mathsf{N}}_{C}\times{}_{C}{\mathsf{M}}\longrightarrow{\mathsf{V}}$. The new
tensor structures $\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}$ are
associative whenever the functors $\otimes$ were left exact (preserved
equalizers).

Now one may wish to iterate this construction, considering a coring object $C$
in the category of $A$-$A$-bimodules ${}_{A}{\mathsf{K}}_{A}$, the categories
of $C$-comodules in the categories of $A$-modules ${}_{A}{\mathsf{M}}$ and
${\mathsf{N}}_{A}$, the category of $C$-$C$-bicomodules in
${}_{A}{\mathsf{K}}_{A}$, etc. Then one encounters the typical phenomenon of
progressive relaxation/worsening of algebraic properties at every step of a
buildup.

The functor $\otimes_{A}$ of tensor product over a ring object $A$ is in most
cases _not_ left exact (being defined as a certain coequalizer, it does not
preserve equalizers). Hence the cotensor product over a coring object
$C\in{}_{A}{\mathsf{K}}_{A}$ will be only associative under certain
(co)flatness conditions imposed on the objects involved. But the associativity
is necessary to even _define_ tensor products over ring objects. So when one
makes the next step and considers a ring object $S$ in the category of
$C$-$C$-bicomodules in ${}_{A}{\mathsf{K}}_{A}$, one discovers that the
functors of tensor product over $S$ are only partially defined.

In this section, we consider coring objects ${\mathcal{C}}$ in the category of
bicomodules over a conventional ring $A$ (i. e., a ring object in the tensor
category of abelian groups ${\mathsf{K}}={\mathsf{Ab}}$). So let $A$ be an
associative ring (with unit).

A _coring_ ${\mathcal{C}}$ over a ring $A$ is an $A$-$A$-bimodule endowed with
a _comultiplication_ map
$\mu\colon{\mathcal{C}}\longrightarrow{\mathcal{C}}\otimes_{A}{\mathcal{C}}$
and a _counit_ map $\varepsilon\colon{\mathcal{C}}\longrightarrow A$
satisfying the following _linearity_ , _coassociativity_ , and _counitality_
equations. First of all, both maps $\mu$ and $\varepsilon$ must be _$A$
-$A$-bimodule morphisms_. Secondly, the two compositions of the
comultiplication map $\mu$ with the two maps $\mu\otimes{\mathrm{id}}$ and
${\mathrm{id}}\otimes\mu\colon\allowbreak{\mathcal{C}}\otimes_{A}{\mathcal{C}}\rightrightarrows{\mathcal{C}}\otimes_{A}{\mathcal{C}}\otimes_{A}{\mathcal{C}}$
induced by the comultiplication map should be equal to each other,

${\mathcal{C}}\longrightarrow{\mathcal{C}}\otimes_{A}{\mathcal{C}}\rightrightarrows{\mathcal{C}}\otimes_{A}{\mathcal{C}}\otimes_{A}{\mathcal{C}}.$

Thirdly, both the compositions of the comultiplication map with the two maps
$\varepsilon\otimes{\mathrm{id}}$ and
${\mathrm{id}}\otimes\varepsilon\colon{\mathcal{C}}\otimes_{A}{\mathcal{C}}\rightrightarrows{\mathcal{C}}$
induced by the counit map $\varepsilon$ should be equal to the identity map
${\mathrm{id}}_{\mathcal{C}}$,

${\mathcal{C}}\longrightarrow{\mathcal{C}}\otimes_{A}{\mathcal{C}}\rightrightarrows{\mathcal{C}}.$

A _left comodule_ ${\mathcal{M}}$ over a coring ${\mathcal{C}}$ over a ring
$A$ is a comodule object in the left module category of left $A$-modules over
the coring object ${\mathcal{C}}$ in the tensor category of $A$-$A$-bimodules.
In other words, it is a left $A$-module endowed with a _left
${\mathcal{C}}$-coaction_ map
$\nu_{\mathcal{M}}\colon{\mathcal{M}}\longrightarrow{\mathcal{C}}\otimes_{A}{\mathcal{M}}$
satisfying the following linearity, _coassociativity_ , and _counitality_
equations. First of all, the map $\nu=\nu_{\mathcal{M}}$ must be a _left
$A$-module morphism_. Secondly, the compositions of the coaction map $\nu$
with the two maps $\mu\otimes{\mathrm{id}}$ and
${\mathrm{id}}\otimes\nu\colon{\mathcal{C}}\otimes_{A}{\mathcal{M}}\rightrightarrows{\mathcal{C}}\otimes_{A}{\mathcal{C}}\otimes_{A}{\mathcal{M}}$
induced by the comultiplication and coaction maps should be equal to each
other,

${\mathcal{M}}\longrightarrow{\mathcal{C}}\otimes_{A}{\mathcal{M}}\rightrightarrows{\mathcal{C}}\otimes_{A}{\mathcal{C}}\otimes_{A}{\mathcal{M}}.$

Thirdly, the composition of the coaction map with the map
$\epsilon\otimes{\mathrm{id}}\colon{\mathcal{C}}\otimes_{A}{\mathcal{M}}\longrightarrow{\mathcal{M}}$
induced by the counit map should be equal to the identity map
${\mathrm{id}}_{\mathcal{M}}$,

${\mathcal{M}}\longrightarrow{\mathcal{C}}\otimes_{A}{\mathcal{M}}\longrightarrow{\mathcal{M}}.$

Similarly, a _right comodule_ ${\mathcal{N}}$ over ${\mathcal{C}}$ is a right
$A$-module endowed with a right ${\mathcal{C}}$-coaction map
$\nu_{\mathcal{N}}\colon{\mathcal{N}}\longrightarrow{\mathcal{N}}\otimes_{A}{\mathcal{C}}$,
which must be a right $A$-module morphism satisfying the coassociativity and
counitality equations

$\displaystyle{\mathcal{N}}\longrightarrow{\mathcal{N}}\otimes_{A}{\mathcal{C}}\rightrightarrows{\mathcal{N}}\otimes_{A}{\mathcal{C}}\otimes_{A}{\mathcal{C}},$
$\displaystyle{\mathcal{N}}\longrightarrow{\mathcal{N}}\otimes_{A}{\mathcal{C}}\longrightarrow{\mathcal{N}}.$

These definitions can be found in [14, Sections 17.1 and 18.1] or [52, Section
1.1.1]. Corings and comodules also appear in noncommutative geometry, or more
specifically, in connection with noncommutative semi-separated stacks [40,
41].

Before introducing _${\mathcal{C}}$ -contramodules_, let us discuss a bit more
abstract nonsense. The conventional tensor calculus over a ring $A$ includes,
in addition to the tensor product functor $\otimes_{A}$, the functor
$\operatorname{Hom}_{A}$ of homomorphisms of (say, left) $A$-modules. Applying
the functor $\operatorname{Hom}_{A}$ to an $A$-$A$-bimodule $E$ and a left
$A$-module $P$ produces a left $A$-module $\operatorname{Hom}_{A}(E,P)$. In
fact, the functor $\operatorname{Hom}_{A}$ endows the category
$A{\operatorname{\mathsf{--mod}}}^{\mathsf{op}}$ _opposite to_ the category of
left $A$-modules with a _right_ module category structure over the tensor
category of $A$-$A$-bimodules $A{\operatorname{\mathsf{--mod--}}}A$. Indeed,
for any $A$-$A$-bimodules $K$ and $L$ and a left $A$-module $P$ one has

$\operatorname{Hom}_{A}(L,\operatorname{Hom}_{A}(K,P))\simeq\operatorname{Hom}_{A}(K\otimes_{A}L,\medspace
P),$

or, denoting temporarily
$P^{\mathsf{op}}*_{A}K=\operatorname{Hom}_{A}(K,P)^{\mathsf{op}}$,

$(P^{\mathsf{op}}*_{A}K)*_{A}L\simeq P^{\mathsf{op}}*_{A}(K\otimes_{A}L).$

(cf. the discussion of Hom space identification rules (1) and (2) in Section
1.1). In other words, one can say that the functor $\operatorname{Hom}_{A}$
makes the category of left $A$-modules a _left Hom category_ over the tensor
category of $A$-$A$-bimodules. The same functor
$\operatorname{Hom}_{A}({-},{-})$ provides a pairing between the left module
category $A{\operatorname{\mathsf{--mod}}}$ and the right module category
$A{\operatorname{\mathsf{--mod}}}^{\mathsf{op}}$ over the tensor category
$A{\operatorname{\mathsf{--mod--}}}A$ taking values in the category of abelian
groups ${\mathsf{Ab}}$.

A _left contramodule_ ${\mathfrak{P}}$ over a coring ${\mathcal{C}}$ over a
ring $A$ is an object of the category opposite to the category of module
objects in the right module category
$A{\operatorname{\mathsf{--mod}}}^{\mathsf{op}}$ over the coring object
${\mathcal{C}}$ in the tensor category $A{\operatorname{\mathsf{--mod--}}}A$.
In other words, it is a left $A$-module endowed with a _left
${\mathcal{C}}$-contraaction_ map
$\pi_{\mathfrak{P}}\colon\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$
satisfying the following linearity, _contraassociativity_ , and
_contraunitality_ equations. First of all, the map $\pi=\pi_{\mathfrak{P}}$
must be a left $A$-module morphism. Secondly, the compositions of the maps
$\operatorname{Hom}(\mu,{\mathfrak{P}})\colon\operatorname{Hom}_{A}({\mathcal{C}}\otimes_{A}{\mathcal{C}},\medspace{\mathfrak{P}})\longrightarrow\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})$
and
$\operatorname{Hom}({\mathcal{C}},\pi)\colon\allowbreak\operatorname{Hom}_{A}({\mathcal{C}},\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}}))\longrightarrow\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})$
induced by the comultiplication and contraaction maps with the contraaction
map should be equal to each other,

$\operatorname{Hom}_{A}({\mathcal{C}},\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}}))\simeq\operatorname{Hom}_{A}({\mathcal{C}}\otimes_{A}{\mathcal{C}},\medspace{\mathfrak{P}})\rightrightarrows\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}.$

Thirdly, the composition of the map
${\mathfrak{P}}\longrightarrow\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})$
induced by the counit map $\varepsilon$ with the contraaction map should be
equal to the identity map,

${\mathfrak{P}}\longrightarrow\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}.$

This definition can be found in [52, Section 3.1.1]. In a slightly lesser
generality of coassociative coalgebras over commutative rings, it was first
given, together with the definition of a comodule, in the memoir [24, Section
III.5].

For any right ${\mathcal{C}}$-comodule ${\mathcal{N}}$ endowed with a left
action of a ring $B$ by right ${\mathcal{C}}$-comodule endomorphisms and any
left $B$-module $V$ the abelian group
$\operatorname{Hom}_{B}({\mathcal{N}},V)$ has a natural left
${\mathcal{C}}$-contramodule structure. Here the left action of $A$ in
$\operatorname{Hom}_{B}({\mathcal{N}},V)$ is induced by the right action of
$A$ in ${\mathcal{N}}$, and the left ${\mathcal{C}}$-contraaction morphism
$\pi\colon\operatorname{Hom}_{A}({\mathcal{C}},\operatorname{Hom}_{B}({\mathcal{N}},V))\longrightarrow\operatorname{Hom}_{B}({\mathcal{N}},V)$
is obtained by applying the contravariant functor
$\operatorname{Hom}_{B}({-},V)$ to the right ${\mathcal{C}}$-coaction morphism
$\nu\colon{\mathcal{N}}\longrightarrow{\mathcal{N}}\otimes_{A}{\mathcal{C}}$,

$\operatorname{Hom}_{A}({\mathcal{C}},\operatorname{Hom}_{B}({\mathcal{N}},V))\simeq\operatorname{Hom}_{B}({\mathcal{N}}\otimes_{A}{\mathcal{C}},\medspace
V)\longrightarrow\operatorname{Hom}_{B}({\mathcal{N}},V).$

The left ${\mathcal{C}}$-comodule ${\mathcal{C}}\otimes_{A}V$, where $V$ is a
left $A$-module, is called the ${\mathcal{C}}$-comodule _coinduced_ from an
$A$-module $V$. For any left ${\mathcal{C}}$-comodule ${\mathcal{L}}$ there is
a natural isomorphism of abelian groups

$\operatorname{Hom}_{\mathcal{C}}({\mathcal{L}},\medspace{\mathcal{C}}\otimes_{A}V)\simeq\operatorname{Hom}_{A}({\mathcal{L}},V),$

where $\operatorname{Hom}_{\mathcal{C}}({\mathcal{L}},{\mathcal{M}})$ denotes
the group of morphisms from a ${\mathcal{C}}$-comodule ${\mathcal{L}}$ to a
${\mathcal{C}}$-comodule ${\mathcal{M}}$ in the category
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ of left
${\mathcal{C}}$-comodules [52, Section 1.1.2].

The left ${\mathcal{C}}$-contramodule
$\operatorname{Hom}_{A}({\mathcal{C}},V)$, where $V$ is a left $A$-module, is
called the ${\mathcal{C}}$-contramodule _induced_ from an $A$-module $V$. For
any left ${\mathcal{C}}$-contramodule ${\mathfrak{Q}}$ there is a natural
isomorphism of abelian groups

$\operatorname{Hom}^{\mathcal{C}}(\operatorname{Hom}_{A}({\mathcal{C}},V),{\mathfrak{Q}})\simeq\operatorname{Hom}_{A}(V,{\mathfrak{Q}}),$

where $\operatorname{Hom}^{\mathcal{C}}({\mathfrak{P}},{\mathfrak{Q}})$
denotes the group of morphisms from a ${\mathcal{C}}$-contramodule
${\mathfrak{P}}$ to a ${\mathcal{C}}$-contramodule ${\mathfrak{Q}}$ in the
category ${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ of left
${\mathcal{C}}$-contramodules [52, Section 3.1.2].

######  Proposition.

(a) The following two conditions on a coring ${\mathcal{C}}$ are equivalent:

  * •

the category of left ${\mathcal{C}}$-comodules is abelian and the forgetful
functor
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}\allowbreak\longrightarrow
A{\operatorname{\mathsf{--mod}}}$ is exact;

  * •

the coring ${\mathcal{C}}$ is a flat right $A$-module.

(b) The following two conditions on a coring ${\mathcal{C}}$ are equivalent:

  * •

the category of left ${\mathcal{C}}$-contramodules is abelian and the
forgetful functor
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}\longrightarrow
A{\operatorname{\mathsf{--mod}}}$ is exact;

  * •

the coring ${\mathcal{C}}$ is a projective left $A$-module.

###### Proof.

One defines a ${\mathcal{C}}$-comodule or ${\mathcal{C}}$-contramodule
structure on the kernel and cokernel of any morphism of left
${\mathcal{C}}$-comodules or left ${\mathcal{C}}$-contamodules computed in the
category of abelian groups/left $A$-modules, assuming respectively that the
functor ${\mathcal{C}}\otimes_{A}{-}\colon
A{\operatorname{\mathsf{--mod}}}\longrightarrow
A{\operatorname{\mathsf{--mod}}}$ is exact (preserves kernels) or the functor
$\operatorname{Hom}_{A}({\mathcal{C}},{-})\colon
A{\operatorname{\mathsf{--mod}}}\longrightarrow
A{\operatorname{\mathsf{--mod}}}$ is exact (preserves cokernels). This allows
to show that the second condition implies the first one in either part (a) or
(b).

