# K-theory cohomology of associative algebra twisted bundles

A. Zuevsky  Institute of Mathematics
Czech Academy of Sciences
Zitna 25, Prague
Czech Republic<EMAIL_ADDRESS>
###### Abstract.

We introduce and study a $K$-theory of twisted bundles for associative
algebras $A(\mathfrak{g})$ of formal series with an infinite-Lie algebra
coefficients over arbitrary compact topological spaces. Fibers of such bundles
are given by elements of algebraic completion of the space of all formal
series in complex parameters, sections are provided by rational functions with
prescribed analytic properties. In this paper we introduce and study K-groups
$K(A(\mathfrak{g}),X)$ of twisted $A(\mathfrak{g})$-bundles as equivalence
classes $[\mathcal{E}]$ of $A(\mathfrak{g})$-bundle $\mathcal{E}$. We show
that for any twisted $A(\mathfrak{g})$-bundle $\mathcal{E}$ there exist
another bundle $\widetilde{\mathcal{E}}$ such that an element of
$K(A(\mathfrak{g}),X)$ for $\mathcal{E}$ can be represented in the form
$[\mathcal{E}]/[\widetilde{\mathcal{E}}]$. The group $K(A(\mathfrak{g}),X)$
homomorphism properties with respect to tensor product, and splitting
properties with respect to reductions of $X$ into base points. We determine
also cohomology of cells of K-groups for the factor $X/Y$ of two compact
spaces $X$ and $Y$.

###### Key words and phrases:

Associative algebras, fiber bundles, rational functions with prescribed
properties, K-groups

##  1\. Data availability statement

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##  2\. Introduction

The equivalence classes of bundles [1, 6, 12] associated to various algebraic
structures defined on topological spaces allow to use combinations of
algebraic and topological properties of non-commutative objects in terms of
abelian groups. It is natural to consider bundles using modules of associative
algebras. Such bundles are important for computations in cohomology theory on
smooth manifolds, [21, 20, 19]. In [23] we introduced and studied fiber
twisted bundles related to modules of associative algebras for infinite-
dimensional Lie algebras in the form of group of automorphisms torsors
originating from local geometry. Our original motivation for this work was to
understand continuous cohomology [3, 4, 7, 9, 10, 17, 19] of non-commutatieve
structrues over compact topological spaces. In particular [4], one hopes to
relate cohomology of infinite-dimensional Lie algebras-valued series
considered on complex manifolds with fiber bundles on auxiliary topological
spaces [17]. Let $\mathfrak{g}$ be an infinite-dimensional Lie algebra [14].
Starting from algebraic completion $G_{(z_{1},\ldots,z_{n})}$ of the space of
$\mathfrak{g}$-valued series in a few formal complex parameters
$(z_{1},\ldots,z_{n})$, we introduce the category
${\mathcal{O}}_{A(\mathfrak{g})}$ of associative algebra modules for the
associative algebra $A(\mathfrak{g})$ originating from
$G_{(z_{1},\ldots,z_{n})}$ by means of factorization with respect to two
natural multiplications [22]. Local parts of twisted bundles are constructed
as principal bundles of products of ${\rm Aut}(\mathfrak{g})$-modules and
spaces of all sets of local parameters of a $X$-covering. As in the untwisted
case [6], this result is crucial in defining $A(\mathfrak{g})$ K-groups and
studying the cohomology their properties. In [6] they explored the vertex
operator algebra approach to K-theory of compact topological spaces. Vertex
operator algebra $V$ bundles and associative algebra bundles related to $V$
gave rise to a series of exact sequences ot K-groups for a compact topological
space. An alternative approach to vertex operator algebra bundles was given in
[2].

In this paper we introduce and determine properties of K-groups
$K(A(\mathfrak{g}),X)$ defined for twisted $A(\mathfrak{g})$-bundles for the
associative algebras $A(\mathfrak{g})$ for ${\mathfrak{g}}$ on compact
topological space $X$. In order to formulate the K-theory, we use the
axiomatics of prescribed rational functions. We show, in particular, that all
elements of $K(A(\mathfrak{g}),X)$ for a twisted $A(\mathfrak{g})$-bundle
$\mathcal{E}$ can be represented in the form
$[\mathcal{E}]/[\widetilde{\mathcal{E}}]$ with another bundle
$\widetilde{\mathcal{E}}$. The group $K(A(\mathfrak{g}),X)$ exhibits natural
homomorphism properties with respect to tensor product of associative
algebras, and possesses splitting properties with respect to reductions of $X$
into a point. We study also cohomology of cells of K-groups for the factor
$X/Y$ of two compact spaces $X$ and $Y$. The cells as short exact sequences of
K-groups of $X/Y$, $X$ and $Y$. The cohomology of K-cells is determined
explicitly. K-groups of twisted associative algebra bundles possess non-
vanishing cohomology in contrast to K-groups for vertex operator algebras.
Studies of K-groups of bundles considered in this paper find their
applications in conformal field theory [2, 18], deformation theory [11, 15,
16], vertex algebras [8, 13], and algebraic topology [3, 7].

##  3\. Prescribed rational functions originating from matrix elements

In this Section the space of prescribed rational functions is defined as
functions with certain analytical and symmetry properties [13]. They depend on
an infinite number of non-commutative parameters. Let
$X_{(\alpha)}=\left\\{X_{\alpha},\alpha\in{\mathbb{Z}}_{>0}\right\\}$ be an
open covering of a compact topological space $X$ which gives a local
trivialization of the $A(\mathfrak{g})$ fiber bundle. Let $\mathfrak{g}$, be
an infinite-dimensional Lie algebra. Denote by $G=G_{(z_{1},\ldots,z_{n})}$ be
the graded (with respect to a grading operator $K_{G}$) algebraic completion
of the space of all formal series in each of complex formal parameters
$(z_{1},\ldots,z_{n})$ individually, with expansion coefficients as elements
$g\in G$, and satisfying certain properties described below. We denote by
$(x_{1},\ldots,x_{n})=(g_{1},z_{1};\ldots;g_{n},z_{n})$ for
$(g_{1},\ldots,g_{n})\in G^{\otimes n}$. It is assumed that on $G$ there
exists a non-degenerate bilinear pairing $(.,.)$.
$G^{\prime}=\coprod_{{\lambda}\in{\mathbb{C}}}G_{{\lambda}}^{*}$ denotes the
gaded dual for $G=\bigoplus_{\lambda\in{{\mathbb{C}}}}G_{(\lambda)}$ with
respect to $(.,.)$. For a fixed $\theta\in G^{*}$, and varying
$(x_{1},\ldots,x_{n})$ consider the matrix elements

$F(x_{1},\ldots,x_{n})=\left(\theta,f(x_{1},\ldots,x_{n})\right),$ (3.1)

where $F(x_{1},\ldots,x_{n})\in{\mathbb{C}}(z_{1},\ldots,z_{n})$ depends
implicitly on $(g_{1},\ldots,g_{n})$. In this paper we consider meromorphic
functions of defined on a compact topological space which are extendable to
rational functions $R(f(z_{1},\ldots,z_{n}))$ on larger domains of several
complex formal parameters $(z_{1},\ldots,z_{n})$. Denote by $F_{n}\mathbb{C}$
the configuration space of $n\geq 1$ ordered coordinates in $\mathbb{C}^{n}$,
$F_{n}\mathbb{C}=\\{(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}\;|\;z_{i}\neq
z_{j},i\neq j\\}$. In order to work with $G$-elements $(g_{1},\ldots,g_{n})$
objects on $X$, we consider converging rational functions
$f(x_{1},\ldots,x_{n})\in G$ of $(z_{1},\ldots,z_{n})\in F_{n}\mathbb{C}$. For
an arbitrary fixed $\theta\in G^{*}$, we call a map linear in
$(g_{1},\ldots,g_{n})$ and $(z_{1},\ldots,z_{n})$,
$F:(x_{1},\ldots,x_{n})\mapsto R((\theta,f(x_{1},\ldots,x_{n}))$, a rational
function in $(z_{1},\ldots,z_{n})$. The poses are only allowed at at
$z_{i}=z_{j}$, $i\neq j$. We define left action of the permutation group
$S_{n}$ on $F(z_{1},\ldots,z_{n})$ by
$\sigma(F)(x_{1},\ldots,x_{n})=F\left(g_{1},z_{\sigma(1)};\ldots;g_{n},z_{\sigma(n)}\right)$.