To prove the converse implication in part (a), notice that the functor
${\mathcal{C}}\otimes_{A}{-}\colon\allowbreak
A{\operatorname{\mathsf{--mod}}}\longrightarrow
A{\operatorname{\mathsf{--mod}}}$ is the composition of the coinduction
functor
$A{\operatorname{\mathsf{--mod}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--comod}}}$
and the forgetful functor
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}\longrightarrow
A{\operatorname{\mathsf{--mod}}}$, the former of which is right adjoint to the
latter one. Since any right adjoint functor between abelian categories is left
exact, one concludes that the functor ${\mathcal{C}}\otimes_{A}{-}$ is left
exact whenever the forgetful functor is exact. Similarly, in part (b) the
functor $\operatorname{Hom}_{A}({\mathcal{C}},{-})\colon
A{\operatorname{\mathsf{--mod}}}\longrightarrow
A{\operatorname{\mathsf{--mod}}}$ is the composition of the induction functor
$A{\operatorname{\mathsf{--mod}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--contra}}}$
and the forgetful functor
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}\longrightarrow
A{\operatorname{\mathsf{--mod}}}$, the former of which is left adjoint to the
latter one. Since any left adjoint functor is right exact, the functor
$\operatorname{Hom}_{A}({\mathcal{C}},{-})$ is right exact whenever the
forgetful functor is exact. ∎

Generally speaking, the cokernels of arbitrary morphisms exist in
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ and are preserved by the
forgetful functor
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}\longrightarrow
A{\operatorname{\mathsf{--mod}}}$, but the kernels in
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ may be problematic when
${\mathcal{C}}$ is not a flat right $A$-module. Similarly, the kernels of
arbitrary morphisms exist in ${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$
and are preserved by the forgeftul functor
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}\longrightarrow
A{\operatorname{\mathsf{--mod}}}$, but the cokernels in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ may be problematic when
${\mathcal{C}}$ is not a projective left $A$-module. Counterexamples showing
that the categories ${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ and
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ are _not_ abelian in general
can be found in [57, Example B.1.1].

Assume that the coring ${\mathcal{C}}$ is a flat right $A$-module; then,
according to Proposition, the category
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ is abelian and the forgetful
functor ${\mathcal{C}}{\operatorname{\mathsf{--comod}}}\longrightarrow
A{\operatorname{\mathsf{--mod}}}$ is exact. Both the infinite direct sums and
infinite products exist in ${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$;
the infinite direct sums are exact and are preserved by the forgetful functor.
Filtered inductive limits are exact in the category of left
${\mathcal{C}}$-comodules; so it satisfies the axioms Ab5 and Ab3*, but _not_
in general Ab4*. The category ${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$
also has a set of generators [14, Sections 3.13 and 18.14]; moreover, when $A$
is a left Noetherian ring or ${\mathcal{C}}$ is a projective right $A$-module,
every left ${\mathcal{C}}$-comodule is the union of its subcomodules that are
finitely generated as $A$-modules [14, Sections 18.16 and 19.12].

The coaction map
$\nu\colon{\mathcal{M}}\longrightarrow{\mathcal{C}}\otimes_{A}{\mathcal{M}}$
embeds every left ${\mathcal{C}}$-comodule ${\mathcal{M}}$ as a subcomodule
into the coinduced ${\mathcal{C}}$-comodule
${\mathcal{C}}\otimes_{A}{\mathcal{M}}$. Infinite products of coinduced
${\mathcal{C}}$-comodules are computed by the rule
$\prod_{\alpha}{\mathcal{C}}\otimes_{A}V_{\alpha}={\mathcal{C}}\otimes_{A}\prod_{\alpha}V_{\alpha}$
[14, Section 18.13]; to compute the product of an arbitrary family of left
${\mathcal{C}}$-comodules, one can present them as the kernels of morphisms of
coinduced ${\mathcal{C}}$-comodules and use the fact that infinite products
always commute with the kernels [52, Section 1.1.2]. There are enough
injective objects in the category
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$; a left
${\mathcal{C}}$-comodule is injective if and only if it is isomorphic to a
direct summand of a ${\mathcal{C}}$-comodule ${\mathcal{C}}\otimes_{A}J$
coinduced from an injective left $A$-module $J$ (see [14, Section 18.19] or
[52, Section 5.1.5]).

Assume that the coring ${\mathcal{C}}$ is a projective left $A$-module; then,
according to Proposition, the category
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$ is abelian and the forgetful
functor ${\mathcal{C}}{\operatorname{\mathsf{--contra}}}\longrightarrow
A{\operatorname{\mathsf{--mod}}}$ is exact. Both the infinite direct sums and
infinite products exist in ${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$;
the infinite products are exact and are preserved by the forgetful functor. So
the category of left ${\mathcal{C}}$-contramodules satisfies the axioms Ab3
and Ab4*, but _not_ in general Ab4 or Ab5*.

The contraaction map
$\pi\colon\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$
presents every left ${\mathcal{C}}$-contramodule as a quotient contramodule of
the induced ${\mathcal{C}}$-contramodule
$\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}})$. Infinite direct sums
of induced ${\mathcal{C}}$-contramodules are computed by the rule
$\bigoplus_{\alpha}\operatorname{Hom}_{A}({\mathcal{C}},V_{\alpha})=\operatorname{Hom}_{A}({\mathcal{C}},\medspace\bigoplus_{\alpha}V_{\alpha})$;
to compute the direct sum of an arbitrary family of left
${\mathcal{C}}$-contramodules, one can present them as the cokernels of
morphisms of induced ${\mathcal{C}}$-contramodules and use the fact that
infinite direct sums always commute with the cokernels [52, Section 3.1.2].
There are enough projective objects in
${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$; a left
${\mathcal{C}}$-contramodule is projective if and only if it is a direct
summand of a ${\mathcal{C}}$-contramodule
$\operatorname{Hom}_{A}({\mathcal{C}},F)$ induced from a projective left
$A$-module $F$ [52, Section 5.1.5].

The discussion in the beginning of this section suggests that one should
consider, in addition to the categories of left ${\mathcal{C}}$-comodules,
right ${\mathcal{C}}$-comodules, and left ${\mathcal{C}}$-contramodules, the
pairing functors of _cotensor product_ and _cohomomorphisms_ acting from those
categories to the category of abelian groups. Let us define these functors of
two co/contramodule arguments now.

The _cotensor product_
${\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathcal{M}}$
of a right ${\mathcal{C}}$-comodule ${\mathcal{N}}$ and a left
${\mathcal{C}}$-comodule ${\mathcal{M}}$ is an abelian group defined as the
kernel of the difference of the pair of maps

$\nu_{\mathcal{N}}\otimes{\mathrm{id}},\
\,{\mathrm{id}}\otimes\nu_{\mathcal{M}}\colon{\mathcal{N}}\otimes_{A}{\mathcal{M}}\rightrightarrows{\mathcal{N}}\otimes_{A}{\mathcal{C}}\otimes_{A}{\mathcal{M}}$

one of which is induced by the ${\mathcal{C}}$-coaction in ${\mathcal{N}}$ and
the other one by the ${\mathcal{C}}$-coaction in ${\mathcal{M}}$. For any
right ${\mathcal{C}}$-comodule ${\mathcal{N}}$ and any left $A$-module $V$
there is a natural isomorphism of abelian groups

${\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}({\mathcal{C}}\otimes_{A}V)\simeq{\mathcal{N}}\otimes_{A}V;$

the similar formula holds for the cotensor product of a coinduced right
${\mathcal{C}}$-comodule and an arbitrary left ${\mathcal{C}}$-comodule. In
particular, one has
${\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathcal{C}}\simeq{\mathcal{N}}$
and
${\mathcal{C}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathcal{M}}\simeq{\mathcal{M}}$
(see [14, Section 21] or [52, Sections 0.2.1 and 1.2.1]).

The abelian group of _cohomomorphisms_
$\operatorname{Cohom}_{\mathcal{C}}({\mathcal{M}},{\mathfrak{P}})$ from a left
${\mathcal{C}}$-comodule ${\mathcal{M}}$ to a left
${\mathcal{C}}$-contramodule ${\mathfrak{P}}$ is defined as the cokernel of
(the difference of) the pair of maps

$\operatorname{Hom}(\nu_{\mathcal{M}},{\mathrm{id}}),\
\,\operatorname{Hom}({\mathrm{id}},\pi_{\mathfrak{P}})\colon\operatorname{Hom}_{A}({\mathcal{C}}\otimes_{A}{\mathcal{M}},\medspace{\mathfrak{P}})\\\
\simeq\operatorname{Hom}_{A}({\mathcal{M}},\operatorname{Hom}_{A}({\mathcal{C}},{\mathfrak{P}}))\rightrightarrows\operatorname{Hom}_{A}({\mathcal{M}},{\mathfrak{P}}).$

For any left ${\mathcal{C}}$-comodule ${\mathcal{M}}$, and left
${\mathcal{C}}$-contramodule ${\mathfrak{P}}$, and any left $A$-module $V$,
there are natural isomorphisms of abelian groups

$\displaystyle\operatorname{Cohom}_{\mathcal{C}}({\mathcal{C}}\otimes_{A}V,\medspace{\mathfrak{P}})\simeq\operatorname{Hom}_{A}(V,{\mathfrak{P}})$
$\displaystyle\operatorname{Cohom}_{\mathcal{C}}({\mathcal{M}},\operatorname{Hom}_{A}({\mathcal{C}},V))\simeq\operatorname{Hom}_{A}({\mathcal{M}},V);$

in particular, one has
$\operatorname{Cohom}_{\mathcal{C}}({\mathcal{C}},{\mathfrak{P}})\simeq{\mathfrak{P}}$
[52, Sections 0.2.4 and 3.2.1].

Notice that the functor of cotensor product
$\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}$ over a coring
${\mathcal{C}}$, being defined as the kernel of a morphism of cokernels, is
_neither_ left _nor_ right exact in general. Similarly, the functor
$\operatorname{Cohom}_{\mathcal{C}}$, being defined as the cokernel of a
morphism of kernels, is neither left _nor_ right exact (even when all the
categories involved are abelian and all the forgetful functors are exact).

###  2.6. Semicontramodules over semialgebras

The notion of a _semialgebra_ over a coalgebra over a field is dual to that of
a coring in the same way as the notion of a coalgebra over a field is dual to
that of an (associative) ring [1, 13, 51, 52]. In this section we present the
related piece of theory, aiming to define semimodules and semicontramodules
over semialgebras and interpret contramodules over topological groups as
semicontramodules over certain semialgebras, as it was promised in Section
1.8.

Let ${\mathcal{C}}$ be a (coassociative) coalgebra (with counit) over a field
$k$. In addition to the definitions of left ${\mathcal{C}}$-comodules, right
${\mathcal{C}}$-comodules and left ${\mathcal{C}}$-contramodules given in
Section 1.1 and then repeated, in the greater generality of a coring
${\mathcal{C}}$, in the previous Section 2.5, we will also need the definition
of a ${\mathcal{C}}$-${\mathcal{C}}$-bicomodule.

Let ${\mathcal{D}}$ be another coalgebra over $k$. A _${\mathcal{C}}$
-${\mathcal{D}}$-bicomodule_ ${\mathcal{K}}$ is a $k$-vector space endowed
with a left ${\mathcal{C}}$-comodule and a right ${\mathcal{D}}$-comodule
structures
$\nu^{\prime}\colon{\mathcal{K}}\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{K}}$
and
$\nu^{\prime\prime}\colon{\mathcal{K}}\longrightarrow{\mathcal{K}}\otimes_{k}{\mathcal{D}}$
which commute with each other in the following sense. The composition of the
left coaction map
$\nu^{\prime}\colon{\mathcal{K}}\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{K}}$
with the map
${\mathrm{id}}\otimes\nu^{\prime\prime}:{\mathcal{C}}\otimes_{k}{\mathcal{K}}\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{K}}\otimes_{k}{\mathcal{D}}$
induced by the right coaction map $\nu^{\prime\prime}$ should be equal to the
composition of the right coaction map
$\nu^{\prime\prime}\colon{\mathcal{K}}\longrightarrow{\mathcal{K}}\otimes_{k}{\mathcal{D}}$
with the map
$\nu^{\prime}\otimes{\mathrm{id}}\colon{\mathcal{K}}\otimes_{k}{\mathcal{D}}\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{K}}\otimes_{k}{\mathcal{D}}$
induced by the left coaction map $\nu^{\prime}$. Equivalently, the vector
space ${\mathcal{K}}$ should be endowed with a _${\mathcal{C}}$
-${\mathcal{D}}$-bicoaction_ map
$\nu\colon{\mathcal{K}}\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{K}}\otimes_{k}{\mathcal{D}}$
satisfying the coassociativity and counitality equations
$(\mu_{\mathcal{C}}\otimes{\mathrm{id}}_{\mathcal{K}}\otimes\mu_{\mathcal{D}})\circ\nu=({\mathrm{id}}_{\mathcal{C}}\otimes\nu\otimes{\mathrm{id}}_{\mathcal{D}})\circ\nu$
and
$(\varepsilon_{\mathcal{C}}\otimes{\mathrm{id}}_{\mathcal{K}}\otimes\varepsilon_{\mathcal{D}})\circ\nu={\mathrm{id}}_{\mathcal{K}}$,

$\displaystyle{\mathcal{K}}\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{K}}\otimes_{k}{\mathcal{D}}\rightrightarrows{\mathcal{C}}\otimes_{k}{\mathcal{C}}\otimes_{k}{\mathcal{K}}\otimes_{k}{\mathcal{D}}\otimes_{k}{\mathcal{D}}$
$\displaystyle{\mathcal{K}}\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{K}}\otimes_{k}{\mathcal{D}}\longrightarrow{\mathcal{K}}$

(see [14, Sections 11.1 or 22.1] or [52, Sections 0.3.1 or 1.2.4]).

Recall from the end of the previous section that the _cotensor product_
${\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathcal{M}}$
of a right ${\mathcal{C}}$-comodule ${\mathcal{N}}$ and a left
${\mathcal{C}}$-comodule ${\mathcal{M}}$ is the $k$-vector space constructed
as the kernel of (the difference of) the pair of maps

$\nu_{\mathcal{N}}\otimes{\mathrm{id}},\
\,{\mathrm{id}}\otimes\nu_{\mathcal{M}}\colon{\mathcal{N}}\otimes_{k}{\mathcal{M}}\rightrightarrows{\mathcal{N}}\otimes_{k}{\mathcal{C}}\otimes_{k}{\mathcal{M}}$

induced by the ${\mathcal{C}}$-coactions maps in ${\mathcal{N}}$ and
${\mathcal{M}}$. Similarly, the $k$-vector space of _cohomomorphisms_ from a
left ${\mathcal{C}}$-comodule ${\mathcal{M}}$ to a left
${\mathcal{C}}$-contramodule ${\mathfrak{P}}$ is construced as the cokernel of
the pair of maps

$\operatorname{Hom}_{k}({\mathcal{C}}\otimes_{k}{\mathcal{M}},\medspace{\mathfrak{P}})\simeq\operatorname{Hom}_{k}({\mathcal{M}},\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}}))\rightrightarrows\operatorname{Hom}_{k}({\mathcal{M}},{\mathfrak{P}})$

one of which is induced by the ${\mathcal{C}}$-coaction in ${\mathcal{M}}$ and
the other one by the ${\mathcal{C}}$-contraaction in ${\mathfrak{P}}$. The
functor of cotensor product of comodules over a coalgebra ${\mathcal{C}}$ over
a field $k$, being defined as the kernel of a morphism of exact functors, is
left exact; while the functor of cohomomorphisms of comodules and
contramodules over ${\mathcal{C}}$, defined as the cokernel of a morphism of
exact functors, is right exact. For left ${\mathcal{C}}$-comodule
${\mathcal{M}}$, right ${\mathcal{C}}$-comodule ${\mathcal{N}}$, and
$k$-vector space $V$, there is a natural isomorphism of $k$-vector spaces [52,
Sections 0.2.4 and 3.2.2, and Proposition 3.2.3.1]

$\operatorname{Cohom}_{\mathcal{C}}({\mathcal{M}},\operatorname{Hom}_{k}({\mathcal{N}},V))\simeq\operatorname{Hom}_{k}({\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathcal{M}},\medspace
V),$

where the $k$-vector space $\operatorname{Hom}_{k}({\mathcal{N}},V)$ is
endowed with a left ${\mathcal{C}}$-contramodule structure as explained in
Section 1.2.