We assume [13] that $G_{({\lambda})}=\\{w\in
G|K_{0}w={\lambda}w\,\;{\lambda}={\rm wt}(w)\\}$, such that
$G_{({\lambda})}=0$ when the real part of $\alpha$ is sufficiently negative,
Moreover we require that $\dim G_{({\lambda})}<\infty$, i.e., it is finite,
and for fixed ${\lambda}$, $G_{(n+{\lambda})}=0$, for all small enough
integers $n$. An admissible $G$ is a ${\mathbb{C}}$-graded vector space with
$G_{(0)}\neq 0$. which satisfies the following conditions. Assume that $G$ is
equipped with a map $\omega_{g}:G[(z_{1},\ldots,z_{n})\to
G[[(z_{1},\ldots,z_{n}),(z_{1}^{-1},\ldots,z^{-1}_{n})]]$,
$g\mapsto\omega_{g}(z_{1},\ldots,z_{n})\equiv\sum_{l\in{\mathbb{C}}}g_{l}\;z^{l}$.
For each element $g\in G$, and $(z_{1},\ldots,z_{n})\in{\mathbb{C}}^{\otimes
n}$ let us associate a formal series
$\omega_{g}(x)=\omega_{g}(z_{1},\ldots,z_{n})=\sum\limits_{(s_{1},\ldots,s_{n})\in{\mathbb{C}}^{\otimes
n}}g_{(s_{1},\ldots,s_{n})}\;z^{s_{1}}_{1}\ldots z^{s_{n}}_{n}$. For $g\in G$,
$w\in$, $n\in{\mathbb{C}}$, $g_{n}w=0$, $n\gg 0$, $\omega_{\bf
1}(z_{1},\ldots,z_{n})={\rm Id}$, For $g\in G$,
$\omega_{g}(z_{1},\ldots,z_{n})w$ contains only finitely many negative power
terms, that is, $\omega_{g}(z_{1},\ldots,z_{n})w\in G$. The locality and
associativity properties are assumed for matrix elements (3.1), for $g_{1}$,
$g_{2}\in\mathfrak{g}$, $w\in G$, $\theta\in G^{\prime}$, the series
$\left(\theta,\omega_{g_{1}}(z_{1})\;\omega_{g_{2}}(z_{2})w\right)$,
$\left(\theta,\omega_{g_{2}}(z_{2})\;\omega_{g_{1}}(z_{1})w\right)$,
$\left(\theta,\omega_{\omega_{g_{1}}(z_{1}-z_{2})g_{2}}(z_{2})w\right)$, are
absolutely convergent in the regions $|z_{1}|>|z_{2}|>0$, $|z_{2}|>|z_{1}|>0$,
$|z_{2}|>|z_{1}-z_{2}|>0$, respectively, to a common rational function in
$z_{1}$ and $z_{2}$. The poles are only allowed at $z_{1}=0=z_{2}$,
$z_{1}=z_{2}$. If $g$ is homogeneous then $g_{m}G_{(n)}\subset G_{({\rm
wt}u-m-1+n)}$. For a subgroup ${\mathfrak{G}}\subset{\rm Aut}\;G$,
$\mathfrak{G}$ acts on $G$ as automorphisms if
$g\;\omega_{h}(z_{1},\ldots,z_{n})\;g^{-1}=\omega_{gh}(z_{1},\ldots,z_{n})$,
for all $g$, $h\in{\mathfrak{G}}$. For an admissible $G$ the operator $K_{G}$
satisfies the derivation property
$\omega_{K_{G}g}(z_{1},\ldots,z_{n})=\frac{d}{dz}\omega_{g}(z_{1},\ldots,z_{n})$.
An admissible ${\mathbb{C}}$-graded
$G=\bigoplus_{\lambda\in{{\mathbb{C}}}}G_{(\lambda)}$,
$G_{({\lambda})}=\\{w\in G|K_{0}w={\lambda}w\\}$ is called an ordinary. We
require that $\dim G_{({\lambda})}$ is finite and for fixed ${\lambda}$,
$G_{(s+{\lambda})}=0$, for all small enough integers $s$. Let us assume that
$G_{(0)}={\mathbb{C}}{\bf 1}$. $G$ is called rational if it is a direct sum of
irreducible admissible $G^{i}$. From [5, 22] we know that, all of finite
number (up to isomorphisms) irreducible admissible $G^{i}$ of a rational $G$,
are ordinary modules. For each ${\lambda}\in{\mathbb{C}}$ denote
$D({\lambda})=\\{{\lambda}+n|0\leq n\in{\mathbb{Z}}\\}$. The category
${\mathcal{O}}_{\mathfrak{g}}$ of ordinary $G$ is that for there exist
finitely many complex weights $({\lambda}_{1},...,{\lambda}_{p})$ such that
with $P(G)=\\{{\lambda}\in{\mathbb{C}}|G_{{\lambda}}\neq 0\\}$,
$P(G)\subset\bigcup_{i=1}^{p}D({\lambda}_{i})$. Any irreducible module is in
${\mathcal{O}}_{\mathfrak{g}}$. If $G$ is rational then
${\mathcal{O}}_{\mathfrak{g}}$ is exactly the category of ordinary modules.
For two $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ the functor
${\mathcal{O}}_{\mathfrak{g}}\times{\mathcal{O}}_{\mathfrak{g}^{\prime}}\to{\mathcal{O}}_{\mathfrak{g}\otimes\mathfrak{g}^{\prime}}$
such that $G^{1}\times G^{2}\to G^{1}\otimes G^{2}$. Next recall the notion of
a contragredient module [8]. We denote the natural bilinear pairing on
$G^{\prime}\times G$ by $(w^{\prime},w)$ for $w^{\prime}\in G^{\prime}$ and
$w\in G$. It is called invariant if
$(w^{\prime},\omega_{g}(z_{1},\ldots,z_{n})w)=(\omega_{g^{\prime}}w^{\prime},w)$,
where $g^{\prime}=e^{z^{3}\partial_{z}}(-z^{-2})^{K_{0}}z^{-1}g$, for
$\omega_{g^{\prime}}w^{\prime}_{i}\in G^{\prime}$, and $g\in G$. Then [8, 6]
$G^{\prime}$ with $\omega_{g}G)$ is also a module. It is irreducible if and
only if $G$ is irreducible. For any $G$, $G^{\prime}\oplus G$ has a natural
non-degenerate symmetric invariant bilinear pairing defined by
$(u+u^{\prime},w+w^{\prime})=(u,w^{\prime})+(w,u^{\prime})$ for any $u$, $w\in
G$ and $u^{\prime}$, $w^{\prime}\in G^{\prime}$. In particular, any $G$ can be
embedded into a module with a nondegenerate symmetric invariant bilinear
pairing.

Let $(z_{1},\ldots,z_{n})\in F_{n}{\mathbb{C}}$. Denote by $T_{G}$ the
translation operator [13]. Denote by $(T_{G})_{i}$ the operator acting on the
$i$-th entry. We then define the action of partial derivatives on an element
$F(x_{1},\ldots,x_{n})$

$\displaystyle\partial_{z_{i}}F(x_{1},\ldots,x_{n})$ $\displaystyle=$
$\displaystyle F((T_{G})_{i}\;(x_{1},\ldots,x_{n})),$
$\displaystyle\sum\limits_{i\geq 1}\partial_{z_{i}}F(x_{1},\ldots,x_{n})$
$\displaystyle=$ $\displaystyle T_{G}F(x_{1},\ldots,x_{n}),$ (3.2)

and call it $T_{G}$-derivative property. For $z\in{\mathbb{C}}$, let

$\displaystyle
e^{zT_{G}}F(x_{1},\ldots,x_{n})=F(g_{1},z_{1}+z;\ldots;g_{n},z_{n}+z).$ (3.3)

Let us denote by ${\rm Ins}_{i}(A)$ the operator of multiplication by
$A\in{\mathbb{C}}$ at the $i$-th position. Then we assume that both sides of
the expression $F\left((g_{1},\ldots,g_{n}),{\rm
Ins}_{i}(z_{1},\ldots,z_{n})\;(z_{1},\ldots,z_{n})\right)=F\left({\rm
Ins}_{i}(e^{zT_{G}})\;(x_{1},\ldots,x_{n})\right)$, are absolutely convergent
on the open disk $|z|<\min_{i\neq j}\\{|z_{i}-z_{j}|\\}$, and equal as power
series expansions in $z$. A rational function has $K_{G}$-property if for
$z\in{\mathbb{C}}^{\times}$ satisfies $(zz_{1},\ldots,zz_{n})\in
F_{n}{\mathbb{C}}$,

$\displaystyle
z^{K_{G}}F(x_{1},\ldots,x_{n})=F\left(z^{K_{G}}(g_{1},zz_{1};\ldots;g_{n},zz_{n})\right).$
(3.4)