For any three coalgebras ${\mathcal{C}}$, ${\mathcal{D}}$, and
${\mathcal{E}}$, any ${\mathcal{C}}$-${\mathcal{D}}$-bicomodule
${\mathcal{N}}$, and any ${\mathcal{D}}$-${\mathcal{E}}$-bicomodule
${\mathcal{M}}$, the cotensor product
${\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{D}}{\mathcal{M}}$
has a natural ${\mathcal{C}}$-${\mathcal{E}}$-bicomodule structure.
Furthermore, for any right ${\mathcal{C}}$-comodule ${\mathcal{N}}$, any
${\mathcal{C}}$-${\mathcal{D}}$-bicomodule ${\mathcal{K}}$, and any left
${\mathcal{C}}$-comodule ${\mathcal{M}}$ there is a natural associativity
isomorphism

$({\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathcal{K}})\mathbin{\text{\smaller$\square$}}_{\mathcal{D}}{\mathcal{M}}\simeq{\mathcal{N}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}({\mathcal{K}}\mathbin{\text{\smaller$\square$}}_{\mathcal{D}}{\mathcal{M}}).$

To put it simply, both the iterated cotensor products are identified with one
and the same subspace in the vector space
${\mathcal{N}}\otimes_{k}{\mathcal{K}}\otimes_{k}{\mathcal{M}}$ (cf. the
beginning of Section 2.5).

Similarly, for any ${\mathcal{C}}$-${\mathcal{D}}$-bicomodule ${\mathcal{K}}$
and any left ${\mathcal{C}}$-contramodule ${\mathfrak{P}}$, the space of
cohomomorphisms
$\operatorname{Cohom}_{\mathcal{C}}({\mathcal{K}},{\mathfrak{P}})$ has a
natural left ${\mathcal{D}}$-contramodule structure. One can define it by
noticing that
$\operatorname{Cohom}_{\mathcal{C}}({\mathcal{K}},{\mathfrak{P}})$ is a
quotient contramodule of the left ${\mathcal{D}}$-contramodule
$\operatorname{Hom}_{k}({\mathcal{K}},{\mathfrak{P}})$, whose contramodule
structure is induced by the right ${\mathcal{D}}$-comodule structure on
${\mathcal{K}}$ via the construction described in Section 1.2. For any
${\mathcal{C}}$-${\mathcal{D}}$-bicomodule ${\mathcal{K}}$, any left
${\mathcal{D}}$-comodule ${\mathcal{M}}$, and any left
${\mathcal{C}}$-contramodule ${\mathfrak{P}}$, there is a natural
associativity isomorphism

$\operatorname{Cohom}_{\mathcal{C}}({\mathcal{K}}\mathbin{\text{\smaller$\square$}}_{\mathcal{D}}{\mathcal{M}},\medspace{\mathfrak{P}})\simeq\operatorname{Cohom}_{\mathcal{D}}({\mathcal{M}},\medspace\operatorname{Cohom}_{\mathcal{C}}({\mathcal{K}},{\mathfrak{P}})).$

Both the (iterated) Cohom spaces are identified with the quotient space of the
vector space
$\operatorname{Hom}_{k}({\mathcal{K}}\otimes_{k}{\mathcal{M}},\medspace{\mathfrak{P}})\simeq\operatorname{Hom}_{k}({\mathcal{M}},\operatorname{Hom}_{k}({\mathcal{K}},{\mathfrak{P}}))$
by one and the same vector subspace [52, Sections 0.3.4 or 3.2.4].

In particular, it follows from these associativity isomorphisms for a
coalgebra ${\mathcal{C}}={\mathcal{D}}$ that the category of
${\mathcal{C}}$-${\mathcal{C}}$-bicomodules
${\mathcal{C}}{\operatorname{\mathsf{--comod--}}}{\mathcal{C}}$ is an
associative tensor category with respect to the cotensor product functor
$\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}$, the category of left
${\mathcal{C}}$-comodules ${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$ is
a left module category over
${\mathcal{C}}{\operatorname{\mathsf{--comod--}}}{\mathcal{C}}$, and the
category ${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{\mathsf{op}}$
opposite to the category of left ${\mathcal{C}}$-contramodules is a right
module category over
${\mathcal{C}}{\operatorname{\mathsf{--comod--}}}{\mathcal{C}}$ with respect
to the cohomomorphism functor $\operatorname{Cohom}_{\mathcal{C}}$.

A _semialgebra_ ${\boldsymbol{\mathcal{S}}}$ over a coalgebra ${\mathcal{C}}$
over a field $k$ is an associative ring object in the tensor category of
${\mathcal{C}}$-${\mathcal{C}}$-bicomodules. In other words, it is a
${\mathcal{C}}$-${\mathcal{C}}$-bicomodule endowed with a _semimultiplication_
map
${\mathbf{m}}\colon{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}\longrightarrow{\boldsymbol{\mathcal{S}}}$
and a _semiunit_ map
${\mathbf{e}}\colon{\mathcal{C}}\longrightarrow{\boldsymbol{\mathcal{S}}}$
satisfying the following _colinearity_ , _semiassociativity_ and
_semiunitality_ equations. First of all, the maps ${\mathbf{m}}$ and
${\mathbf{e}}$ must be ${\mathcal{C}}$-${\mathcal{C}}$-bicomodule morphisms.
Secondly, the compositions of the two maps
${\mathbf{m}}\mathbin{\text{\smaller$\square$}}{\mathrm{id}}_{\boldsymbol{\mathcal{S}}}$
and
${\mathrm{id}}_{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}{\mathbf{m}}\colon{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}\rightrightarrows{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}$
induced by the semimultiplication map ${\mathfrak{m}}$ with the
semimultiplication map

${\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}\rightrightarrows{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}\longrightarrow{\boldsymbol{\mathcal{S}}}$

should be equal to each other,
${\mathbf{m}}\circ({\mathbf{m}}\mathbin{\text{\smaller$\square$}}{\mathrm{id}}_{\boldsymbol{\mathcal{S}}})={\mathbf{m}}\circ({\mathrm{id}}_{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}{\mathbf{m}})$.
Thirdly, both the compositions of the maps
${\mathbf{e}}\mathbin{\text{\smaller$\square$}}{\mathrm{id}}_{\boldsymbol{\mathcal{S}}}$
and
${\mathrm{id}}_{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}{\mathbf{e}}\colon{\boldsymbol{\mathcal{S}}}\rightrightarrows{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}$
induced by the semiunit map ${\mathbf{e}}$ with the semimultiplication map
${\mathbf{m}}$

${\boldsymbol{\mathcal{S}}}\rightrightarrows{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}\longrightarrow{\boldsymbol{\mathcal{S}}}$

should be equal to the identity map,
${\mathbf{m}}\circ({\mathbf{e}}\mathbin{\text{\smaller$\square$}}{\mathrm{id}}_{\boldsymbol{\mathcal{S}}})={\mathrm{id}}_{\boldsymbol{\mathcal{S}}}={\mathbf{m}}\circ({\mathrm{id}}_{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}{\mathbf{e}})$.

A _left semimodule_ ${\boldsymbol{\mathcal{M}}}$ over a semialgebra
${\boldsymbol{\mathcal{S}}}$ over a coalgebra ${\mathcal{C}}$ is a module
object in the left module category of left ${\mathcal{C}}$-comodules over the
ring object ${\boldsymbol{\mathcal{S}}}$ in the tensor category of
${\mathcal{C}}$-${\mathcal{C}}$-bicomodules. In other words, it is a left
${\mathcal{C}}$-comodule endowed with a _left semiaction_ map
${\mathbf{n}}\colon{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{M}}}\longrightarrow{\boldsymbol{\mathcal{M}}}$
satisfying the following colinearity, _semiassociativity_ and _semiunitality_
equations. First of all, the map ${\mathbf{n}}$ must be a left
${\mathcal{C}}$-comodule morphism. Secondly, the compositions of the two maps
${\mathbf{m}}\mathbin{\text{\smaller$\square$}}{\mathrm{id}}_{\boldsymbol{\mathcal{M}}}$
and
${\mathrm{id}}_{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}{\mathbf{n}}\colon{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{M}}}\rightrightarrows{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{M}}}$
induced by the semimultiplication and semiaction maps with the semiaction map

${\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{M}}}\rightrightarrows{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{M}}}\longrightarrow{\boldsymbol{\mathcal{M}}}$

should be equal to each other,
${\mathbf{n}}\circ({\mathbf{m}}\mathbin{\text{\smaller$\square$}}{\mathrm{id}}_{\boldsymbol{\mathcal{M}}})={\mathbf{n}}\circ({\mathrm{id}}_{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}{\mathbf{n}})$.
Thirdly, the composition of the map
${\mathbf{e}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathrm{id}}_{\boldsymbol{\mathcal{M}}}\colon{\boldsymbol{\mathcal{M}}}\longrightarrow{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{M}}}$
induced by the semiunit map ${\mathbf{e}}$ with the semiaction map
${\mathbf{n}}$

${\boldsymbol{\mathcal{M}}}\longrightarrow{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{M}}}\longrightarrow{\boldsymbol{\mathcal{M}}}$

should be equal to the identity map,
${\mathbf{n}}\circ({\mathbf{e}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathrm{id}}_{\boldsymbol{\mathcal{M}}})={\mathrm{id}}_{\boldsymbol{\mathcal{M}}}$.
A _right semimodule_ ${\boldsymbol{\mathcal{N}}}$ over
${\boldsymbol{\mathcal{S}}}$ is a right ${\mathcal{C}}$-comodule endowed with
a right semiaction map
${\mathbf{n}}\colon{\boldsymbol{\mathcal{N}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}\longrightarrow{\boldsymbol{\mathcal{N}}}$
satisfying the similar equations

$\displaystyle{\boldsymbol{\mathcal{N}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}\rightrightarrows{\boldsymbol{\mathcal{N}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}\longrightarrow{\boldsymbol{\mathcal{N}}},$
$\displaystyle{\boldsymbol{\mathcal{N}}}\longrightarrow{\boldsymbol{\mathcal{N}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}\longrightarrow{\boldsymbol{\mathcal{N}}}.$

These definitions can be found in [1, Sections 2.3 and 6.1], [13, Section 6],
[15, Section 8], and [52, Sections 0.3.2 and 1.3.1]; see [52, Section 0.3.10]
for some further references.

Before defining semicontramodules, let us recall from the discussion in
Section 1.1 that there are two ways to define the conventional modules over
associative algebras over $k$ in tensor/polylinear algebra terms. In addition
to the familiar definition of a left $A$-module $M$ as a $k$-vector space
endowed with a $k$-linear map $n\colon A\otimes_{k}M\allowbreak\longrightarrow
M$ satisfying the associativity and unitality equations, one can also say that
a left $A$-module structure on $M$ is defined by a linear map
$p\colon\operatorname{Hom}_{k}(A,M)\longrightarrow M$ satisfying the
correspodingly rewritten equations.

A _left semicontramodule_ ${\boldsymbol{\mathfrak{P}}}$ over a semialgebra
${\boldsymbol{\mathcal{S}}}$ over a coalgebra ${\mathcal{C}}$ is an object of
the category opposite to the category of module objects in the right module
category ${\mathcal{C}}{\operatorname{\mathsf{--contra}}}^{\mathsf{op}}$ over
the ring object ${\boldsymbol{\mathcal{S}}}$ in the tensor category
${\mathcal{C}}{\operatorname{\mathsf{--comod--}}}{\mathcal{C}}$. In other
words, it is a left ${\mathcal{C}}$-contramodule endowed with a _left
semiaction_ map
${\mathbf{p}}\colon{\boldsymbol{\mathfrak{P}}}\longrightarrow\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},{\boldsymbol{\mathfrak{P}}})$
satisfying the following _contralinearity_ , _semicontraassociativity_ , and
_semicontraunitality_ equations. First of all, the map ${\mathbf{p}}$ must be
a left ${\mathcal{C}}$-contramodule morphism. Secondly, the compositions of
the semicontraaction map ${\mathbf{p}}$ with the two maps
$\operatorname{Cohom}({\mathbf{m}},{\boldsymbol{\mathfrak{P}}})\colon\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},{\boldsymbol{\mathfrak{P}}})\longrightarrow\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}},\medspace{\boldsymbol{\mathfrak{P}}})$
and
$\operatorname{Cohom}({\boldsymbol{\mathcal{S}}},{\mathbf{p}})\colon\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},{\boldsymbol{\mathfrak{P}}})\longrightarrow\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},{\boldsymbol{\mathfrak{P}}}))$

${\boldsymbol{\mathfrak{P}}}\longrightarrow\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},{\boldsymbol{\mathfrak{P}}})\rightrightarrows\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}},\medspace{\boldsymbol{\mathfrak{P}}})\simeq\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},{\boldsymbol{\mathfrak{P}}}))$

should be equal to each other,
$\operatorname{Cohom}({\mathbf{m}},{\boldsymbol{\mathfrak{P}}})\circ{\mathbf{p}}=\operatorname{Cohom}({\boldsymbol{\mathcal{S}}},{\mathbf{p}})\circ{\mathbf{p}}$,
where the above identification
$\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}},\medspace{\boldsymbol{\mathfrak{P}}})\simeq\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},{\boldsymbol{\mathfrak{P}}}))$
is presumed. Thirdly, the composition of the semicontraaction map with the map
$\operatorname{Cohom}_{\mathcal{C}}({\mathbf{e}},{\boldsymbol{\mathfrak{P}}})\colon\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},{\boldsymbol{\mathfrak{P}}})\longrightarrow{\boldsymbol{\mathfrak{P}}}$
induced by the semiunit map ${\mathbf{e}}$

${\boldsymbol{\mathfrak{P}}}\longrightarrow\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},{\boldsymbol{\mathfrak{P}}})\longrightarrow{\boldsymbol{\mathfrak{P}}}$

should be equal to the identity map,
$\operatorname{Cohom}_{\mathcal{C}}({\mathbf{e}},{\boldsymbol{\mathfrak{P}}})\circ{\mathbf{p}}={\mathrm{id}}_{\boldsymbol{\mathfrak{P}}}$.
This definition can be found in [52, Sections 0.3.5 or 3.3.1].

For any right ${\boldsymbol{\mathcal{S}}}$-semimodule
${\boldsymbol{\mathcal{N}}}$ and any $k$-vector space $V$, the left
${\mathcal{C}}$-contramodule
$\operatorname{Hom}_{k}({\boldsymbol{\mathcal{N}}},V)$ has a natural left
${\boldsymbol{\mathcal{S}}}$-semicontramodule structure. The left
semicontraaction map
${\mathbf{p}}\colon\operatorname{Hom}_{k}({\boldsymbol{\mathcal{N}}},V)\longrightarrow\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},\operatorname{Hom}_{k}({\boldsymbol{\mathcal{N}}},V))$
is constructed by applying the functor $\operatorname{Hom}_{k}({-},V)$ to the
right semiaction map ${\mathbf{n}}$ of the
${\boldsymbol{\mathcal{S}}}$-semimodule ${\boldsymbol{\mathcal{N}}}$

$\operatorname{Hom}_{k}({\boldsymbol{\mathcal{N}}},V)\longrightarrow\operatorname{Hom}_{k}({\boldsymbol{\mathcal{N}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}},\medspace
V)\simeq\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},\operatorname{Hom}_{k}({\boldsymbol{\mathcal{N}}},V)).$

Generally speaking, the kernels of arbitrary morphisms exist in the category
of left ${\boldsymbol{\mathcal{S}}}$-semimodules
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}$ and are
preserved by the forgetful functors
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--comod}}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$, but the cokernels in
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}$ may be
problematic when ${\mathcal{C}}$ is not an injective right
${\mathcal{C}}$-comodule. Similarly, the cokernels of arbitrary morphisms
exist in the category of left ${\boldsymbol{\mathcal{S}}}$-semicontramodules
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}$ and are
preserved by the forgetful functors
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--contra}}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$, but the kernels in
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}$ may be
problematic when ${\mathcal{C}}$ is not an injective left
${\mathcal{C}}$-comodule.