Now we recall the definition of rational functions with prescribed analytical
behavior on a domain of $X$. We denote by $P_{k}:G\to G_{(k)}$,
$k\in{\mathbb{C}}$, the projection of $G$ on $G_{(k)}$. Following [13], we
formulate the following definition. For $i$, $j=1,\dots,(l+k)n$, $k\geq 0$,
$i\neq j$, $1\leq l^{\prime},l^{\prime\prime}\leq n$, let
$(l_{1},\ldots,l_{n})$ be a partition of $(l+k)n=\sum\limits_{i\geq 1}l_{i}$,
and $k_{i}=l_{1}+\cdots+l_{i-1}$. For $\zeta_{i}\in{\mathbb{C}}$, define
$h_{i}=F(\omega_{g_{k_{1}+l_{1}}}(z_{k_{1}+l_{1}}-\zeta_{1})\ldots\omega_{g_{k_{l}+l_{l}}}(z_{k_{l}+l_{l}}-\zeta_{l}))$,
for $i=1,\dots,l$. A rational function $F$ is a rational function with
prescribed analytical behavior if it satisfies properties (3)–(3.4). In
addition to that, it is assumed that the function
$\sum\limits_{(r_{1},\ldots,r_{l})\in{\mathbb{Z}}^{l}}F(P_{r_{1}}h_{1},\zeta_{1};\ldots;P_{r_{l}}h_{l},\zeta_{l})$,
is absolutely convergent to an analytically extension in
$(z_{1},\ldots,z_{n})_{l+k}$ in the domains
$|z_{k_{i}+p}-\zeta_{i}|+|z_{k_{j}+q}-\zeta_{j}|<|\zeta_{i}-\zeta_{j}|$, for
$i$, $j=1,\dots,k$, $i\neq j$, and for $p=1,\dots$, $l_{i}$, $q=1$, $\dots$,
$l_{j}$. The convergence and analytic extention do not depend on complex
parameters $(\zeta)_{l}$. On the diagonal of $(z_{1},\ldots,z_{n})_{l+k}$ the
order of poles is bounded from above by escribed ppositive numbers
$\beta(g_{l^{\prime},i},g_{l^{\prime\prime},j})$. For
$(g_{1},\ldots,g_{l+k})\in G^{\otimes(l+k)}$, $z_{i}\neq z_{j}$, $i\neq j$
$|z_{i}|>|z_{s}|>0$, for $i=1,\dots,k$, $s=k+1,\dots,l+k$ the sum
$\sum_{q\in{\mathbb{C}}}$
$F(\omega_{g_{1}}(x_{1})\ldots\omega_{g_{k}}(x_{k})P_{q}(\omega_{g_{1+k}}(z_{1+k})\ldots\omega_{g_{l+k}}(x_{l+k})))$,
is absolutely convergent and analytically extendable to a rational function in
variables $(z_{1},\ldots,z_{n})_{l+k}$. The order of pole that is allowed at
$z_{i}=z_{j}$ is bounded from above by the numbers
$\beta(g_{l^{\prime},i},g_{l^{\prime\prime},j})$.

For $m\geq 1$ and $1\leq p\leq m-1$, let $J_{m;p}$ be the set of elements of
the group $S_{m}$ which preserve the order of the first $p$ numbers and the
order of the last $m-p$ numbers, i.e., $J_{m,p}=\\{\sigma\in
S_{m}\;|\;\sigma(1)<\cdots<\sigma(p),\;\sigma(p+1)<\cdots<\sigma(m)\\}$.
Denote by $J_{m;p}^{-1}=\\{\sigma\;|\;\sigma\in J_{m;p}\\}$. In what follows,
we apply the condition

$\sum_{\sigma\in
J_{n;p}^{-1}}(-1)^{|\sigma|}\sigma(F(g_{\sigma(1)},z_{1};\ldots;g_{\sigma(n)},z_{n}))=0.$
(3.5)

on certain rational functions. The space
$\Theta\left(n,k,G_{(z_{1},\ldots,z_{n})},U\right)$ of $n$ formal complex
parameters matrix elements $F(x_{1},\ldots,x_{n})$ as the space of restricted
rational functions with prescribed analytical behavior on a
$F_{n}{\mathbb{C}}$-domain $U\subset X$, and satisfying $T_{G}$\- and
$K_{G}$-properties (3)–(3.4), (3.5).

##  4\. The associative algebra $A(\mathfrak{g})$

In this Section we remind [5, 22] the definition and properties of the
associative algebra $A(\mathfrak{g})$ for $\mathfrak{g}$. For any homogeneous
vectors $h$, $\widetilde{h}\in G$, one defines the bilinear extention to
$G\times G$ of the multiplications $h*_{\kappa}\widetilde{h}={\rm
Res}_{z}\left((1+z)^{{\rm
wt}(h)}\sum\limits_{l\in{\mathbb{C}}}h_{n}z^{l-\kappa}\right).\widetilde{h}$,
for $\kappa=1$, $2$. Here ${\rm Res}_{z}$ denotes the coefficient in front of
$z^{-1}$. For $h$, $\widetilde{h}\in G$, define
$A(\mathfrak{g})=G_{(z_{1},\ldots,z_{n})}/({\rm
span}(h*_{2}\widetilde{h}))^{\theta}$, $\theta=0$, $1$. For $\theta=0$ we get
back to $G_{(z_{1},\ldots,z_{n})}$ with associativity property $G\times G$ and
expressed via matrix elements. For $\theta=1$ we obtain an associative algebra
associated with ordinary associativity. In [22, 6] we find the following

######  Theorem 1.

The bilinear operation $*_{1}$ turns $A(\mathfrak{g})$ into an associative
algebra with the linear map
$\phi:g\mapsto\exp\left(-z^{2}\partial_{z}\right)(-1)^{-z\partial_{z}}g$,
inducing an anti-involution $\nu$ on $A(\mathfrak{g})$.

In what follows, let us denote by $W=W_{(z_{1},\ldots,z_{n})}\subset
G_{(z_{1},\ldots,z_{n})}$ an $A(\mathfrak{g})$-module. The space of lowest
weight vectors of $G$ is then defined as $L(W)=\\{g\in G,w\in W|g_{{\rm
wt}(h)+m}w=0,h\in W,m\geq 0\\}$. With
$W=\bigoplus_{{\lambda}\in{\mathbb{C}}}W_{({\lambda})}$, each homogeneous
subspace $L(W)_{({\lambda})}=L(W)\cap W_{({\lambda})}$ of the natural grading
$L(W)=\bigoplus_{{\lambda}\in{\mathbb{C}}}L(W)_{({\lambda})}$ is finite
dimensional [6, 22]. It is easy to see the following

######  Lemma 1.

Let $W$, $\widetilde{W}$ be two $A(\mathfrak{g})$-modules with an
$A(\mathfrak{g})$-module homomorphism $\varphi:W\to\widetilde{W}$. Then
$\varphi(L(W))\subset L(\widetilde{W})$. In particular, if $\varphi$ is an
isomorphism then $\varphi(L(W))=L(\widetilde{W})$.

An associative algebra $A(\mathfrak{g})$ is called semisimple if it is a
direct sum of full matrix algebras. Let $A(\mathfrak{g})$ and
$A(\widetilde{\mathfrak{g}})$ be two associative algebras with anti-
involutions $\nu_{A(\mathfrak{g})}$ and $\nu_{A(\widetilde{\mathfrak{g}})}$
respectively. Then one has

######  Lemma 2.

$A(\mathfrak{g})\otimes_{\mathbb{C}}A(\widetilde{\mathfrak{g}})$ is an
associative algebra with anti-involution
$\nu_{A(\mathfrak{g})}\otimes\nu_{A(\widetilde{\mathfrak{g}})}$.

We denote by $W^{\prime}$ the dual space to $W$ with respect to the form
$(.,.)$. The following lemma is obvious.

######  Lemma 3.

$W^{\prime}$ is an $A(\mathfrak{g})$-module such that
$(a\;m^{\prime},m)=(m^{\prime},\nu(a)\;m)$, for $a\in A(\mathfrak{g})$,
$m^{\prime}\in W^{\prime}$, $m\in W$, and $\nu$ is an anti-involution.

For $w_{i}\in W$, $i=1$, $2$, and $a\in A(\mathfrak{g})$, a form $(.,.)$
defined on $W$ is called invariant if
$(a\;w_{1},w_{2})=(w_{1},\nu(a)\;w_{2})$. The category
${\mathcal{O}}_{A(\mathfrak{g})}$ consists of $A(\mathfrak{g})$-modules $W$
with ${\lambda}_{i}\in{\mathbb{C}}$, $1\leq i\leq p$, such that
$W=\bigoplus_{i=1,\atop n\geq 0}^{p}W_{({\lambda}_{i})}$, is a direct sum of
finite dimensional $A(\mathfrak{g})$-modules, and ${\rm
Hom}_{A(\mathfrak{g})}\left(W_{({\lambda})},W_{(\mu)}\right)=0$, if
$\mu\neq{\lambda}$. From [6] we have

######  Theorem 2.

For homogeneous $g\in G$ extended linearly to all $G$ and $W_{(0)}\neq 0$,
$L(W)$ is a left $A(\mathfrak{g})$-module, and the linear map $g\mapsto
g_{({\rm wt}(g)-1)}|L(W)$: $W\rightarrow{\rm End}(L(W))$, induces a
homomorphism from $W$ to ${\rm End}(L(W))$. For all
${\lambda}\in{\mathbb{C}}$, $L(W)_{({\lambda})}$ is an finite-dimensional
$A(\mathfrak{g})$-module.