Now let us assume that the semialgebra ${\boldsymbol{\mathcal{S}}}$ is an
injective right ${\mathcal{C}}$-comodule. Then the cotensor product functor
${\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{-}\colon{\mathcal{C}}{\operatorname{\mathsf{--comod}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--comod}}}$
is exact, so the category
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}$ of left
${\boldsymbol{\mathcal{S}}}$-semimodules is abelian and the forgetful functors
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--comod}}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ are exact. Both the infinite direct sums
and infinite products exist in
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}$ and both are
preserved by the forgetful functor
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--comod}}}$,
though only the infinite direct sums are preserved by the full forgetful
functor
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$.

Indeed, let ${\boldsymbol{\mathcal{M}}}_{\alpha}$ be a family of left
${\boldsymbol{\mathcal{S}}}$-semimodules and
$\prod_{\alpha}{\boldsymbol{\mathcal{M}}}_{\alpha}$ be their infinite product
in the category of left ${\mathcal{C}}$-comodules
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}$; then one can easily
construct a left semiaction map
${\mathbf{m}}\colon{\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}\prod_{\alpha}{\boldsymbol{\mathcal{M}}}_{\alpha}\longrightarrow\prod_{\alpha}{\boldsymbol{\mathcal{M}}}_{\alpha}$
and show that the left ${\boldsymbol{\mathcal{S}}}$-semimodule so obtained is
the product of the family of objects ${\boldsymbol{\mathcal{M}}}_{\alpha}$ in
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}$. So the category
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}$ satisfies the
axioms Ab5 and Ab3*, but not in general Ab4*. It also has a set of generators,
for which one can take the ${\boldsymbol{\mathcal{S}}}$-semimodules
${\boldsymbol{\mathcal{S}}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathcal{L}}$
induced from finite-dimensional left ${\mathcal{C}}$-comodules
${\mathcal{L}}$. Hence there are enough injective objects in
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--simod}}}$; we will see in
Section 3.5 below how one can construct them.

Assume that the semialgebra ${\boldsymbol{\mathcal{S}}}$ is an injective left
${\mathcal{C}}$-comodule. Then the cohomomorphism functor
$\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},{-})\colon{\mathcal{C}}{\operatorname{\mathsf{--contra}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--contra}}}$
is exact, so the category
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}$ of left
${\boldsymbol{\mathcal{S}}}$-semicontramodules is abelian and the forgetful
functors
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--contra}}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ are exact. Both the infinite direct sums
and infinite products exist in the category
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}$ and both are
preserved by the forgetful functor
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--contra}}}$,
though only the infinite products are preserved by the full forgetful functor
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}\longrightarrow
k{\operatorname{\mathsf{--vect}}}$.

Indeed, let ${\boldsymbol{\mathfrak{P}}}_{\alpha}$ be a family of left
${\boldsymbol{\mathcal{S}}}$-semicontramodules and
$\bigoplus_{\alpha}{\boldsymbol{\mathfrak{P}}}_{\alpha}$ be their infinite
direct sum in the category ${\mathcal{C}}{\operatorname{\mathsf{--contra}}}$.
Then one can easily construct a left semicontraaction map
${\mathbf{p}}:\bigoplus_{\alpha}{\boldsymbol{\mathfrak{P}}}_{\alpha}\longrightarrow\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}},\medspace\bigoplus_{\alpha}{\boldsymbol{\mathfrak{P}}}_{\alpha})$
and show that the left ${\boldsymbol{\mathcal{S}}}$-semicontramodule so
obtained is the direct sum of the family of objects
${\boldsymbol{\mathfrak{P}}}_{\alpha}$ in
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}$. So the
category ${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}$
satisfies the axioms Ab3 and Ab4*, but not in general Ab4 or Ab5*. There are
enough projective objects in
${\boldsymbol{\mathcal{S}}}{\operatorname{\mathsf{--sicntr}}}$; we will see in
Section 3.5 how to construct them.

######  Example.

Let us explain the construction of the semialgebra
${\boldsymbol{\mathcal{S}}}$ for which the category of
${\boldsymbol{\mathcal{S}}}$-semicontramodules is equivalent to the category
of contramodules over a (locally compact totally disconnected) topological
group $G$, as it was promised in Section 1.8. In fact, we will see that for
any given group $G$ there is a whole family of such semialgebras
${\boldsymbol{\mathcal{S}}}$ depending on the choice of a compact (i. e.,
profinite) open subgroup $H\subset G$. All of them are Morita equivalent to
each other in the sense of [52, Section 8.4.5], i. e., the categories of (say,
left) semimodules over all of them are equivalent, as are the categories of
semicontramodules.

Given a commutative ring $k$, by a _discrete $G$-module over $k$_ we mean a
$k$-module ${\mathcal{M}}$ endowed with a $k$-linear discrete $G$-module
structure ${\mathcal{M}}\longrightarrow{\mathcal{M}}\\{G\\}$; similarly, a
_$G$ -contramodule over $k$_ is a $k$-module endowed with a $k$-linear
$G$-contramodule structure ${\mathfrak{P}}[[G]]\longrightarrow{\mathfrak{P}}$.
In other words, a discrete $G$-module over $k$ is a $k$-linear object in the
additive category $G{\operatorname{\mathsf{--discr}}}$ and a $G$-contramodule
over $k$ is a $k$-linear object in the additive category
$G{\operatorname{\mathsf{--contra}}}$.

For the beginning, let $k$ be a field. We will freely use the terminology and
notation of Section 1.8; in particular, $k(X)$ denotes the vector space of
locally constant compactly supported $k$-valued functions on a (locally
compact totally disconnected) topological space $X$. Then for any topological
spaces $X$ and $Y$ there is a natural isomorphism $k(X\times Y)\simeq
k(X)\otimes_{k}k(Y)$. For any profinite group $H$, the inverse image map
$k(H)\longrightarrow k(H\times H)$ with respect to the multiplication map
$H\times H\longrightarrow H$, together with the map $k(H)\longrightarrow k$ of
evaluation at the unit element $e\in H$, endow the vector space $k(H)$ with
the structure of a coassociative coalgebra over $k$. For any $k$-vector space
$A$ one has $A\\{H\\}=A(H)\simeq k(H)\otimes_{k}A$ and
$A[[H]]\simeq\operatorname{Hom}_{k}(k[[H]],A)$, so one can easily identify
discrete $G$-modules over $k$ with (left or right) $k(H)$-comodules and
$G$-contramodules over $k$ with $k(H)$-contramodules.

Let $H$ be a compact open subgroup in a topological group $G$; then the left
and right actions of $H$ in $G$ endow $k(G)$ with a natural structure of
bicomodule over $k(H)$. Denote by $G\times^{H}G$ the quotient space of the
Cartesian square $G\times G$ by the equivalence relation
$(g^{\prime}h,g^{\prime\prime})\sim(g^{\prime},hg^{\prime\prime})$ for all
$g^{\prime}$, $g^{\prime\prime}\in G$ and $h\in H$. Being a disjoint union of
$G/H$ copies of $G$, this quotient is also a locally compact and totally
disconnected topological space. The inverse image of functions with respect to
the factorization map $G\times G\longrightarrow G\times^{H}G$ identifies the
vector space $k(G\times^{H}G)$ with the cotensor product
$k(G)\mathbin{\text{\smaller$\square$}}_{k(H)}k(G)\subset
k(G)\otimes_{k}k(G)=k(G\times G)$.

For any étale map (local homeomorphism) of topological spaces $p\colon
X\longrightarrow Y$ and any abelian group $A$, the push-forward map
$A(X)\longrightarrow A(Y)$, assigning to a function $f\colon X\longrightarrow
A$ the function $p_{*}(f)\colon Y\longrightarrow A$,

$(p_{*}f)(y)=\sum_{p(x)=y}f(x),\qquad y\in Y,\ \ x\in X,$

is defined [52, Section E.1.1]. In particular, the push-forwards with respect
to the multiplication map $G\times^{H}G\longrightarrow G$ and the embedding
map $H\longrightarrow G$ provide natural left and right $H$-equivariant
$k$-linear maps
$k(G)\mathbin{\text{\smaller$\square$}}_{k(H)}k(G)\longrightarrow k(G)$ and
$k(H)\longrightarrow k(G)$ endowing the vector space
${\boldsymbol{\mathcal{S}}}_{k}(G,H)=k(G)$ with the structure of a semialgebra
over the coalgebra ${\mathcal{C}}_{k}(H)=k(H)$ [52, Section E.1.2].

It is claimed that the category of (left or right)
${\boldsymbol{\mathcal{S}}}_{k}(G,H)$-semimodules is isomorphic to the
category of discrete $G$-modules over $k$; the datum of a
${\boldsymbol{\mathcal{S}}}_{k}(G,H)$-semimodule structure on a $k$-vector
space ${\mathcal{M}}$ is equivalent to the datum of a discrete $G$-module
structure on ${\mathcal{M}}$. Indeed, according to [52, Sections E.1.3 and
10.2.2], the datum of a ${\boldsymbol{\mathcal{S}}}_{k}(G,H)$-semimodule
structure on ${\mathcal{M}}$ is equivalent to that of two structures of a
$G$-module and a ${\mathcal{C}}_{k}(H)$-comodule satisfying two compatibility
equations; essentially, this is the same as an action of $G$ in
${\mathcal{M}}$ whose restriction to $H$ comes from a
${\mathcal{C}}_{k}(H)$-coaction. It remains to notice that an action of $G$
whose restriction to $H$ is discrete is the same thing as a discrete action of
$G$.

Similarly, the category of
${\boldsymbol{\mathcal{S}}}_{k}(G,H)$-semicontramodules is isomorphic to the
category of $G$-contramodules over $k$; the datum of a
${\boldsymbol{\mathcal{S}}}_{k}(G,H)$-semicontramodule structure on a
$k$-vector space ${\mathfrak{P}}$ is equivalent to the datum of a
$G$-contramodule structure. Indeed, according to _loc. cit._ the datum of a
${\boldsymbol{\mathcal{S}}}_{k}(G,H)$-semicontramodule structure on
${\mathcal{M}}$ is equivalent to that of two structures of a $G$-module and a
${\mathcal{C}}_{k}(H)$-contramodule satisfying two compatibility equations.
Essentially, it is claimed that a contraassociative $G$-contraction map
${\mathfrak{P}}[[G]]\longrightarrow{\mathfrak{P}}$ can be uniquely recovered
from its restriction to the point measures in $G$ and the measures supported
inside $H$, provided that the two compatibility equations are satisfied. One
notices that there is an external product map
$k[[G]]\otimes_{k}{\mathfrak{P}}[[G]]\longrightarrow{\mathfrak{P}}[[G\times
G]]$ or
${\mathfrak{P}}[[G]]\otimes_{k}k[[G]]\longrightarrow{\mathfrak{P}}[[G\times
G]]$ assiging to a $k$-valued and a ${\mathfrak{P}}$-valued measures on $G$ a
${\mathfrak{P}}$-valued measure on $G\times G$. The contraassociativity
equation in the definition of a $G$-contramodule, restricted to the external
products of $k$-valued point measures in $G$ and ${\mathfrak{P}}$-valued
measures in $H$, taken in any fixed order, provides a prescription for the
desired recovering of the $G$-contraaction map from its restrictions to the
two specific kinds of measures.

Notice that the underlying $k$-vector space $k(G)$ of the semialgebras
${\boldsymbol{\mathcal{S}}}_{k}(G,H)$ does not depend on the choice of a
compact open subgroup $H\subset G$, but the semialgebra structure does,
${\boldsymbol{\mathcal{S}}}_{k}(G,H)$ being a semialgebra over the coalgebra
${\mathcal{C}}_{k}(H)=k(H)$ depending on $H$. Still, the abelian categories of
semimodules and semicontramodules over ${\boldsymbol{\mathcal{S}}}_{k}(G,H)$
do not depend on $H$; but their semiderived categories and the functors of
semi-infinite (co)homology and the derived semimodule-semicontramodule
correspondence, whose construction is the aim of the book [52], _do_ depend on
$H$ in a quite essential way [52, Section 8.4.6 and Remark E.3.2].

Now one would like to replace a field $k$ with an arbitrary commutative ring,
including first of all $k={\mathbb{Z}}$. This was one of the motivating
examples for developing the theory of semimodules and semicontramodules in the
generality of semialgebras over corings over (generally speaking,
noncommutative) rings rather than just over coalgebras over fields in the main
body of the book [52]. One unpleasant technical complication that arises in
this connection is the possible nonassociativity of cotensor product over a
coring (see the discussion in the beginning of Section 2.5). Hence the
importance of various sufficient conditions guaranteeing such associativity;
see [14, Sections 11.6 and 22.5–6] and [52, Section 1.2.5].

In particular, the results of [14, 11.6(iv)] or [52, Proposition 1.2.5(f)]
ensure that the notions of a semialgebra ${\boldsymbol{\mathcal{S}}}_{k}(G,H)$
and arbitrary semimodules over it are unproblematic for any commutative ring
$k$. To consider semicontramodules, one also needs associativity of the
cohomomorphism functor, which holds in this case by the result of [52,
Proposition 3.2.5(j)]. All the assertions mentioned above in this example
still hold in this setting. The more advanced homological constructions and
results of [52] in the application to the semialgebras
${\boldsymbol{\mathcal{S}}}_{k}(G,H)$ depend on the assumption of the ring $k$
having finite homological dimension, though.

###  2.7. The category ${\mathsf{O}}^{\mathsf{ctr}}$

The conventional concept of representations of an algebraic group $G$ is that
of comodules over the coalgebra of regular functions ${\mathcal{C}}(G)$ on
$G$. Since every comodule over a coassociative coalgebra is the union of its
finite-dimensional subcomodules [62, Section 2.1] (cf. the discussion in
Sections 1.3–1.4), it means that infinite-dimensional representations of $G$
are simply the unions of finite-dimensional representations, or the _ind-
finite-dimensional_ representations.

In particular, while every finite-dimensional representation of the Lie
algebra ${\mathfrak{g}}$ of a simply connected semisimple complex algebraic
group $G$ can be integrated to a representation of $G$, this is _no_ longer
true for infinite-dimensional representations. Indeed, for any nonzero Lie
algebra ${\mathfrak{g}}$ one can easily construct a module that is not a union
of its finite-dimensional submodules (it suffices to consider the enveloping
algebra $U({\mathfrak{g}})$ with the action of ${\mathfrak{g}}$ by left
multiplications).

The Lie correspondence takes a particularly simple form in the case of
unipotent algebraic groups and nilpotent Lie algebras over a field of
characteristic zero, which are two equivalent categories [20, Corollaire
VI.2.4.5] (see [52, Section D.6.1] for further references and a discussion
including the finite characteristic case). A module over a finite-dimensional
nilpotent Lie algebra ${\mathfrak{g}}$ comes from an (always unique)
representation of the corresponding unipotent algebraic group $G$ if and only
if it is a union of finite-dimensional ${\mathfrak{g}}$-modules where all the
vectors from ${\mathfrak{g}}$ act by nilpotent linear operators [50, Sections
3.3.5–7].

It is a classical idea to work with categories intermediate between those of
representations of a Lie or algebraic group $G$ and modules over its Lie
algebra ${\mathfrak{g}}$. For this purpose, one starts from a Lie algebra
${\mathfrak{g}}$ with a chosen _subgroup_ $H$, i. e., an algebraic group
corresponding to a Lie subalgebra ${\mathfrak{h}}\subset{\mathfrak{g}}$. Then
one considers ${\mathfrak{g}}$-modules ${\mathcal{M}}$ for which the
restriction to ${\mathfrak{h}}$ of the action of ${\mathfrak{g}}$ in
${\mathcal{M}}$ can be/has been integrated to an algebraic action of $H$. As
to the choice of the subgroup $H$, there are two basic approaches: given a
complex (or real) semisimple Lie group $G$ and its Lie algebra
${\mathfrak{g}}$, one can take $H$ to be a maximal compact subgroup of $G$; or
otherwise one can use a Borel (or maximal unipotent) subgroup of $G$ in the
role of $H$.

Modules over a semisimple Lie algebra ${\mathfrak{g}}$ integrable to
representations of a maximal compact Lie subgroup $H$ are classically known as
_Harish-Chandra modules_ [21, 65], while ${\mathfrak{g}}$-modules which can be
integrated to an algebraic action of the Borel subgroup form what has been
called the _BGG_ (Bernstein–Gelfand–Gelfand) _category ${\mathsf{O}}$_ [4,
35]. Both can be united under an umbrella notion of _algebraic Harish-Chandra
modules_ , which means simply “modules over a pair consisting of a Lie algebra
and an algebraic subgroup” (see a terminological discussion in [52, Remark
D.2.5]).

An _algebraic Harish-Chandra pair_ [5, Sections 1.8.2 and 3.3.2] is a set of
data consisting of a Lie algebra ${\mathfrak{g}}$ over a field $k$, a finite-
dimensional Lie subalgebra ${\mathfrak{h}}\subset H$, an algebraic group $H$
over $k$ whose Lie algebra is identified with ${\mathfrak{h}}$, and an action
of $H$ by automorphisms of the Lie algebra ${\mathfrak{g}}$. The following two
compatibility conditions have to be satisfied. Firstly, the subalgebra
${\mathfrak{h}}$ must be an $H$-invariant subspace in ${\mathfrak{g}}$, and
the restriction of the action of $H$ in ${\mathfrak{g}}$ to ${\mathfrak{h}}$
should coincide with the adjoint action of $H$ in ${\mathfrak{h}}$. Secondly,
the action of ${\mathfrak{h}}$ in ${\mathfrak{g}}$ obtained by taking the
derivative of the action of $H$ in ${\mathfrak{g}}$ should coincide with the
adjoint action of ${\mathfrak{h}}$ in ${\mathfrak{g}}$.

A _Harish-Chandra module_ ${\mathcal{M}}$ over an algebraic Harish-Chandra
pair $({\mathfrak{g}},H)$ is a $k$-vector space endowed with a
${\mathfrak{g}}$-module structure and an action of $H$ satisfying the
following compatibility conditions. Firstly, the restriction of the
${\mathfrak{g}}$-action in ${\mathcal{M}}$ to the Lie subalgebra
${\mathfrak{h}}$ should coincide with the derivative of the action of $H$ in
${\mathcal{M}}$. Secondly, the ${\mathfrak{g}}$-action map
${\mathfrak{g}}\otimes_{k}{\mathcal{M}}\longrightarrow{\mathcal{M}}$ should be
$H$-equivariant.

In algebraic (rather than algebro-geometric) terms, an (affine) algebraic
group $G$ over a field $k$ is described by the $k$-vector space
${\mathcal{C}}(G)$ of regular functions on $G$, endowed with a noncommutative
convolution comultiplication
$\mu\colon{\mathcal{C}}(G)\longrightarrow{\mathcal{C}}(G)\otimes_{k}{\mathcal{C}}(G)$
induced by the multiplication map $G\times G\longrightarrow G$ and a
commutative pointwise multiplication
$m\colon{\mathcal{C}}(G)\otimes_{k}{\mathcal{C}}(G)\longrightarrow{\mathcal{C}}(G)$.
Together with the antipode map
$s\colon{\mathcal{C}}(G)\longrightarrow{\mathcal{C}}(G)$ induced by the
inverse element map $G\longrightarrow G$, these structures make
${\mathcal{C}}(G)$ a commutative Hopf algebra [62]. By a representation of $G$
one conventionally means a ${\mathcal{C}}(G)$-comodule, while the
multiplication on ${\mathcal{C}}(G)$ is being used in order to define a
${\mathcal{C}}(G)$-comodule structure on the tensor product
${\mathcal{L}}\otimes_{k}{\mathcal{M}}$ of any two $k$-vector spaces endowed
with ${\mathcal{C}}(G)$-comodule structures. The antipode map $s$, being
(always) an anti-automorphism for (both the multiplication and) the
comultiplication of ${\mathcal{C}}={\mathcal{C}}(G)$ (or any other Hopf
algebra ${\mathcal{C}}$), allows to identify the categories of left and right
${\mathcal{C}}$-comodules, so the difference between them is inessential here.

Over a field $k$ of characteristic $0$, the enveloping algebra
$U({\mathfrak{g}})$ of the Lie algebra ${\mathfrak{g}}$ of an algebraic group
$G$ is interpreted as the algebra of left or right invariant differential
operators on $G$ or, simpler yet, the algebra of distributions (“delta
functions”) on $G$ supported at the unit element $e\in G$. The $k$-vector
space of distributions here is defined as the discrete dual vector space to
the linearly compact $k$-vector space of functions on the formal completion of
$G$ at $e$. The noncommutative convolution multiplication $m\colon
U({\mathfrak{g}})\otimes_{k}U({\mathfrak{g}})\longrightarrow
U({\mathfrak{g}})$ in the Hopf algebra $U({\mathfrak{g}})$ is induced by the
multiplication map $G\times G\longrightarrow G$, while the commutative
comultiplication $\mu\colon U({\mathfrak{g}})\longrightarrow
U({\mathfrak{g}})\otimes_{k}U({\mathfrak{g}})$ is induced by the diagonal
embedding $G\longrightarrow G\times G$ and the antipode map $s\colon
U({\mathfrak{g}})\longrightarrow U({\mathfrak{g}})$ simply multiplies every
vector from ${\mathfrak{g}}$ by $-1$.

Evaluating a $\\{e\\}$-supported distribution at a regular function on $G$
defines a natural pairing
$\phi\colon{\mathcal{C}}(G)\otimes_{k}U({\mathfrak{g}})\longrightarrow k$. For
example, the pairing with an element of ${\mathfrak{g}}$ assigns to a regular
function on $G$ the value of its derivative along the corresponding tangent
vector at the origin $e\in G$. The pairing $\phi$ is compatible with the Hopf
algebra structures on ${\mathcal{C}}(G)$ and $U({\mathfrak{g}})$, transforming
the comultiplication in the former into the multiplication in the latter and
vice versa. In our left-right conventions (see Section 1.4 for a discussion),
this compatibility is expressed by the equations

$\displaystyle\phi(f,uv)$ $\displaystyle=\phi(f_{(2)},u)\phi(f_{(1)},v)$
$\displaystyle\phi(fg,u)$ $\displaystyle=\phi(f,u_{(2)})\phi(g,u_{(1)}),$

for any $f$, $g\in{\mathcal{C}}(G)$ and $u$, $v\in U({\mathfrak{g}})$, where
$\mu(f)=f_{(1)}\otimes f_{(2)}$ and $\mu(u)=u_{(1)}\otimes u_{(2)}$ is
Sweedler’s symbolic notation for the comultiplication maps.

Given a ${\mathcal{C}}(G)$-comodule ${\mathcal{M}}$, one defines the
“derivative” $U({\mathfrak{g}})$-module structure $m\colon
U({\mathfrak{g}})\otimes_{k}{\mathcal{M}}\longrightarrow{\mathcal{M}}$ on
${\mathcal{M}}$ as given by the composition
$U({\mathfrak{g}})\otimes_{k}{\mathcal{M}}\longrightarrow
U({\mathfrak{g}})\otimes_{k}{\mathcal{C}}(G)\otimes_{k}{\mathcal{M}}\longrightarrow{\mathcal{M}}$
of the maps induced by the coaction map and the pairing $\phi$ (with the
arguments’ positions inverted). Alternatively, one can obtain a left action
map in the form
$p\colon{\mathcal{M}}\longrightarrow\operatorname{Hom}_{k}(U(g),{\mathcal{M}})$
as the composition
${\mathcal{M}}\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{M}}\longrightarrow\operatorname{Hom}_{k}(U({\mathfrak{g}}),{\mathcal{M}})$
of the left coaction map and the map induced by the pairing $\phi$.
Furthermore, the adjoint action of $G$ in itself preserves the origin, so,
_unlike_ the left and right actions of ${\mathfrak{g}}$ in the enveloping
algebra $U({\mathfrak{g}})$, the adjoint action of ${\mathfrak{g}}$ can be
integrated to a representation of $G$ in $U({\mathfrak{g}})$ as well as in
${\mathfrak{g}}$. Hence both ${\mathfrak{g}}$ and $U({\mathfrak{g}})$ are
endowed with natural ${\mathcal{C}}(G)$-comodule structures.

Now an algebraic Harish-Chandra pair $({\mathfrak{g}},H)$ is described in pure
algebraic terms as a set of data consisting of a Lie algebra ${\mathfrak{g}}$,
a Hopf algebra ${\mathcal{C}}(H)$, a Lie subalgebra
${\mathfrak{h}}\subset{\mathfrak{g}}$, a pairing
$\phi\colon{\mathcal{C}}(H)\otimes_{k}U({\mathfrak{h}})\longrightarrow k$
compatible with the Hopf algebra structures, and a coaction of
${\mathcal{C}}(H)$ in ${\mathfrak{g}}$ satisfying the following compatibility
conditions. Firstly, the coaction of ${\mathcal{C}}(H)$ in ${\mathfrak{g}}$
should be compatible with the Lie algebra structure on ${\mathfrak{g}}$; then
there is also the induced coaction of ${\mathcal{C}}(H)$ in
$U({\mathfrak{g}})$. Secondly, the Lie subalgebra
${\mathfrak{h}}\subset{\mathfrak{g}}$ should be preserved by the
${\mathcal{C}}(H)$-coaction and the pairing $\phi$ should be compatible with
the induced coaction of ${\mathcal{C}}(H)$ in $U({\mathfrak{h}})$ and the
adjoint coaction of ${\mathcal{C}}(H)$ in itself (equivalently, the
restriction $\psi\colon{\mathcal{C}}(H)\times{\mathfrak{h}}\longrightarrow k$
of the pairing $\phi$ should be compatible with the adjoint
${\mathcal{C}}(H)$-coaction in ${\mathcal{C}}(H)$ and
${\mathcal{C}}(H)$-coaction in ${\mathfrak{g}}$ restricted to
${\mathfrak{h}}$). Thirdly, the adjoint action of ${\mathfrak{h}}$ in
${\mathfrak{g}}$ should coincide with the derivative ${\mathfrak{h}}$-action
of the ${\mathcal{C}}(H)$-coaction, which is defined in terms of the given
pairing $\phi$.

Two generalizations of this setting, in two different directions, are
discussed at length in the book [52]. A “quantum” version, with two
noncommutative, noncocommutative Hopf algebras ${\mathcal{C}}$ and $K$ in
place of ${\mathcal{C}}(H)$ and $U({\mathfrak{h}})$, an associative algebra
$R$ in the role of $U({\mathfrak{g}})$, and “adjoint” coactions of
${\mathcal{C}}$ in $K$ and $R$, is introduced in [52, Section C.1]. A “Tate”
version, with the Hopf algebra ${\mathcal{C}}(H)$ of regular functions on an
infinite-dimensional pro-affine pro-algebraic group $H$ and a linearly compact
open subalgebra ${\mathfrak{h}}$ in a Tate (locally linearly compact) Lie
algebra ${\mathfrak{g}}$, can be found in [52, Section D.2.1] (see the
overview in Section 2.8 below).

To repeat a previously given definition in the slightly new language, a
Harish-Chandra module ${\mathcal{M}}$ over an algebraic Harish-Chandra pair
$({\mathfrak{g}},H)$ is a $k$-vector space endowed with a
${\mathfrak{g}}$-module and a ${\mathcal{C}}(H)$-comodule structures
satisfying two compatibility conditions. The restriction of the
${\mathfrak{g}}$-action in ${\mathcal{M}}$ to ${\mathfrak{h}}$ should coincide
with the derivative ${\mathfrak{h}}$-action of the
${\mathcal{C}}(H)$-coaction, and the action map
${\mathfrak{g}}\otimes_{k}{\mathcal{M}}\longrightarrow{\mathcal{M}}$ should be
a ${\mathcal{C}}(H)$-comodule morphism (where the coaction in the tensor
product is defined in terms of the multiplication in the Hopf algebra
${\mathcal{C}}(H)$); equivalently, the action map
$U({\mathfrak{g}})\otimes_{k}{\mathcal{M}}\longrightarrow{\mathcal{M}}$ should
be a ${\mathcal{C}}(H)$-comodule morphism.

Before defining _Harish-Chandra contramodules_ , let us say a few words about
contramodules over the coalgebra ${\mathcal{C}}(G)$ of regular functions on an
algebraic group $G$. Unfortunately, there does _not_ seem to be any particular
way to _interpret_ a ${\mathcal{C}}(G)$-contramodule structure on a $k$-vector
space ${\mathfrak{P}}$ in any terms more explicit than the general definition
of a contramodule over a coalgebra over a field $k$. The only known exception
is the case of a reductive algebraic group $H$ over a field $k$ of
characteristic $0$, when [52, Lemma A.2.2] or the last sentence of Section 1.2
apply and the semisimple abelian categories of ${\mathcal{C}}(H)$-comodules
and ${\mathcal{C}}(H)$-contramodules are equivalent. Otherwise there is only
the general intuition of contramodules as modules with infinite summation
operations, supported by examples such as that of comodules and contramodules
over the coalgebra ${\mathcal{C}}(H)$ of regular functions on the one-
dimensional unipotent algebraic group $H=\mathbb{G}_{a}$ considered in Section
1.3.