##  5\. Twisted $A(\mathfrak{g})$-bundles

In this Section we recall [23] the construction of associative algebra
$A(\mathfrak{g})$ twisted bundles. Here we show how to organize elements of
the space
$\Theta\left(n,k,W_{(z_{1},\ldots,z_{n}),{\lambda}},X_{\alpha}\right)$ of
prescribed rational functions into sections of a twisted
$A(\mathfrak{g})$-bundle on $X$. Let $\mathcal{H}$ be a subgroup of the group
${\rm Aut}_{(z_{1},\ldots,z_{n})}\;{\mathcal{O}}_{X}$ of $n$ independent
formal parameters $(z_{1},\ldots,z_{n})$ automorphisms on $X$. A non-empty set
$\mathcal{X}$ is called a group $\mathfrak{H}$-torsor [2] if it is endowed
with a simply transitive right action of $\mathfrak{H}$. This means that for
$\xi$, $\widetilde{\xi}\in\mathcal{X}$, there exists a unique
$h\in\mathfrak{H}$ such that $\xi\cdot h=\widetilde{\xi}$, where for $h$,
$\widetilde{h}\in\mathfrak{H}$ the right action is given by
$\xi\cdot(h\cdot\widetilde{h})=(\xi\cdot h)\cdot\widetilde{h}$. This
construction allows us to identify $\mathcal{X}$ with $\mathfrak{H}$ by
sending $\xi\cdot h$ to $h$.

Similar to [2], one sees that certain subspaces $W\subset
G_{(z_{1},\ldots,z_{n})}$ are $\mathcal{H}$-modules. For $\mathcal{H}$-torsor
$X_{\alpha}$ of its module $W$ and $X_{\alpha}$, let us associate the
$X_{\alpha}$-twist of $W$
$\mathcal{E}_{X_{\alpha}}=W\;{{}_{\bigtimes}\atop{}^{\mathcal{H}}}X_{\alpha}=W\times
X_{\alpha}/\left\\{(w,a\cdot\xi)\sim(aw,\xi)\right\\}$, for $\xi\in
X_{\alpha}$, $a\in\mathcal{H}$, and $w\in W$. The isomorphisms
$i_{(z_{1},\ldots,z_{n}),X_{\alpha}}:W\;\widetilde{\to}\;\mathcal{E}_{X_{\alpha}}$
of $W$ define a representation of $\mathcal{H}$ on $W$. Then
$\mathcal{E}_{X_{\alpha}}$ is canonically identified with the twist of $W$ by
the $\mathcal{H}$-torsor of $X_{\alpha}$. Elements of
$\Theta\left(n,k,W_{(z_{1},\ldots,z_{n})},X_{\alpha}\right)$ form sections
$F(x_{1},\ldots,x_{n})$. The principal bundle for the group $\mathcal{H}$ on
$X$ provides the construction of local parts of a twisted
$A(\mathfrak{g})$-bundle. Let ${\it Aut}_{X_{\alpha}}$ be the space of all
sets of local formal parameters on $X_{\alpha}$. We define the
$\mathcal{H}$-twist of $W$
$\mathcal{E}_{(z_{1},\ldots,z_{n})}=W\;{{}_{\bigtimes}\atop{}^{\mathcal{H}}}\;{\it
Aut}_{X_{\alpha}}$.

Let us assume that a ${\mathbb{C}}$-grading on $W$ is induced by a
$K_{0}$-graidng defined on $G$ Let ${\mathfrak{G}}\subset{\rm Aut}(G)$ be a
$W$-grading preserving subgroup. Denote by
${\mathcal{O}}_{\mathfrak{G},A(\mathfrak{g})}$ a subcategory of
${\mathcal{O}}_{A(\mathfrak{g})}$ consisting of $A(\mathfrak{g})$-modules $W$
such that $\mathfrak{G}$ acts on $W$ as automorphisms. By using ideas of [2],
let us define the local part $\mathcal{E}(W_{(z_{1},\ldots,z_{n}),(\lambda)})$
of a tweisted $A(\mathfrak{g})$-bundle via matrix elements
$F(x_{1},\ldots,x_{n})$ that belong to
$\Theta\left(n,k,W_{(z_{1},\ldots,z_{n}),(\lambda)}\right)$ for all $n$,
$k\geq 0$. on a finite part $\left\\{X_{\alpha},\alpha\in I_{0}\right\\}$ of a
covering $\left\\{X_{\alpha}\right\\}$ of $X$. By using the properties of
prescribed rational functions we form a principal $\mathcal{H}$-bundle, which
is a fiber bundle $\mathcal{E}(W_{(z_{1},\ldots,z_{n}),({\lambda})})$ with the
fiber space provided by elements $f(x_{1},\ldots,x_{n})\in W$, and defined by
trivializations
$i_{(z_{1},\ldots,z_{n})}:F(x_{1},\ldots,x_{n})=(\theta,f(x_{1},\ldots,x_{n}))\to
X_{\alpha}$, with a continuous free and transitive
$F(x_{1},\ldots,x_{n})$-preserving right action
$F(x_{1},\ldots,x_{n})\times\mathcal{H}\to F(x_{1},\ldots,x_{n})$. The
projection ${\it Aut}_{X_{\alpha}}\rightarrow X$ is a principal
$\mathcal{H}$-bundle similar to [2]. $\mathcal{H}$-torsor ${\it
Aut}_{X_{\alpha}}$ is the fiber of this bundle over $X_{\alpha}$. For a
finite-dimensional $\mathcal{H}$-module
$W_{i_{(z_{1},\ldots,z_{n})},\lambda}$, let
$\mathcal{E}(W_{(z_{1},\ldots,z_{n}),{\lambda}})=\bigoplus_{n,k\geq
0}W_{i_{(z_{1},\ldots,z_{n})},\lambda}\;{{}_{\bigtimes}\atop{}^{\mathcal{H}}}\;{\it
Aut}_{X_{\alpha}}$, be the fiber bundle associated to
$W_{i_{(z_{1},\ldots,z_{n})},\lambda}$, ${\it Aut}_{X_{\alpha}}$, with
sections provided by elements of $\Theta(n,k,W,X_{\alpha})$, for $n$, $k\geq
0$. On $X$ we can choose $\left\\{X_{\alpha}\right\\}$ such that the bundle
$\mathcal{E}(W_{(z_{1},\ldots,z_{n}),{\lambda}})$ over $X_{\alpha}$ is
$X_{\alpha}\times F(x_{1},\ldots,x_{n})$. The map
$\mathcal{E}(W_{(z_{1},\ldots,z_{n}),{\lambda}}):{\mathbb{C}}^{n}\rightarrow
X$ is the fiber bundle $\mathcal{E}(W_{(z_{1},\ldots,z_{n}),{\lambda}})$ with
fiber $f(x_{1},\ldots,x_{n})$, the total space ${\mathbb{C}}^{n}$ of
$\mathcal{E}(W_{(z_{1},\ldots,z_{n}),{\lambda}})$, and $X$ is its base space.
For every $X_{\alpha}$ of $X$ $i_{(z_{1},\ldots,z_{n})}^{-1}$ is homeomorphic
to $X_{a}\times{\mathbb{C}}^{n}$. For
$f((x_{1},\ldots,x_{n})):i_{(z_{1},\ldots,z_{n})}^{-1}\rightarrow
X_{\alpha}\times{\mathbb{C}}^{n}$, that $\mathcal{P}\circ
f((x_{1},\ldots,x_{n}))=i_{(z_{1},\ldots,z_{n})}\circ{i^{-1}_{(z_{1},\ldots,z_{n})}}\left(X_{\alpha}\right)$,
where $\mathcal{P}$ is the projection map on $X_{\alpha}$.

A twisted fiber bundle $\mathcal{E}$ over $X$ associated to
$A(\mathfrak{g})$-module with the fiber
$W\in{\mathcal{O}}_{\mathfrak{G},A(\mathfrak{g})}$ and
$\Theta\left(n,k,W,X\right)$, $n$, $k\geq 0$-valued sections is a direct sum
of vector bundles
$\mathcal{E}=\bigoplus_{{\lambda}\in{\mathbb{C}}}\mathcal{E}(W_{(z_{1},\ldots,z_{n}),\lambda})$,
such that all transition functions are $A(\mathfrak{g})$-module isomorphisms.
It includes a family of continuous isomorphisms
$H_{\alpha}=\left\\{H_{\alpha,{\lambda}},{\lambda}\in{\mathbb{C}}\right\\}$,
$H_{\alpha,\lambda}:\mathcal{E}(W_{(\lambda}))|_{X_{\alpha}}\to
W_{i_{(z_{1},\ldots,z_{n})},{\lambda}}\;{{}_{\bigtimes}\atop{}^{\mathfrak{G}}}\;{\it
Aut}_{X_{\alpha}}$, of fiber bundles such that for transition functions
$g_{\alpha\beta,{\lambda}}=H_{\alpha,\lambda}*_{2}H_{\beta,\lambda}^{-1}$, for
all $\lambda\in{\mathbb{C}}$. Then
$g_{\alpha\beta}(x)=\left(g_{\alpha\beta,{\lambda}}(\xi)\right):W\to W$, are
$A(\mathfrak{g})$-module isomorphisms for any $\xi\in X_{\alpha}\bigcap
X_{\beta}$ where the transition functions $g_{\alpha,\beta}(x)$ are
$G$-valued.