However, the vectors spaces
${\mathfrak{P}}=\operatorname{Hom}_{k}({\mathcal{M}},V)$ for all
${\mathcal{C}}(H)$-comodules ${\mathcal{M}}$ (i. e., vector spaces with the
algebraic group $H$ acting in them in the conventional sense) and all
$k$-vector spaces $V$ provide _examples_ of ${\mathcal{C}}(H)$-contramodules
for _any_ algebraic group $H$ (see Section 1.2). Moreover, for any commutative
(for simplicity) Hopf algebra ${\mathcal{C}}$, a ${\mathcal{C}}$-comodule
${\mathcal{M}}$, and a ${\mathcal{C}}$-contramodule ${\mathfrak{P}}$, the
$k$-vector space
${\mathfrak{Q}}=\operatorname{Hom}_{k}({\mathcal{M}},{\mathfrak{P}})$ has a
natural ${\mathcal{C}}$-contramodule structure. To construct the desired left
${\mathcal{C}}$-contraaction map
$\pi_{\mathfrak{Q}}\colon\operatorname{Hom}_{k}({\mathcal{C}},\operatorname{Hom}_{k}({\mathcal{M}},{\mathfrak{P}}))\longrightarrow\operatorname{Hom}_{k}({\mathcal{M}},{\mathfrak{P}})$,
suppose that we are given a $k$-linear map
$g\colon{\mathcal{C}}\longrightarrow\operatorname{Hom}_{k}({\mathcal{M}},{\mathfrak{P}})$
and a vector $m\in{\mathcal{M}}$. Consider the $k$-linear map
$f\colon{\mathcal{C}}\longrightarrow{\mathfrak{P}}$ assigning to any element
$c\in{\mathcal{C}}$ the vector

$f(c)=g\big{(}s(m_{(-1)})c\big{)}(m_{(0)})\in{\mathfrak{P}},$

and set

$\pi_{\mathfrak{Q}}(g)=\pi_{\mathfrak{P}}(f),$

where $m\longmapsto m_{(-1)}\otimes m_{(0)}$ is the Sweedler notation for the
left coaction map
$\nu_{\mathcal{M}}\colon{\mathcal{M}}\longrightarrow{\mathcal{C}}\otimes_{k}{\mathcal{M}}$,
and
$\pi_{\mathfrak{P}}\colon\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$
is the left contraaction map of the original ${\mathcal{C}}$-contramodule
${\mathfrak{P}}$ [52, Section C.4.2].

Furthermore, given an algebraic group $G$ and a
${\mathcal{C}}(G)$-contramodule ${\mathfrak{P}}$, one defines the
“contraderivative” $U({\mathfrak{g}})$-module structure
$U({\mathfrak{g}})\otimes_{k}{\mathfrak{P}}\longrightarrow{\mathfrak{P}}$ on
${\mathfrak{P}}$ as given by the composition
$U({\mathfrak{g}})\otimes_{k}{\mathfrak{P}}\longrightarrow\operatorname{Hom}_{k}({\mathcal{C}}(G),{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$
of the map

$u\otimes p\longmapsto(c\mapsto\phi(c,u)p),\qquad u\in U({\mathfrak{g}}),\
\,c\in{\mathcal{C}}(G),\ \,p\in{\mathfrak{P}}$

induced by the pairing
$\phi\colon{\mathcal{C}}(G)\otimes_{k}U({\mathfrak{g}})\longrightarrow k$ with
the left contraaction map $\pi$ (cf. [52, Sections 10.1.2]).

A _Harish-Chandra contramodule_ ${\mathfrak{P}}$ over an algebraic Harish-
Chandra pair $({\mathfrak{g}},H)$ is a $k$-vector space endowed with a
${\mathfrak{g}}$-module and a ${\mathcal{C}}(H)$-contramodule structures
satisfying the following two compatibility conditions. Firstly, the
restriction of the ${\mathfrak{g}}$-module structure on ${\mathfrak{P}}$ to
the Lie subalgebra ${\mathfrak{h}}\subset{\mathfrak{g}}$ should coincide with
the contraderivative ${\mathfrak{h}}$-module structure of the
${\mathcal{C}}(H)$-contramodule structure on ${\mathfrak{P}}$. Secondly, the
$U({\mathfrak{g}})$-action map in the form
${\mathfrak{P}}\longrightarrow\operatorname{Hom}_{k}(U({\mathfrak{g}}),{\mathfrak{P}})$
should be a ${\mathcal{C}}(H)$-contramodule morphism, where the
${\mathcal{C}}(H)$-contramodule structure on
$\operatorname{Hom}_{k}(U({\mathfrak{g}}),{\mathfrak{P}})$ is obtained from
the ${\mathcal{C}}(H)$-comodule structure on $U({\mathfrak{g}})$ and the
${\mathcal{C}}(H)$-contramodule structure on ${\mathfrak{P}}$ as described
above.

The latter condition is equivalent to the ${\mathfrak{g}}$-action map
${\mathfrak{P}}\longrightarrow\operatorname{Hom}_{k}({\mathfrak{g}},{\mathfrak{P}})$
being a ${\mathcal{C}}(H)$-contramodule morphism. To convince oneself that
this is so, one can present the space
$\operatorname{Hom}_{k}(U({\mathfrak{g}}),{\mathfrak{P}})$ as the projective
limit of the Hom spaces
$\operatorname{Hom}_{k}(F_{n}U({\mathfrak{g}}),{\mathfrak{P}})$, where $F$
denotes the Poincaré–Birkhoff–Witt filtration of $U({\mathfrak{g}})$, and
further present every space
$\operatorname{Hom}_{k}(F_{n}U({\mathfrak{g}}),{\mathfrak{P}})$ as a subspace
of the Hom space
$\operatorname{Hom}_{k}(\bigoplus_{i=0}^{n}{\mathfrak{g}}^{\otimes
n},\medspace{\mathfrak{P}})$. Then it remains to use the fact that the
${\mathcal{C}}(H)$-comodule structure on $U({\mathfrak{g}})$ is compatible
with the associative algebra structure, i. e., the multiplication map
$U({\mathfrak{g}})\otimes_{k}U({\mathfrak{g}})\longrightarrow
U({\mathfrak{g}})$ is a ${\mathcal{C}}(H)$-comodule morphism, together with
the assumption of associativity of the $U({\mathfrak{g}})$-action in
${\mathfrak{P}}$.

Viewing the case of a semisimple Lie algebra ${\mathfrak{g}}$ with a Borel or
maximal unipotent subgroup $H$ as our main example, we denote by
${\mathsf{O}}({\mathfrak{g}},H)$ the category of Harish-Chandra modules over
an algebraic Harish-Chandra pair $({\mathfrak{g}},H)$ and by
${\mathsf{O}}^{\mathsf{ctr}}({\mathfrak{g}},H)$ the category of Harish-Chandra
contramodules. These are abelian categories with exact forgetful functors
${\mathsf{O}}({\mathfrak{g}},H)\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ and
${\mathsf{O}}^{\mathsf{ctr}}({\mathfrak{g}},H)\longrightarrow
k{\operatorname{\mathsf{--vect}}}$, the former of which preserves infinite
direct sums, while the latter preserves infinite products. The category
${\mathsf{O}}({\mathfrak{g}},H)$ satisfies the axioms Ab5 and Ab3* (but not
Ab4*), while the category ${\mathsf{O}}^{\mathsf{ctr}}({\mathfrak{g}},H)$
satisfies the axioms Ab3 and Ab4* (but not Ab4 or Ab5*).

Now it is claimed that there is a semialgebra ${\boldsymbol{\mathcal{S}}}$
over the coalgebra ${\mathcal{C}}={\mathcal{C}}(H)$ such that the categories
${\mathsf{O}}({\mathfrak{g}},H)$ and
${\mathsf{O}}^{\mathsf{ctr}}({\mathfrak{g}},H)$ are identified with the
categories of semimodules and semicontramodules over
${\boldsymbol{\mathcal{S}}}$. More precisely, there are naturally _two_ such
semialgebras ${\boldsymbol{\mathcal{S}}}^{l}({\mathfrak{g}},H)$ and
${\boldsymbol{\mathcal{S}}}^{r}({\mathfrak{g}},H)$, differing by the left-
right symmetry. The category of Harish-Chandra modules
${\mathsf{O}}({\mathfrak{g}},H)$ is isomorphic to the categories of left
semimodules over ${\boldsymbol{\mathcal{S}}}^{l}({\mathfrak{g}},H)$ and right
semimodules over ${\boldsymbol{\mathcal{S}}}^{r}({\mathfrak{g}},H)$, while the
category of Harish-Chandra contramodules
${\mathsf{O}}^{\mathsf{ctr}}({\mathfrak{g}},H)$ is isomorphic to the category
of left semicontramodules over
${\boldsymbol{\mathcal{S}}}^{r}({\mathfrak{g}},H)$.

The semialgebra ${\boldsymbol{\mathcal{S}}}^{l}({\mathfrak{g}},H)$ is
constructed as the tensor product
$U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}{\mathcal{C}}(H)$, where the left
$U({\mathfrak{h}})$-module structure on ${\mathcal{C}}(H)$ is obtained by
deriving the left coaction of ${\mathcal{C}}(H)$ in itself. The right
${\mathcal{C}}$-comodule structure on
$U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}{\mathcal{C}}(H)$ is induced by
the right coaction of ${\mathcal{C}}$ in itself, while the left
${\mathcal{C}}$-comodule structure on this tensor product is defined in terms
of the multiplication in ${\mathcal{C}}$, as the tensor product of the left
coaction of ${\mathcal{C}}$ in itself and the adjoint coaction of
${\mathcal{C}}$ in $U({\mathfrak{g}})$. The semiunit map
${\mathbf{e}}\colon{\mathcal{C}}\longrightarrow{\boldsymbol{\mathcal{S}}}^{l}=U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}{\mathcal{C}}$
is induced by the embedding of algebras $U({\mathfrak{h}})\longrightarrow
U({\mathfrak{g}})$. Finally, the semimultiplication map
${\mathbf{m}}\colon{\boldsymbol{\mathcal{S}}}^{l}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}^{l}\longrightarrow{\boldsymbol{\mathcal{S}}}^{l}$
is defined as the composition

$\big{(}U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}{\mathcal{C}}\big{)}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}\big{(}U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}{\mathcal{C}}\big{)}\text{$\mskip
1.5mu$}\simeq\text{$\mskip
1.5mu$}U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}\big{(}{\mathcal{C}}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}(U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}{\mathcal{C}})\big{)}\\\
\text{$\mskip 1.5mu$}\simeq\text{$\mskip
1.5mu$}U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}{\mathcal{C}}\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}{\mathcal{C}}$

of the mutual associativity isomorphism of the tensor and cotensor products,
whose natural existence in this case can be easily established from the fact
that $U({\mathfrak{g}})$ is a projective right $U({\mathfrak{h}})$-module (by
the Poincaré–Birkhoff–Witt theorem) [52, Section 1.2.3], and the map induced
by the multiplication map
$U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}U({\mathfrak{g}})\longrightarrow
U({\mathfrak{g}})$. This construction can be found in [52, Section 10.2.1].

Notice that for any left ${\mathcal{C}}$-comodule ${\mathcal{M}}$ there is a
natural isomorphism

${\boldsymbol{\mathcal{S}}}^{l}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathcal{M}}=\big{(}U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}{\mathcal{C}}\big{)}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\mathcal{M}}\simeq
U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}{\mathcal{M}},$

where the $U({\mathfrak{h}})$-module structure on ${\mathcal{M}}$ is obtained
by deriving the ${\mathcal{C}}(H)$-comodule structure. Hence it follows that
the datum of a left ${\boldsymbol{\mathcal{S}}}^{l}$-semimodule structure on a
$k$-vector space ${\boldsymbol{\mathcal{M}}}$ is equivalent to that of a left
${\mathcal{C}}(H)$-comodule and a left $U({\mathfrak{g}})$-module structures
on ${\boldsymbol{\mathcal{M}}}$ satisfying two compatibility equations [52,
Section 10.2.2]. These are easily seen to express the definition of a
structure of Harish-Chandra module over $({\mathfrak{g}},H)$ on the vector
space ${\boldsymbol{\mathcal{M}}}$.

Similarly, the semialgebra
${\boldsymbol{\mathcal{S}}}^{r}={\boldsymbol{\mathcal{S}}}^{r}({\mathfrak{g}},H)$
is constructed as the tensor product
${\mathcal{C}}(H)\otimes_{U({\mathfrak{h}})}U({\mathfrak{g}})$, where the
right $U({\mathfrak{h}})$-module structure on ${\mathcal{C}}(H)$ is obtained
by deriving the right coaction of ${\mathcal{C}}(H)$ in itself. The left
${\mathcal{C}}$-comodule structure on this tensor product is induced by the
left coaction of ${\mathcal{C}}$ in itself, while the right
${\mathcal{C}}$-comodule structure is obtained by multiplying the right
coaction of ${\mathcal{C}}$ in itself and the adjoint coaction of
${\mathcal{C}}$ in $U({\mathfrak{g}})$. The semimultiplication and semiunit
maps of the semialgebra ${\boldsymbol{\mathcal{S}}}^{r}$ are induced by the
multiplication map
$U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}U({\mathfrak{g}})\longrightarrow
U({\mathfrak{g}})$ and the embedding $U({\mathfrak{h}})\longrightarrow
U({\mathfrak{g}})$, as above.

For any left ${\mathcal{C}}$-contramodule ${\mathfrak{P}}$ there is a natural
isomorphism

$\operatorname{Cohom}_{\mathcal{C}}({\boldsymbol{\mathcal{S}}}^{r},{\mathfrak{P}})=\operatorname{Cohom}_{\mathcal{C}}({\mathcal{C}}\otimes_{U({\mathfrak{h}})}U({\mathfrak{g}}),\medspace{\mathfrak{P}})\simeq\operatorname{Hom}_{U({\mathfrak{h}})}(U({\mathfrak{g}}),{\mathfrak{P}}),$

where the $U({\mathfrak{h}})$-module structure on ${\mathfrak{P}}$ is obtained
by contraderiving the ${\mathcal{C}}(H)$-contramodule structure. Hence one
concludes that the datum of a left
${\boldsymbol{\mathcal{S}}}^{r}$-semicontramodule structure on a $k$-vector
spaced ${\boldsymbol{\mathfrak{P}}}$ is equivalent to that of a left
${\mathcal{C}}$-contramodule and left $U({\mathfrak{g}})$-module structures on
${\boldsymbol{\mathfrak{P}}}$ satisfying two compatibility equations (cf.
Example in Section 2.6). These are easily seen to express the definition of a
structure of Harish-Chandra contramodule over $({\mathfrak{g}},H)$ on the
vector space ${\boldsymbol{\mathfrak{P}}}$.

In addition to these descriptions of left
${\boldsymbol{\mathcal{S}}}^{l}$-semimodules and left
${\boldsymbol{\mathcal{S}}}^{r}$-semicontramodules, one would like to have an
explicit interpretation of what it means to have a left
${\boldsymbol{\mathcal{S}}}^{r}$-semimodule structure on a $k$-vector space
${\boldsymbol{\mathcal{M}}}$. Such a description of left
${\boldsymbol{\mathcal{S}}}^{r}$-semimodules is indeed obtained in [52,
Sections C.2 and C.4.3–4] under certain mild assumptions, which we will now
discuss.

So far we used the notions of an “algebraic group $H$” and “a commutative Hopf
algebra ${\mathcal{C}}(H)$” interchangeably, but in fact there are several
differences between the two (cf. [19]). First of all, it is only _affine_
algebraic groups that can be described by the algebras of global functions on
them. As our aim is to consider linear representations of our algebraic
groups, we can safely assume all of them to be affine.

Secondly, it is only _finitely generated_ commutative algebras over a field
that correspond to algebraic varieties; the spectrum of an arbitrary
commutative algebra is, generally speaking, a _pro-affine pro-algebraic
variety_. Considering Harish-Chandra pairs with pro-affine pro-algebraic
groups $H$ presumes also having an infinite-dimensional linearly compact Lie
subalgebra ${\mathfrak{h}}\subset{\mathfrak{g}}$. Postponing this discussion
to Section 2.8, we for now assume all our algebraic groups $H$ to be finite-
dimensional, or the commutative Hopf algebras ${\mathcal{C}}(H)$ to be
finitely generated as algebras over $k$.