Now we recall properties of twisted $A(\mathfrak{g})$-bundles [23]. For
$A(\mathfrak{g})={\mathbb{C}}$, the twisted $A(\mathfrak{g})$-bundle is a
classical complex vector bundle over $X$. We have

######  Lemma 4.

For $\mathcal{E}_{1}$, $\mathcal{E}_{2}$ $A(\mathfrak{g}_{1})$\- and
$A(\mathfrak{g}_{2})$-bundles $\mathcal{E}_{1}\otimes\mathcal{E}_{2}$ is
$A(\mathfrak{g}_{1})\otimes_{{\mathbb{C}}}A(\mathfrak{g}_{2})$-bundle over
$X$.

For $\mathcal{E}$, let us introduce
$\mathcal{E}^{\prime}=\oplus_{{\lambda}\in{\mathbb{C}}}(\mathcal{E}(W_{(\lambda)}))^{*}$
which is, due to properties of the non-degenerate bilinear pairing $(.,.)$, is
also a $A(\mathfrak{g})$-bundle. For two bundles $\mathcal{E}$ and
$\mathcal{E}^{\prime}$ on $X$, a map
$\eta:\mathcal{E}\to\mathcal{E}^{\prime}$, is called a bundle morphism if
there exist a family of continuous morphisms of fiber bundles
$\eta_{\lambda}:\mathcal{E}(W_{(\lambda)})\to\mathcal{E}^{\prime}(W_{(\lambda)})$,
such that with $\eta=\left(\eta_{\lambda}\right)$, for all
${\lambda}\in{\mathbb{C}}$, and
$\eta_{\lambda}:\mathcal{E}\to\mathcal{E}^{\prime}$, is an
$A(\mathfrak{g})$-module morphism. From [6, 23] we find

######  Lemma 5.

$\mathcal{E}\bigoplus\mathcal{E}^{\prime}$ is a twisted
$A(\mathfrak{g})$-bundle, endowed with non-degenerate symmetric invariant
bilinear pairing
$\left(g_{\alpha\beta}^{*}(\xi)\theta,g_{\alpha\beta}(\xi)u\right)=\left(\theta,u\right)$,
independent of $g_{\alpha\beta}$ for all $\alpha$, $\beta\in I$, $\xi\in
X_{\alpha}\cap X_{\beta}$, $\xi\in G$, $\theta\in G^{\prime}$, and induced by
the natural bilinear pairing on $G\oplus G^{\prime}$.

A twisted $A(\mathfrak{g})$-bundle $\mathcal{E}$ is called trivial if there
exists an $A(\mathfrak{g})$-bundle isomorphism $\varphi:\mathcal{E}\to W\times
X$, here $W\times X$ is the $A(\mathfrak{g})$-bundle on $X$ with $W$ as
fibers. For any $A(\mathfrak{g})$-module $M\in{\mathcal{O}}_{A(\mathfrak{g})}$
we denote the trivial $A(\mathfrak{g})$-bundle on $X$ by ${\mathcal{M}}$. Let
us now understand how the notion of a trivial bundle is determined by the
definition of a twisted $A(\mathfrak{g})$-bundle. Let
$W\in{\mathcal{O}}_{A(\mathfrak{g})}$. A subgroup $\mathcal{H}\subset{\rm
Aut}_{(z_{1},\ldots,z_{n})}{\mathcal{O}}_{X}$ determines a trivial bundle
$W\times X$ if $\mathcal{H}$ satisfies the following properties. $W$ is a
$\mathcal{H}$-module. The $\mathcal{H}$ twist $\mathcal{E}(X_{\alpha})$
preserves $W$ under ${\rm
Aut}_{(z_{1},\ldots,z_{n})}{\mathcal{O}}_{X_{\alpha}}$-actions by isomorphisms
$i_{(z_{1},\ldots,z_{n})}$. The local part of the bundle $W\times X$ is given
by the principle $\mathcal{H}$-bundle with $\mathcal{H}$-actions preserving
sections $F(x_{1},\ldots,x_{n})\to X_{\alpha}$. The trivial bundle $W\times X$
is the direct sum of trivial bundles with transition functions $g$ given by
$W$-preserving triansition functions of $\left\\{X_{\alpha}\right\\}$,
$\alpha\in I$. From [23] we have

######  Proposition 1.

For any twisted $A(\mathfrak{g})$-bundle $\mathcal{E}$, there exists a twisted
$A(\mathfrak{g})$-bundle $\widetilde{\mathcal{E}}$ such that
$\mathcal{E}\bigoplus\widetilde{\mathcal{E}}$ is a trivial twisted
$A(\mathfrak{g})$-bundle with a $A(\mathfrak{g})$-module $W$-preserving action
of a subgroup of ${\rm Aut}_{(z_{1},\ldots,z_{n})}\;{\mathcal{O}}_{X}$.

Twisted $A(\mathfrak{g})$-bundles possess the following homotopy-stability
property:

######  Proposition 2.

For a homotopy $\tau_{t}:\widetilde{X}\to X$, $0\leq t\leq 1$, of a compact
Hausdorff space $\widetilde{X}$, and a twisted $A(\mathfrak{g})$-bundle
$\mathcal{E}$ over $X$,
$\tau^{*}_{0}(\mathcal{E})\simeq\tau^{*}_{1}(\mathcal{E})$.

We then have

######  Proposition 3.

For a rational $A(\mathfrak{g})$-module $W$ with a decomposition into
irreducible modules $W^{i}$, any twisted $A(\mathfrak{g})$-bundle over $X$
$\mathcal{E}\simeq\oplus_{i=1}^{p}\mathcal{V}(\mathcal{E})^{i}\otimes\mathcal{W}^{i}$
with trivial twisted bundles $\mathcal{W}^{i}$ associated to $W^{i}$, and
vector bundles $\mathcal{V}(\mathcal{E})^{i}$.

###### Proof.

Let $\mathcal{E}$ be a twisted $A(\mathfrak{g})$-bundle with fiber
$W=\oplus_{i=1}^{p}M_{i}\otimes W^{i}$, where $M_{i}$ is the space of
multiplicity of $W^{i}$ in $W$. Then each set of transition functions
$\left\\{g_{\alpha\beta}\right\\}$ defines a map
$h_{\alpha\beta}:X_{\alpha}\cap X_{\beta}\to\bigoplus_{i=1}^{p}{\rm
End}(M_{i})$. For each $i$ we define a vector bundle
$\mathcal{V}(\mathcal{E})^{i}$ over $X$ with fiber $M_{i}$ and transition
functions $\left\\{h_{\alpha\beta}|\alpha,\beta\in I\right\\}$. Then we have a
$A(\mathfrak{g})$-bundle $\mathcal{V}(\mathcal{E})^{i}\otimes\mathcal{W}^{i}$.
$\square$

##  6\. $K\left(A(\mathfrak{g}),X\right)$-group for twisted
$A(\mathfrak{g})$-bundles

In this Seciton, for an associative algebra $A(\mathfrak{g})$, we introduce
the K-group $K(A(\mathfrak{g}),X)$ of a twisted $A(\mathfrak{g})$-bundle on a
compact topological space $X$, and study corresponding properties. We follow
the set-ups of [1, 6] with certain necessary extensions. According to the
definition of a twisted $A(\mathfrak{g})$-bundle $\mathcal{E}$, the set of its
equivalence classes $[\mathcal{E}]$ is an abelian semigroup with addition
given by the direct sum. We denote by $K(A(\mathfrak{g}),X)$ the abelian group
generated by the set of equivalence classes $[\mathcal{E}]$ of a twisted
$A(\mathfrak{g})$-bundle $\mathcal{E}$. For $\mathfrak{g}={\mathbb{C}}$, the
group $K(A(\mathfrak{g}),X)$ becomes the ordinary group $K(X)$ as in [1]. Let
us define by $\Omega_{0}=\left(\mathfrak{g},W\subset
G_{(z_{1},\ldots,z_{n})},K_{G},(.,.)\right.$,
$\left.\theta,\beta(g^{\prime},g^{\prime\prime}),K_{0};\mathcal{H},i_{(z_{1},\ldots,z_{n})},X_{\alpha},\alpha\in
I\right)$ the extented moduli space for a twisted $A(\mathfrak{g})$-bundle
defined on a compact topological space $X$. Let us now describe the set of
parameters the isomorphism classes of twisted $A(\mathfrak{g})$-bundles depend
on. Recall that $\mathcal{H}\subset{\rm
Aut}_{(z_{1},\ldots,z_{n})}\;{\mathcal{O}}_{X}$ of independent formal
parameters $(z_{1},\ldots,z_{n})$ automorphisms on $X$. The construction
involves the category of subsets $W\subset G$ which is a $\mathcal{H}$-module.
$\mathcal{H}$-torsors and $X_{\alpha}$-twists of $W$ on $X_{\alpha}$ are
determined by $\mathcal{H}$, $Aut_{X_{\alpha}}$, and $W$. The elements of
$\Theta\left(n,k,W,X_{\alpha}\right)$ give rise to a collection of sections
$F(x_{1},\ldots,x_{n})$ as prescribed rational functions. The space
$\Theta\left(n,k,W_{(z_{1},\ldots,z_{n}),\lambda}\right)$ for all $n$, $k\geq
0$, of prescribed rational functions is fixed by assumptions of Lemma. The
module $W\subset{\mathcal{O}}(A(\mathfrak{g}))$ is endowed with a
${\mathbb{C}}$-grading generated by $K_{0}$. Note that the trivializations
$i_{(}z_{1},\ldots,z_{n}):F(x_{1},\ldots,x_{n})=(\theta,f(x_{1},\ldots,x_{n}))\to
X_{a}$ are chosen in such way that they preserve the space
$\Theta\left(n,k,W_{(z_{1},\ldots,z_{n}),\lambda}\right)$. The choice of
trivializations
$i_{(z_{1},\ldots,z_{n}),X_{\alpha}}:W\;\widetilde{\to}\;\mathcal{E}_{X_{\alpha}}$,
is coherent with the choice of $\left\\{X_{\alpha}\right\\}$. Suppose we take
another system of domains $\left\\{X^{\prime}_{\alpha^{\prime}}\right\\}$. One
shows that the construction of the bundle does not depend on the choice of
transition functions on the intersections $X_{\alpha}\cap
X^{\prime}_{\alpha^{\prime}}$. Thus, the equivalence classes do not depend on
the choice of a covering. Since, by construction, the transition functions
$g_{\alpha\beta}(x)=\left(g_{\alpha\beta,{\lambda}}(\xi)\right):W\to W$, are
$A(\mathfrak{g})$-module isomorphisms for any $\xi\in X_{\alpha}\bigcap
X_{\beta}$, then the construction of the bundle is invariant of the choice of
transition functions. The twisted $A(\mathfrak{g})$-bundle
$\widetilde{\mathcal{E}}$ in Proposition 1 is constructed as follows [23].
Define an $A(\mathfrak{g})$-bundle injective and bilinear form preserving
homomorphism $\psi:\mathcal{E}\to W^{\;\oplus\;s}\times X$ for some $s$,
sending $\mathcal{E}$ to a trivial $A(\mathfrak{g})$-bundle. Let us take
$\widetilde{\mathcal{E}}=\psi(\mathcal{E})^{\dagger}$ with respect to the
bilinear form $(.,.)$. As the dual to $\mathcal{E}$, $\widetilde{\mathcal{E}}$
is an $A(\mathfrak{g})$-bundle on $X$. From Proposition 1 we infer

######  Lemma 6.

For the fixed set of data $(\mathfrak{g}$, $W\subset G$, $K_{G}$-grading,
$(.,.)$, $\theta\in W^{\prime}_{(z_{1},\ldots,z_{n})}$,
$\beta(g^{\prime},g^{\prime\prime}))$ of $\Omega_{0}$, the elements of
$K(A(\mathfrak{g}),X)$ are of the form
$[\mathcal{E}]/[\widetilde{}\mathcal{E}]|_{\mathcal{H}.W}$ up to
$W$-preserving action $\mathcal{H}.W$ of a subgroup $\mathcal{H}\subset{\rm
Aut}_{(z_{1},\ldots,z_{n})}$ ${\mathcal{O}}_{\mathcal{H}.X}$, where
${\mathcal{H}.X}$ denotes the $\mathcal{H}$ preserving section and transition
functions.

The statement of following Lemma follows from Lemma 4 and the construction of
a twisted $A(\mathfrak{g})$-bundle.

######  Lemma 7.

For two infinite-dimensional algebras $\mathfrak{g}$ and
$\widetilde{\mathfrak{g}}$, the tensor product of the twisted
$A(\mathfrak{g})$\- and $A(\widetilde{\mathfrak{g}})$-bundles induces a
natural group homomorphism
$K(A(\mathfrak{g}),X)\otimes_{{\mathbb{Z}}}K(A(\widetilde{\mathfrak{g}}),X)\to
K(A(\mathfrak{g})\otimes A(\widetilde{\mathfrak{g}}),X)$.

According to the definition of a twisted $A(\mathfrak{g})$-bundle
$\mathcal{E}$, the set of isomorphis classes of $\mathcal{E}$ is determined in
particular by $\mathcal{H}$-invariant modules $W$ and $\mathcal{H}$-invariant
sections $F(z_{1},\ldots,z_{n})$. The axioms of prescribed rational functions
$F(x_{1},\ldots,x_{n})$ form a functional representation of $G$ with
additional analytic behavior properties. Lemma 7 shows that for each $s$
elements of the K-group is the tensor product of $A(\mathfrak{g})$ where each
element if represented by commutative elements of $F(z_{1},\ldots,z_{n})$.
Therefore, from Lemma 7 we obtain

######  Lemma 8.

With $A(\mathfrak{g})^{\otimes 0}\simeq{\mathbb{C}}$, the group
$K=\bigoplus\limits_{s\geq 0}$ $K\left(A(\mathfrak{g})^{\otimes s},X\right)$
form a commutative algebra over $K(A(\mathfrak{g}),X)$.

Next we state

######  Lemma 9.

Any element of $K(A(\mathfrak{g}),X)$ has the form
$[\mathcal{E}]/[{\mathcal{M}}]$, where $\mathcal{E}$ is a twisted
$A(\mathfrak{g})$-bundle and $M$ is a $A(\mathfrak{g})$-module. For two
equivalent classes $[\mathcal{E}]$, $[\mathcal{E}^{\prime}]\in
K(A(\mathfrak{g}),X)$, there exists a $A(\mathfrak{g})$-module $M$ such that
$\mathcal{E}\cong\mathcal{E}^{\prime}$ up to the trivial
$A(\mathfrak{g})$-bundle ${\mathcal{M}}$.

###### Proof.

As we have shown in Lemma 6, for every $\mathcal{E}$ one constructs a bundle
$\mathcal{E}^{\prime}$ such that every element of $K(A(\mathfrak{g}),X)$ is of
the form $[\mathcal{E}]/[\mathcal{E}^{\prime}]$ up to $A(\mathfrak{g})$-module
$M$-preserving action of a subgroup $\mathcal{H}\subset{\rm
Aut}_{(z_{1},\ldots,z_{n})}\;{\mathcal{O}}_{X}$. By Proposition 1, for any
twisted $A(\mathfrak{g})$-bundle $\mathcal{E}$, there exists a twisted
$A(\mathfrak{g})$-bundle $\widetilde{\mathcal{E}}$ and a
$A(\mathfrak{g})$-module $M$ such that
$\mathcal{E}\oplus\widetilde{\mathcal{E}}\cong{\mathcal{M}}$. Thus we have
$[\mathcal{E}^{\prime}]/[\mathcal{E}]=[\mathcal{E}^{\prime}\oplus\widetilde{\mathcal{E}}]/[\mathcal{E}\oplus\widetilde{\mathcal{E}}]=[\mathcal{E}^{\prime}\oplus\widetilde{E}]/[{\mathcal{M}}]$.
If $[\mathcal{E}]=[\mathcal{E}^{\prime}]$ then there exists a
$A(\mathfrak{g})$-bundle $\mathcal{E}^{\prime\prime}$ such that
$\mathcal{E}\oplus\mathcal{E}^{\prime\prime}\cong\widetilde{\mathcal{E}}\oplus\mathcal{E}^{\prime\prime}$.
Let $\mathcal{E}$ be a $A(\mathfrak{g})$-bundle such that
$\mathcal{E}^{\prime\prime}\oplus\mathcal{E}\cong{\mathcal{M}}$ for some
$A(\mathfrak{g})$-module $M$. Then we have
$E\oplus{\mathcal{M}}\cong\mathcal{E}^{\prime}\oplus{\mathcal{M}}$.  $\square$