Thirdly and finally, _algebraic varieties_ are generally assumed to be
_reduced schemes_ , i. e., to have no nilpotent elements in their structure
sheaves (or, if they are affine, in the algebras of global functions on them).
Now, over a field $k$ of characteristic $0$, every algebraic _group_ scheme is
a smooth variety, and over any field $k$ every reduced algebraic group scheme
is a smooth variety; but over a field $k$ of finite characteristic there exist
_nonreduced_ algebraic group schemes. It suffices to consider the spectrum of
the finite-dimensional algebra ${\mathcal{C}}=k[x]/(x^{p})$ over a field $k$
of characteristic $p$ and notice that the rule $\mu(x)=1\otimes x+x\otimes 1$
describes a well-defined coassociative, counital comultiplication making
${\mathcal{C}}$ a Hopf algebra over $k$.

Now, assuming the algebraic group $H$ to be a smooth finite-dimensional
variety over a field $k$, there is the one-dimensional vector bundle of
differential forms of the top degree on $H$. The group $H$ acts in itself by
the left and right multiplications, and there are the two induced actions in
the vector space of global differential top forms
${\mathcal{E}}=\Omega^{top}(H)$. The subspace of left $H$-invariant top forms
in ${\mathcal{E}}$ is always one-dimensional, as is the subspace of right
$H$-invariant top forms, but these two subspaces may not coincide. A smooth
algebraic group $H$ admitting a nonzero _biinvariant_ top form is said to be
_unimodular_. An algebraic group $H$ is unimodular if and only if all the
operators of its adjoint representation $\operatorname{Ad}_{H}\colon
H\longrightarrow\operatorname{GL}({\mathfrak{h}})$ belong to the subgroup
$\operatorname{SL}({\mathfrak{h}})\subset\operatorname{GL}({\mathfrak{h}})$.
All the reductive algebraic groups are unimodular, as are all the nilpotent
ones; but many solvable groups are not.

Let us first assume the smooth algebraic group $H$ to be unimodular. Then the
semialgebras ${\boldsymbol{\mathcal{S}}}^{l}({\mathfrak{g}},H)$ and
${\boldsymbol{\mathcal{S}}}^{r}({\mathfrak{g}},H)$ are naturally isomorphic to
each other; the isomorphism is provided by the maps given by the formulas

$c\otimes_{U({\mathfrak{h}})}u\longmapsto
u_{[0]}\otimes_{U({\mathfrak{h}})}cu_{[1]}\quad\text{and}\quad
u\otimes_{U({\mathfrak{h}})}c\longmapsto
s(u_{[1]})c\otimes_{U({\mathfrak{h}})}u_{[0]},$

where $c\in{\mathcal{C}}(H)$, $u\in U({\mathfrak{g}})$, the notation
$c\otimes_{U({\mathfrak{h}})}u$ and $u\otimes_{U({\mathfrak{h}})}c$ stands for
elements of
${\boldsymbol{\mathcal{S}}}^{r}={\mathcal{C}}(H)\otimes_{U({\mathfrak{h}})}U({\mathfrak{g}})$
and
${\boldsymbol{\mathcal{S}}}^{l}=U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}{\mathcal{C}}(H)$,
respectively, and $u\longmapsto u_{[0]}\otimes u_{[1]}$ with $u_{[0]}\in
U({\mathfrak{g}})$ and $u_{[1]}\in{\mathcal{C}}(H)$ is the Sweedler notation
for right coaction map defining the adjoint coaction of ${\mathcal{C}}(H)$ in
$U({\mathfrak{g}})$ [52, Section C.2.6]. Accordingly, the categories of left
${\boldsymbol{\mathcal{S}}}^{r}$-semimodules and left
${\boldsymbol{\mathcal{S}}}^{l}$-semimodules are naturally isomorphic.

In the general case of a nonunimodular smooth algebraic group $H$, the
categories of left ${\boldsymbol{\mathcal{S}}}^{r}$-semimodules and left
${\boldsymbol{\mathcal{S}}}^{l}$-semimodules are still naturally equivalent,
but the equivalence
${\boldsymbol{\mathcal{S}}}^{r}{\operatorname{\mathsf{--simod}}}\simeq{\boldsymbol{\mathcal{S}}}^{l}{\operatorname{\mathsf{--simod}}}$
does _not_ commute with the forgetful functors
${\boldsymbol{\mathcal{S}}}^{r}{\operatorname{\mathsf{--simod}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--comod}}}$
and
${\boldsymbol{\mathcal{S}}}^{l}{\operatorname{\mathsf{--simod}}}\longrightarrow{\mathcal{C}}{\operatorname{\mathsf{--comod}}}$.
Instead, the two forgetful functors differ by a twist with the modular
character $\det\operatorname{Ad}_{H}\colon H\longrightarrow
GL({\mathfrak{h}})\longrightarrow k^{*}$ [52, Sections C.2.4–5].

When that the ground field $k$ has characteristic $0$ and the Harish-Chandra
pair $({\mathfrak{g}},H)$ originates from a closed embedding of algebraic
groups $H\subset G$ over $k$, choosing a biinvariant top form on $H$ allows to
interpret elements of the semialgebra
${\boldsymbol{\mathcal{S}}}^{l}\simeq{\boldsymbol{\mathcal{S}}}^{r}$ as
distributions on the smooth variety $G$, supported in the smooth closed
subvariety $H$ and regular along $H$ [52, Remark C.4.4]. In the nonunimodular
case, the $k$-vector space of all distributions on $G$, supported in $H$ and
regular along $H$, has a natural structure of an
_${\boldsymbol{\mathcal{S}}}^{l}$
-${\boldsymbol{\mathcal{S}}}^{r}$-bisemimodule_ providing the Morita
equivalence between the semialgebras ${\boldsymbol{\mathcal{S}}}^{l}$ and
${\boldsymbol{\mathcal{S}}}^{r}$.

###  2.8. Tate Harish-Chandra pairs

It appears that one cannot integrate the Virasoro Lie algebra to a Lie group,
but _a half_ of it one easily can. Indeed, let $k$ be a field of
characteristic $0$. Denote by $H(k)$ the set of all formal Taylor power series
$a(z)=a_{1}z+a_{2}z^{2}+a_{3}z^{3}+\dotsb$ with a vanishing coefficient in
degree $0$ and a nonvanishing coefficient $a_{1}\neq 0$ in degree $1$. Then
the composition multiplication $(a*b)(z)=a(b(z))$ defines a group structure on
the set $H(k)$. This group is naturally the group of $k$-points of a certain
pro-affine pro-algebraic group, which we denote by $H$. The Lie algebra
${\mathfrak{h}}$ of the pro-algebraic group $H$ can be easily identified with
the algebra of vector fields on the formal disk with a vanishing vector at the
origin $zk[[z]]d/dz$, or with the closed subalgebra in the Virasoro Lie
algebra topologically spanned by the basis vectors $L_{0}$, $L_{1}$, $L_{2}$,
… (see Section 1.7).

Let us say a few words about Lie theory in the pro-algebraic group setting.
For any subcoalgebra ${\mathcal{D}}$ in a commutative Hopf algebra
${\mathcal{C}}$, the subalgebra generated by ${\mathcal{D}}+s({\mathcal{D}})$
in a Hopf subalgebra in ${\mathcal{C}}$. Since ${\mathcal{C}}$ is the union of
its finite-dimensional subcoalgebras (see Section 1.3), it it also the union
of its Hopf subalgebras that are finitely generated as commutative algebras.
Thus there is no difference between the notions of a pro-affine pro-algebraic
variety with a group structure and a pro-object in the category of affine
algebraic groups. The Lie functor on the category of (pro-afffine) pro-
algebraic groups can be simply obtained by passing to the pro-objects on both
sides of the functor assigning a Lie algebra to an algebraic group; so the Lie
algebra of a pro-algebraic group is a filtered projective limit of finite-
dimensional Lie algebras. In particular, it follows that the topological Lie
algebra $k[[z]]d/dz$, which has no closed ideals (see Section 2.4), _cannot_
correspond to any pro-algebraic group, though its subalgebra $zk[[z]]d/dz$
does, as we have just seen.

We are interested in considering Harish-Chandra pairs with a Tate Lie algebra
${\mathfrak{g}}$ and pro-algebraic subgroup $H$ corresponding to a open
linearly compact subalgebra ${\mathfrak{h}}\subset{\mathfrak{g}}$. A precise
definition presents a small technical difficulty in that one has to explain
what it means for the coalgebra ${\mathcal{C}}={\mathcal{C}}(H)$ to act in a
Tate vector space ${\mathfrak{g}}$. Neither the notion of a
${\mathcal{C}}$-comodule _nor_ that of a ${\mathcal{C}}$-contramodule are
suitable for the task; rather, the compact vector space ${\mathfrak{h}}$,
being dual to a ${\mathcal{C}}$-comodule, can be viewed as a
${\mathcal{C}}$-contramodule, while the quotient space
${\mathfrak{g}}/{\mathfrak{h}}$ has a ${\mathcal{C}}$-comodule structure.

However, an action of a algebraic group in a vector space is, of course, _not_
determined by its restriction to an invariant subspace and the induced action
in the quotient space. The authors of the manuscript [6], where (what we call)
Tate Harish-Chandra pairs seem to have first appeared, solve the problem by
working over the field of complex numbers $\mathbb{C}$ and considering an
action of the _group of points_ $H(\mathbb{C})$ of a pro-algebraic group $H$
in a Tate Lie algebra ${\mathfrak{g}}$ [6, Section 3.1]. The approach taken in
[52, Appendix D], which works over an arbitrary field, is to give the
definition of a _continuous coaction_ of a discrete coalgebra in a topological
vector space.

Here we restrict ourselves to a brief sketch. Let ${\mathcal{C}}$ be a
coassociative coalgebra and $V$ be a topological vector space over $k$ (see
Sections 2.3–2.4). A _continuous right coaction_ of ${\mathcal{C}}$ in $V$ is
a continuous linear map $V\longrightarrow V\otimes^{!}{\mathcal{C}}$, where
${\mathcal{C}}$ is considered as a discrete topological vector space,
satisfying the coassociativity and counitality equations. Equivalently, a
continuous right coaction can be defined as a continuous linear map
$V\otimes^{*}{\mathcal{C}}^{*}\longrightarrow V$, where ${\mathcal{C}}^{*}$ is
viewed as a linearly compact vector space, satisfying the associativity and
unitality equations.

For any topological vector space $V$ with a continuous coaction of a
coassociative coalgebra ${\mathcal{C}}$, open subspaces of $V$ invariant under
the continuous coaction form a base of neighborhoods of zero in $V$. Given a
topological vector space $V$ endowed with a continuous coaction of a coalgebra
${\mathcal{C}}$ and a topological vector space $W$ endowed with a continuous
coaction of a coalgebra ${\mathcal{D}}$, all the three topological tensor
products $V\otimes^{!}W$, $V\otimes^{*}W$, and
$V\mathbin{\overset{\leftarrow}{\vphantom{o}\smash{\otimes}}}W$ are naturally
endowed with a continuous coaction of the coalgebra
${\mathcal{C}}\otimes_{k}{\mathcal{D}}$. In particular, when two topological
vector spaces $V$ and $W$ are endowed with continuous coactions of a Hopf
algebra ${\mathcal{C}}$, the three topological tensor products acquire the
tensor product coactions of ${\mathcal{C}}$ [52, Sections D.1.3–4].

A continuous coaction of a commutative Hopf algebra ${\mathcal{C}}$ in a
topological Lie algebra ${\mathfrak{g}}$ is said to be _compatible_ with the
Lie algebra structure if the bracket map
$[\,,\,]\colon{\mathfrak{g}}\otimes^{*}{\mathfrak{g}}\longrightarrow{\mathfrak{g}}$
commutes with the continuous coactions of ${\mathcal{C}}$. Similarly one can
speak about compatibility of topological associative algebra structures,
continuous actions, pairings, etc. with continuous coactions of
${\mathcal{C}}$ [52, Section D.1.5]. A Tate Lie algebra ${\mathfrak{g}}$ with
a continuous coaction of a commutative Hopf algebra ${\mathcal{C}}$ compatible
with the Lie algebra structure has a base of neighborhoods of zero consisting
of ${\mathcal{C}}$-invariant compact open Lie subalgebras
${\mathfrak{h}}\subset{\mathfrak{g}}$ [52, Section D.1.8].

A _Tate Harish-Chandra pair_ $({\mathfrak{g}},{\mathcal{C}})$ is a set of data
consisting of a Tate Lie algebra ${\mathfrak{g}}$, a commutative Hopf algebra
${\mathcal{C}}$, a continuous coaction of ${\mathcal{C}}$ in ${\mathfrak{g}}$
compatible with the Lie algebra structure, a ${\mathcal{C}}$-invariant compact
open subalgebra ${\mathfrak{h}}\subset{\mathfrak{g}}$, and a continuous
pairing $\psi\colon{\mathcal{C}}\times{\mathfrak{h}}\longrightarrow k$, where
${\mathcal{C}}$ is endowed with the discrete topology. These data should
satisfy four compatibility equations: the pairing $\psi$ should be compatible
with the comultiplication in ${\mathcal{C}}$ and the Lie bracket in
${\mathfrak{g}}$, and also with the multiplication in ${\mathcal{C}}$; the
pairing $\psi$ should be compatible with the restriction to ${\mathfrak{h}}$
of the continuous coaction of ${\mathfrak{h}}$ in ${\mathcal{C}}$ and the
adjoint coaction of ${\mathcal{C}}$ in itself; the action of ${\mathfrak{h}}$
in ${\mathfrak{g}}$ obtained by deriving the continuous coaction of
${\mathcal{C}}$ in ${\mathfrak{g}}$ using the paring $\psi$ should coincide
with the adjoint action of ${\mathfrak{h}}$ in ${\mathfrak{g}}$. We refer to
[52, Section D.2.1] for the details.

In particular, let ${\mathfrak{g}}$ be a Tate Lie algebra with a pro-nilpotent
compact open subalgebra ${\mathfrak{h}}\subset{\mathfrak{g}}$, i. e.,
${\mathfrak{h}}$ is the projective limit of a filtered projective system of
finite-dimensional nilpotent Lie algebras. Assume further that the discrete
${\mathfrak{h}}$-module ${\mathfrak{g}}/{\mathfrak{h}}$ is nilpotent or “ind-
nilpotent”, i. e., it is the union of finite-dimensional nilpotent modules
over finite-dimensional quotient Lie algebras of ${\mathfrak{h}}$ by its open
ideals. Then, over a field of characteristic $0$, one can integrate the pair
of Lie algebras $({\mathfrak{g}},{\mathfrak{h}})$ to a Tate Harish-Chandra
pair with the Hopf algebra ${\mathcal{C}}={\mathcal{C}}(H)$ of the pro-
unipotent pro-algebraic group $H$ corresponding to the Lie algebra
${\mathfrak{h}}$. A version of this construction is applicable over fields of
arbitrary characteristic [52, Section D.6]. In particular, let
${\mathfrak{g}}=\bigoplus_{n<0}{\mathfrak{g}}_{n}\oplus\prod_{n\geqslant
0}{\mathfrak{g}}_{n}$ be the Laurent totalization a ${\mathbb{Z}}$-graded Lie
algebra with finite-dimensional components (see Section 2.4); then for any
integer $m\geqslant 1$ the Lie subalgebra ${\mathfrak{h}}=\prod_{n\geqslant
m}{\mathfrak{g}}_{m}\subset{\mathfrak{g}}$ satisfies the above nilpotency
conditions, so there is the corresponding Tate Harish-Chandra pair
$({\mathfrak{g}},{\mathcal{C}})$.