Here we describe the group $K(A(\mathfrak{g}),(\xi_{1},\ldots,\xi_{n}))$ for
$(\xi_{1},\ldots,\xi_{n})\in X$. According to the construction of a twisted
$A(\mathfrak{g})$-bundle, one has a subgroup $\mathcal{H}_{0}\subset{\rm
Aut}_{(z_{\xi_{1}},\ldots,z_{\xi_{n}})}{\mathcal{O}}_{(\xi_{1},\ldots,\xi_{n})}$
over a set $(\xi_{1},\ldots,\xi_{n})\in X$, such that an
$A(\mathfrak{g})$-module $W$ is $\mathcal{H}_{0}$-module. The local part of
$\mathcal{E}$ is defined by $\mathcal{E}_{(\xi_{1},\ldots,\xi_{n})}$
$\mathcal{H}_{0}$-twists of $W_{(z_{\xi_{1}},\ldots,z_{\xi_{n}})}$ attached to
$(\xi_{1},\ldots,\xi_{n})$ via mappings
$i_{(z_{1},\ldots,z_{n})}:W_{(z_{\xi_{1}},\ldots,z_{\xi_{n}})}\to\mathcal{E}_{(\xi_{1},\ldots,\xi_{n})}$.
An invariant $\mathcal{H}_{0}$-action on $F(x_{1},\ldots,x_{n})$ is assumed.
Basically,
$\mathcal{E}_{(\xi_{1},\ldots,\xi_{n})}=W_{(z_{\xi_{1}},\ldots,z_{\xi_{n}})}{\times\atop
S_{n}}Aut_{(\xi_{1},\ldots,\xi_{n})}$. The equivalence classes
$[\mathcal{E}]|_{(\xi_{1},\ldots,\xi_{n})}$ of twisted
$A(\mathfrak{g})$-bundle $\mathcal{E}$ over a set of points
$(\xi_{1},\ldots,\xi_{n})\in X$. are given by equivalence classes of
${\mathcal{H}}_{0}$-invariant ${\rm
Aut}_{(z_{\xi_{1}},\ldots,z_{\xi_{n}})}{\mathcal{O}}_{(\xi_{1},\ldots,\xi_{n})}$
modules $W_{(z_{\xi_{1}},\ldots,z_{\xi_{n}})}$. If
$A(\mathfrak{g})=\bigoplus_{i=1}^{p}A(\mathfrak{g})$ is semisimple up to
isomorphisms, then $K(A(\mathfrak{g}),(\xi_{1},\ldots,\xi_{n}))$ is isomorphic
to the group ${\mathbb{Z}}^{p}\times\ldots\times{\mathbb{Z}}^{p}$ with
generators $[{\mathcal{M}}^{1}],\ldots,[{\mathcal{M}}^{p}]$ for inequivalent
$A(\mathfrak{g})$-modules $M^{1},\ldots,M^{p}$. Generalizing results of [6],
we obtain.

######  Lemma 10.

For a semisimple associative algebra $A(\mathfrak{g})$
$K(A(\mathfrak{g}),X)=K(X)\otimes_{\mathbb{Z}}K(A(\mathfrak{g}),(\xi_{1},\ldots,\xi_{n}))$.

###### Proof.

For rational $G$ correspoding twisted $A(\mathfrak{g})$-bundle $\mathcal{E}$
is determined by certain vector bundles. Assume that its fiber
$M=\oplus_{i=1}^{p}E_{i}\otimes M^{i}$ where $\\{M^{1},\ldots,M^{p}\\}$
contains all inequivalent irreducible $A(\mathfrak{g})$-modules, and $E_{i}$
is the space of multiplicity of $M^{i}$ in $M$. Similar to Proposition 3, we
know that if $A(\mathfrak{g})$ is semisimple then there exist vector bundles
$\mathcal{V}(\mathcal{E})^{i}$ for $i=1,\ldots,p$ such that
$\mathcal{E}\simeq\bigoplus_{i=1}^{p}\mathcal{V}\left(\mathcal{E}\right)^{i}\otimes{\mathcal{M}}^{i}$,
where trivial bundles ${\mathcal{M}}^{i}$ correspond to inequivalent set of
$A(\mathfrak{g})$-modules $M^{i}$. According our description of a twisted
$A(\mathfrak{g})$-bundle over a set $(\xi_{1},\ldots,\xi_{n})\in X$, one has a
map to equivalence classes which gives
$K(\bigoplus_{i=1}^{p}A(\mathfrak{g})^{i},X)=\bigoplus_{i=1}^{p}K(A(\mathfrak{g})^{i},X)=K(\bigoplus_{i=1}^{p}\mathcal{V}\left(\mathcal{E}\right)^{i},X)\otimes_{\mathbb{Z}}K({\mathcal{M}}^{i})=K(X)\otimes_{\mathbb{Z}}K(A(\mathfrak{g}),(\xi_{1},\ldots,\xi_{n}))$.
$\square$

##  7\. Cohomology of $K$-cells

By using results of [6], we first define K-groups for twisted associative
algebra bundles on factor spaces of compact topological spaces. Let
${\mathcal{C}}_{c}$ denote the category of compact spaces, ${\mathcal{C}}_{0}$
the category of compact spaces with distinguished basepoints. We define a
functor ${\mathcal{C}}_{0}\times{\mathcal{C}}_{0}\to{\mathcal{C}}_{0}$, by
associating two compact topological spaces $X$, $Y$ to a compact space $X/Y$
with base points $(\xi_{1},\ldots,\xi_{n})=Y/Y$. In the case
$Y\neq\left\\{\emptyset\right\\}$, $X/Y$ is the disjoint union of $X$ with
points $(\xi_{1},\ldots,\xi_{n})$, and for $X\in{\mathcal{C}}_{c}$, we denote
$X_{0}=X/\left\\{\emptyset\right\\}$. If $X$ is in ${\mathcal{C}}_{0}$, we
define a functor ${\rm Ker}\;i_{0}^{*}(X)$ to be the kernel of the map
$i_{0}^{*}:K(A(\mathfrak{g}),X)\to
K(A(\mathfrak{g}),(\xi_{1},\ldots,\xi_{n}))$ where
$i_{0}:(\xi_{1},\ldots,\xi_{n})\to X$ is the inclusion of the basepoints. If
$c:X\to(\xi_{1},\ldots,\xi_{n})$ is the collapsing map then $c^{*}$ induces a
splitting $K(A(\mathfrak{g}),X)={\rm Ker}\;i_{0}^{*}(X)\oplus
K(A(\mathfrak{g}),(\xi_{1},\ldots,\xi_{n}))$. It is clear that
$K(A(\mathfrak{g}),X)\simeq{\rm Ker}\;i^{*}(X_{0})$. Now we define
$K(A(\mathfrak{g}),X/Y)={\rm Ker}\;i_{0}^{*}(X/Y)$. In particular,
$K(A(\mathfrak{g}),X)\simeq K(X_{0})$.

Now let us introduce [6] the generalized smash product operator in
${\mathcal{C}}_{0}$, for $X$, $Y\in{\mathcal{C}}_{0}$. We put $X\wedge
Y=X\times Y/X\vee Y$ for $X\vee
Y=X\times\\{(\xi_{1},\ldots,\xi_{n})\\}\bigcup\\{(\xi^{\prime}_{1},\ldots,\xi^{\prime}_{n})\\}\times
Y$, being base-points of $X$, $Y$ respectively. For any triple of spaces $X$,
$Y$, $Z\in{\mathcal{C}}_{0}$, one has a natural homeomorphism $X\wedge(Y\wedge
Z)\simeq(X\wedge Y)\wedge Z$. Let $I(t)$, $0\leq t\leq 1$ we denote the
boundary $\partial I(t)=\\{0,1\\}$. Then $I(t)/\partial I(t)\simeq
S^{1}\in{\mathcal{C}}_{0}$. For $X\in{\mathcal{C}}_{0}$ we define the operator
$\mathcal{S}$ of the reduced suspension on $X$ as ${\mathcal{S}}X=S^{1}\wedge
X\in{\mathcal{C}}_{0}$. Then one has the $s$-th power of iterated suspensions
${\mathcal{S}}^{s}X\cong S^{s}\wedge X$. For $X\in{\mathcal{C}}_{0}$ and
$s\geq 0$ we consider ${\rm Ker}\;i_{0}^{*}({\mathcal{S}}^{s}X)$. For $X$,
$Y\in{\mathcal{C}}_{c}$ let us define $K(A(\mathfrak{g}),X/Y)^{-s}$ $=$
$\left({\rm Ker}\;i_{0}^{*}\right)^{-s}$ $(X/Y)={\rm
Ker}\;i_{0}^{*}({\mathcal{S}}^{s}(X/Y))$, $K(A(\mathfrak{g}),X)^{-s}$ $=$
$K(A(\mathfrak{g}),X_{0})^{-s}$ = ${\rm
Ker}\;i_{0}^{*}({\mathcal{S}}^{s}(X_{0}))$. We next define the cone functor
$C:{\mathcal{C}}_{c}\to{\mathcal{C}}_{0}$ on $X$ by $CX=I(t)\times
X/\\{0\\}\times X$, and identify $X$ with the $\\{1\\}\times X\subset CX$. One
calls $CX/X=I(t)\times X/\partial I(t)\times X$ the unreduced suspension of
$X$. In what follows, we assume that $X$ is a finite CW-complex and $Y\subset
X$ is a CW sub-complex. Next, we have a modification of a Lemma from [6] in
our context of twisted $A(\mathfrak{g})$-bundles:

######  Lemma 11.

For $\xi\in{\rm Ker}\;i_{0}^{*}({\mathcal{S}}X)$, $T(t)=1-t$, $0\leq t\leq 1$
such that $(T\wedge 1):{\mathcal{S}}X\to{\mathcal{S}}X$ and the identity on
$X\in{\mathcal{C}}_{0}$. Then $(T\wedge 1)^{*}\xi=-\xi$ for $\xi$.