A _Harish-Chandra module_ ${\mathcal{M}}$ over a Tate Harish-Chandra pair
$({\mathfrak{g}},{\mathcal{C}})$ is a $k$-vector space endowed with a discrete
${\mathfrak{g}}$-module and a ${\mathcal{C}}$-comodule structures satisfying
the following two compatibility equations. Firstly, the derivative
${\mathfrak{h}}$-action of the ${\mathcal{C}}$-coaction in ${\mathcal{M}}$,
which is always a discrete action due to the continuity/discreteness condition
imposed on the pairing $\psi$, should coincide with the restriction of the
${\mathfrak{g}}$-action in ${\mathcal{M}}$ to ${\mathfrak{h}}$. Secondly, the
action map
${\mathfrak{g}}\otimes^{*}{\mathcal{M}}\longrightarrow{\mathcal{M}}$ should be
compatible with the continuous coactions of ${\mathcal{C}}$; equivalently, the
action map
${\mathfrak{g}}/U\otimes_{k}{\mathcal{L}}\longrightarrow{\mathcal{M}}$ should
be a morphism of ${\mathcal{C}}$-comodules for any finite-dimensional
${\mathcal{C}}$-subcomodule ${\mathcal{L}}\subset{\mathcal{M}}$ and any
${\mathcal{C}}$-invariant compact open subspace $U\subset{\mathfrak{g}}$
annihilating ${\mathcal{L}}$ [52, Section D.2.5].

The pairing $\psi\colon{\mathcal{C}}\times{\mathfrak{h}}\longrightarrow k$ can
be viewed as a Lie algebra morphism
${\mathfrak{h}}\longrightarrow{\mathcal{C}}^{*}$ (where the Lie algebra
structure on ${\mathcal{C}}^{*}$ is defined in terms of its associative
algebra structure), and as such, can be uniquely extended to an associative
algebra morphism $U({\mathfrak{h}})\longrightarrow{\mathcal{C}}^{*}$,
providing a pairing
$\phi\colon{\mathcal{C}}\otimes_{k}U({\mathfrak{h}})\longrightarrow k$
compatible with the Hopf algebra structures on both sides. When the pairing
$\phi$ is nondegenerate in ${\mathcal{C}}$ [52, condition D.2.2 (iv)], the
derivative action functor
${\mathcal{C}}{\operatorname{\mathsf{--comod}}}\longrightarrow{\mathfrak{h}}{\operatorname{\mathsf{--discr}}}$
is fully faithful [52, Section 10.1.4], which allows to simplify the
definition of a Harish-Chandra module over $({\mathfrak{g}},{\mathcal{C}})$.
Namely, the second one of the above two compatibility equations holds
automatically in this case and can be dropped, so Harish-Chandra modules over
$({\mathfrak{g}},{\mathcal{C}})$ can be simply defined as discrete
${\mathfrak{g}}$-modules whose discrete ${\mathfrak{h}}$-module structure
comes from a ${\mathcal{C}}$-comodule structure (cf. [52, Section D.2.2]). In
particular, this nondegeneracy condition holds in the above example of a Tate
Harish-Chandra pair associated with a Tate Lie algebra ${\mathfrak{h}}$ with a
pro-nilpotent compact open subalgebra ${\mathfrak{g}}$ acting ind-nilpotently
in ${\mathfrak{g}}/{\mathfrak{h}}$ (over any field $k$).

A _Harish-Chandra contramodule_ ${\mathfrak{P}}$ over a Tate Harish-Chandra
pair $({\mathfrak{g}},{\mathcal{C}})$ is a $k$-vector space endowed with a
${\mathfrak{g}}$-contramodule and a ${\mathcal{C}}$-contramodule structures
satisfying the following two compatibility equations. Firstly, the
contraderivative ${\mathfrak{h}}$-contraaction of the
${\mathcal{C}}$-contraaction in ${\mathfrak{P}}$, which is defined as the
composition

${\mathfrak{h}}\otimes^{^}\,{\mathfrak{P}}\text{$\mskip
1.5mu$}\simeq\text{$\mskip
1.5mu$}\operatorname{Hom}_{k}({\mathfrak{h}}^{\vee},{\mathfrak{P}})\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}\operatorname{Hom}_{k}({\mathcal{C}},{\mathfrak{P}})\text{$\mskip
1.5mu$}\relbar\joinrel\relbar\joinrel\rightarrow\text{$\mskip
1.5mu$}{\mathfrak{P}}$

of the map induced by the pairing $\psi$ and the ${\mathcal{C}}$-contraaction
map, should coincide with the restriction of the ${\mathfrak{g}}$-contraaction
in ${\mathfrak{P}}$ to the subalgebra ${\mathfrak{h}}$ (see the definition of
a contramodule over a Tate Lie algebra ${\mathfrak{g}}$ in Section 2.4).
Secondly, for any ${\mathcal{C}}$-invariant compact open subspace
$U\subset{\mathfrak{g}}$ the ${\mathfrak{g}}$-contraaction map
$\operatorname{Hom}_{k}(U^{\vee},{\mathfrak{P}})\longrightarrow{\mathfrak{P}}$
should be a morphism of ${\mathcal{C}}$-contramodules (where the
${\mathcal{C}}$-contramodule structure on the Hom space from a
${\mathcal{C}}$-comodule $U^{\vee}$ into a ${\mathcal{C}}$-contramodule
${\mathfrak{P}}$ is provided by the construction from Section 2.7). This
definition can be found in [52, Sections D.2.7–8].

For any Harish-Chandra module ${\mathcal{M}}$ over
$({\mathfrak{g}},{\mathcal{C}})$ and any $k$-vector space $E$, the vector
space ${\mathfrak{P}}=\operatorname{Hom}_{k}({\mathcal{M}},E)$ has a natural
Harish-Chandra contramodule structure with the ${\mathcal{C}}$-contraaction in
${\mathfrak{P}}$ provided by the construction from Section 1.2 and the
${\mathfrak{g}}$-contraaction in ${\mathfrak{P}}$ defined according to the
construction from Section 2.4.

The categories ${\mathsf{O}}({\mathfrak{g}},{\mathcal{C}})$ and
${\mathsf{O}}^{\mathsf{ctr}}({\mathfrak{g}},{\mathcal{C}})$ of Harish-Chandra
modules and contramodules over a Tate Harish-Chandra pair
$({\mathfrak{g}},{\mathcal{C}})$ are abelian, and the forgetful functors
${\mathsf{O}}({\mathfrak{g}},{\mathcal{C}})\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ and
${\mathsf{O}}^{\mathsf{ctr}}({\mathfrak{g}},{\mathcal{C}})\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ are exact. Both the infinite direct sums
and infinite products exist in the cagories
${\mathsf{O}}({\mathfrak{g}},{\mathcal{C}})$ and
${\mathsf{O}}^{\mathsf{ctr}}({\mathfrak{g}},{\mathcal{C}})$, but only the
direct sums are preserved by the forgetful functor
${\mathsf{O}}({\mathfrak{g}},{\mathcal{C}})\longrightarrow
k{\operatorname{\mathsf{--vect}}}$ and only the products are preserved by the
functor
${\mathsf{O}}^{\mathsf{ctr}}({\mathfrak{g}},{\mathcal{C}})\longrightarrow
k{\operatorname{\mathsf{--vect}}}$. The category
${\mathsf{O}}({\mathfrak{g}},{\mathcal{C}})$ satisfies the axioms Ab5 and Ab3*
(but not Ab4*), while the category
${\mathsf{O}}({\mathfrak{g}},{\mathcal{C}})$ satisfies the axioms Ab3 and Ab4*
(but not Ab4 or Ab5*).

There are enough injective objects in the category
${\mathsf{O}}({\mathfrak{g}},{\mathcal{C}})$ and enough projective objects in
the category ${\mathsf{O}}^{\mathsf{ctr}}({\mathfrak{g}},{\mathcal{C}})$. The
identification of these categories with categories of semimodules and
semicontramodules over certain semialgebras, which we will now briefly
discuss, will make the explicit constructions of such injective and projective
objects explained below in Section 3.5 applicable in this case.

As in Section 2.7, the semialgebras
${\boldsymbol{\mathcal{S}}}^{l}({\mathfrak{g}},{\mathcal{C}})$ and
${\boldsymbol{\mathcal{S}}}^{r}({\mathfrak{g}},{\mathcal{C}})$ are defined as
the tensor products

${\boldsymbol{\mathcal{S}}}^{l}=U({\mathfrak{g}})\otimes_{U({\mathfrak{h}})}{\mathcal{C}}\quad\text{and}\quad{\boldsymbol{\mathcal{S}}}^{r}={\mathcal{C}}\otimes_{U({\mathfrak{h}})}U({\mathfrak{g}}),$

where, as above, $U({\mathfrak{g}})$ and $U({\mathfrak{h}})$ denote the
universal enveloping algebras of the Lie algebras ${\mathfrak{g}}$ and
${\mathfrak{h}}$ considered as abstract Lie algebras without any topologies.
The left and right $U({\mathfrak{h}})$-module structures on ${\mathcal{C}}$
are obtained by deriving the left and right coactions of ${\mathcal{C}}$ in
itself using the pairing $\phi$. The right coaction of ${\mathcal{C}}$ in
${\boldsymbol{\mathcal{S}}}^{l}$ and the left coaction in
${\boldsymbol{\mathcal{S}}}^{r}$ are induced by the right and left coactions
of ${\mathcal{C}}$ in itself. The construction of the left coaction of
${\mathcal{C}}$ in ${\boldsymbol{\mathcal{S}}}^{l}$ and the right coaction in
${\boldsymbol{\mathcal{S}}}^{r}$ is rather delicate [52, Section D.2.3]. Once
the ${\mathcal{C}}$-${\mathcal{C}}$-bicomodule structures on
${\boldsymbol{\mathcal{S}}}^{l}$ and ${\boldsymbol{\mathcal{S}}}^{r}$ has been
defined, the constructions of the semimultiplication and semiunit maps are
similar to those in Section 2.7 (see [52, Section 10.2.1]), though one still
has to check that the maps
${\boldsymbol{\mathcal{S}}}^{l}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}^{l}\longrightarrow{\boldsymbol{\mathcal{S}}}^{l}$
and
${\boldsymbol{\mathcal{S}}}^{r}\mathbin{\text{\smaller$\square$}}_{\mathcal{C}}{\boldsymbol{\mathcal{S}}}^{r}\longrightarrow{\boldsymbol{\mathcal{S}}}^{r}$
so obtained are morphisms of ${\mathcal{C}}$-${\mathcal{C}}$-bicomodules (the
nontrivial part is to show that the former is a morphism of left
${\mathcal{C}}$-comodules and the latter a morphism of right ones).

Now the category ${\mathsf{O}}({\mathfrak{g}},{\mathcal{C}})$ of Harish-
Chandra modules over a Tate Harish-Chandra pair
$({\mathfrak{g}},{\mathcal{C}})$ is isomorphic to the category of left
semimodules over the semialgebra
${\boldsymbol{\mathcal{S}}}^{l}({\mathfrak{g}},{\mathcal{C}})$; the datum of a
left ${\boldsymbol{\mathcal{S}}}^{l}$-semimodule structure on a $k$-vector
space ${\boldsymbol{\mathcal{M}}}$ is equivalent to that of a Harish-Chandra
module structure on ${\boldsymbol{\mathcal{M}}}$. Similarly, the category
${\mathsf{O}}^{\mathsf{ctr}}({\mathfrak{g}},{\mathcal{C}})$ of Harish-Chandra
contramodules over $({\mathfrak{g}},{\mathcal{C}})$ is isomorphic to the
category of left semicontramodules over the semialgebra
${\boldsymbol{\mathcal{S}}}^{r}({\mathfrak{g}},{\mathcal{C}})$; the datum of a
left ${\boldsymbol{\mathcal{S}}}^{r}$-semicontramodule structure on a
$k$-vector space ${\boldsymbol{\mathfrak{P}}}$ is equivalent to that a Harish-
Chandra contramodule structure on ${\boldsymbol{\mathfrak{P}}}$ [52, Sections
10.2.2, D.2.5, and D.2.8].

As in Section 2.7, one would like to have also an explicit description of what
it means to have a left ${\boldsymbol{\mathcal{S}}}^{r}$-semimodule structure
on a $k$-vector space ${\boldsymbol{\mathcal{M}}}$. In the infinite-
dimensional situation, one cannot hope to have a Morita equivalence between
the semialgebras ${\boldsymbol{\mathcal{S}}}^{l}$ and
${\boldsymbol{\mathcal{S}}}^{r}$. Rather, the determinantal anomaly moves one
step higher in the cohomological degree when one passes to Tate vector spaces,
and what used to be a twist with the modular character in the finite-
dimensional case becomes _a shift of the central charge_ in the Tate Harish-
Chandra situation. Before formulating the precise assertion, let us give the
definition of a _central extension_
$\varkappa\colon({\mathfrak{g}}^{\prime},{\mathcal{C}})\longrightarrow({\mathfrak{g}},{\mathcal{C}})$
of a Tate Harish-Chandra pair $({\mathfrak{g}},{\mathcal{C}})$.

A _morphism of Tate Harish-Chandra pairs_
$({\mathfrak{g}}^{\prime},{\mathcal{C}}^{\prime})\longrightarrow({\mathfrak{g}},{\mathcal{C}})$
can be defined as a set of data consisting of a continuous morphism of Tate
Lie algebras ${\mathfrak{g}}^{\prime}\longrightarrow{\mathfrak{g}}$ and a
morphism of Hopf algebras ${\mathcal{C}}\longrightarrow{\mathcal{C}}^{\prime}$
such that the map ${\mathfrak{g}}^{\prime}\longrightarrow{\mathfrak{g}}$
commutes with the continuous ${\mathcal{C}}^{\prime}$-coactions and takes the
subalgebra ${\mathfrak{h}}^{\prime}\subset{\mathfrak{g}}^{\prime}$ into the
subalgebra ${\mathfrak{h}}\subset{\mathfrak{g}}$, while the maps
${\mathcal{C}}\longrightarrow{\mathcal{C}}^{\prime}$ and
${\mathfrak{h}}^{\prime}\longrightarrow{\mathfrak{h}}$ are compatible with the
pairings $\psi$ and $\psi^{\prime}$. A _central extension of Tate Harish-
Chandra pairs with the kernel $k$_ is a morphism
$({\mathfrak{g}}^{\prime},{\mathcal{C}}^{\prime})\longrightarrow({\mathfrak{g}},{\mathcal{C}})$
such that ${\mathcal{C}}^{\prime}={\mathcal{C}}$ is the same Hopf algebra and
${\mathcal{C}}\longrightarrow{\mathcal{C}}^{\prime}$ the identity map,
${\mathfrak{g}}^{\prime}\longrightarrow{\mathfrak{g}}$ is a quotient map of
topological vector spaces with a one-dimensional kernel
$k\subset{\mathfrak{g}}^{\prime}$ in which a fixed basis vector
$1_{{\mathfrak{g}}^{\prime}}\in k\subset{\mathfrak{g}}^{\prime}$ is chosen,
the map of Lie subalgebras
${\mathfrak{h}}^{\prime}\longrightarrow{\mathfrak{h}}$ is an isomorphism, the
kernel $k=k\cdot 1_{{\mathfrak{g}}^{\prime}}$ lies in the center of the Lie
algebra ${\mathfrak{g}}^{\prime}$, and the coaction of the Hopf algebra
${\mathcal{C}}$ in the subspace $k\subset{\mathfrak{g}}^{\prime}$ is trivial
[52, Section D.2.2]. As it is usual for extensions with a fixed kernel, the
set of all (isomorphism classes of) central extensions of a given Tate Harish-
Chandra pair $({\mathfrak{g}},{\mathcal{C}})$ with the kernel $k$ has a
natural structure of abelian group (and in fact, even of a $k$-vector space)
with respect to the Baer sum operation [52, Section D.3.1].

The Lie algebra $\operatorname{\mathfrak{gl}}(V)$ of continuous endomorphisms