###### Proof.

By construction of a twisted $A(\mathfrak{g})$-bundle and according to Lemma
9, for any $\xi\in{\rm Ker}\;i_{0}^{*}({\mathcal{S}}X)$, there exist a
$A(\mathfrak{g})$-bundle $\mathcal{E}$ and a $A(\mathfrak{g})$-module $M$ such
that $\xi=[\mathcal{E}]/[{\mathcal{M}}]|_{\mathcal{H}.M}$, where
${\mathcal{M}}$ is a trivial bundle associated to $M$. For two cones $C_{1}X$
and $C_{2}X$, on the contructible
$C_{1}X\cup{C_{2}}X=C_{1}X_{C_{2}}X={\mathcal{S}}X$, the restrictions
$\mathcal{E}|_{CX}$, $\mathcal{E}|_{C_{2}X}$ of $\mathcal{E}$ are trivial.
Then we can define maps $\rho_{i}:\mathcal{E}|_{C_{i}X}\to M\times C_{i}X$,
$1\leq i\leq 2$, such that $\rho=\rho_{2}\circ\rho_{1}^{-1}=\rho^{s}:X\to{\rm
Aut}_{G_{(z_{1},\ldots,z_{n})}}(M)$, $s\in{\mathbb{C}}$. According to
definition of the smash product above, $(T\wedge 1)^{*}$ acts on
$[\mathcal{E}]$ $x\to\rho(x)^{-1}=(\rho^{s})^{-1}$ resulting in
$\mathcal{E}^{\prime}\in K(A(\mathfrak{g}),{\mathcal{S}}X)$. Since $\rho_{i}$
are isomorphisms of $M$, from Lemma 5 we obtain that classes of isomorphisms
of $\mathcal{E}$ and $\mathcal{E}^{\prime}$ are the direct sum of classes of
${\mathcal{M}}$ and ${\mathcal{M}}^{\prime}$ in
$K(A(\mathfrak{g}),{\mathcal{S}}X)$.  $\square$

Now we establish the cohomological properties of $K(A(\mathfrak{g}),X/Y)$.

######  Lemma 12.

For inclusions $i:Y\to X$, $j:X_{0}\to X/Y$, the cohomology of the cell
${\mathcal{K}}_{X/Y}=K(A(\mathfrak{g}),X/Y)\stackrel{{\scriptstyle
j^{*}}}{{\to}}K(A(\mathfrak{g}),X)\stackrel{{\scriptstyle
i^{*}}}{{\to}}K(A(\mathfrak{g}),Y)$ is given by
$H_{K(A(\mathfrak{g}),X/Y)}={\rm Ker}\;i^{*}/{\rm Im}\;j^{*}={\rm
Aut}_{(z_{1},\ldots,z_{n})}{\mathcal{O}}_{X/Y}|_{\mathcal{H}.W}$.

###### Proof.

Let $Y_{0}=Y/\left\\{\emptyset\right\\}$. The composition $i^{*}\circ j^{*}$
is induced by the composition $j\circ i:Y_{0}\to X/Y$, and $i^{*}\circ
j^{*}=0$. Suppose now that $\xi\in{\rm Ker}\;i^{*}$. According to Lemma 9,
$\xi$ is represented in the form
$([\mathcal{E}]/[{\mathcal{M}}])|_{\mathcal{H}.W_{(z_{1},\ldots,z_{n})}}$
where $\mathcal{E}$ is a twisted $A(\mathfrak{g})$-bundle over $X$ and $M$ is
a $A(\mathfrak{g})$-module. Since $i^{*}\xi=0$, it follows that
$[\mathcal{E}]|_{Y}=[{\mathcal{M}}]$ in $K(A(\mathfrak{g}),Y)$. According to
Lemma 9, this implies that there exists a $A(\mathfrak{g})$-module
$M^{\prime}|_{\mathcal{H}.W}$ such that
$(\mathcal{E}\oplus{\mathcal{M}}^{\prime})|_{\mathcal{H}.W}=({\mathcal{M}}\oplus{\mathcal{M}}^{\prime})|_{\mathcal{H}.W}$,
and the bundle ${\mathcal{M}}^{\prime}$ is trivial. Now as $Y$ is a CW sub-
complex of $X$, there exists an open neighborhood $\widetilde{Y}$ of $Y$ in
$X$ such that $Y$ satisfies the following condition with respect to
$\widetilde{Y}$. For $0\leq t\leq 1$, one can find a map
$f(t):\widetilde{Y}\to\widetilde{Y}$ such that $f(1)={\rm
Id}_{\widetilde{Y}}$, $f(0)|_{Y}={\rm Id}_{Y}$, and $f(0)(\widetilde{Y})=Y$.
By Lemma 2 the triviality of the blow-up
$(\mathcal{E}\oplus{\mathcal{M}}^{\prime})|_{\widetilde{Y}}$ of
$(\mathcal{E}\oplus{\mathcal{M}}^{\prime})|_{Y}$ is homotopically preserved,
on $\widetilde{Y}$. This defines a bundle
$\mathcal{E}\oplus{\mathcal{M}}^{\prime}/\alpha$ on $X/Y$, and an element
$\tau=[\mathcal{E}\oplus{\mathcal{M}}^{\prime}/\gamma]/[{\mathcal{M}}\oplus{\mathcal{M}}^{\prime}]\in{\rm
Ker}\;i_{0}^{*}(X/Y)=K(A(\mathfrak{g}),X/Y)$, where
$I^{*}:K(A(\mathfrak{g}),X)\to K(A(\mathfrak{g}),\xi)$. Since
${\mathcal{M}}^{\prime}$ is trivial,
$j^{*}(b\tau)=[\mathcal{E}\oplus{\mathcal{M}}^{\prime}/\gamma]/[{\mathcal{M}}\oplus{\mathcal{M}}^{\prime}]|_{\mathcal{H}.W}=[\mathcal{E}]/[{\mathcal{M}}]_{\mathcal{H}.W}=b\xi$,
where $b\in{\rm Aut}_{(z_{1},\ldots,z_{n})}\;{\mathcal{O}}_{X/Y}$. Thus ${\rm
Ker}\;i^{*}=\left({\rm
Aut}_{(z_{1},\ldots,z_{n})}\;{\mathcal{O}}_{X/Y}\right)\;{\rm Im}\;j^{*}$.
$\square$

######  Lemma 13.

For $X/Y\in{\mathcal{C}}_{c}$ and $Y\in{\mathcal{C}}_{0}$, the cohomology of
the sequence $K(A(\mathfrak{g}),X/Y)\to{\rm Ker}\;i_{0}^{*}(X)\to{\rm
Ker}\;i_{0}^{*}(Y)$, is given by ${\rm
Aut}_{(z_{\xi_{1}},\ldots,z_{\xi_{n}})}\;{\mathcal{O}}_{(\xi_{1},\ldots,\xi_{n})}|_{\mathcal{H}.W}$
up to the twist $\mathcal{E}_{\mathcal{H}.W}$.

###### Proof.

The statement follows from Lemma 12 and the decomposition isomorphisms
$K(A(\mathfrak{g}),X)\simeq{\rm Ker}\;i_{0}^{*}(X)\oplus
K(A(\mathfrak{g}),(\xi_{1},\ldots,\xi_{n}))$, $K(A(\mathfrak{g}),Y)\simeq{\rm
Ker}\;i_{0}^{*}(Y)\oplus
K(A(\mathfrak{g}),(\xi^{\prime}_{1},\ldots,\xi^{\prime}_{n}))$. Thus, we
obtain the statement of the lemma with the twist inserted.  $\square$

For $s\geq 0$ we next define the $(-s)$-th power of the cell of K-groups

${\mathcal{K}}_{X/Y}^{-s}=K(A(\mathfrak{g}),X/Y)^{-s}\stackrel{{\scriptstyle
j^{*}}}{{\to}}K(A(\mathfrak{g}),X)^{-s}\stackrel{{\scriptstyle
i^{*}}}{{\to}}K(A(\mathfrak{g}),Y)^{-s}.$

For a map $\nu_{1}:Z\to Z/X$, and the isomorphism
$\vartheta:K(A(\mathfrak{g}),Z/X)\to K(A(\mathfrak{g}),Y)^{-1}$, let us define
the operator $\delta^{(s)}:K(A(\mathfrak{g}),X/Y)^{-s}\to
K(A(\mathfrak{g}),X/Y)^{-s-1}$, as the operator
$\delta=\nu_{1}^{*}\circ\vartheta^{-1}$ acting on $K(A(\mathfrak{g}),Y)^{-s}$.
The main proposition of this paper is

######  Proposition 4.

The $s$-th cell cohomology $H^{s}_{{\mathcal{K}}_{X/Y}}={\rm
Ker}\;\delta^{(-s)}/{\rm Im}\;\delta^{-(s+1)}$, of the left-infinite sequence

$\cdots\stackrel{{\scriptstyle\delta^{(-s-2)}}}{{\longrightarrow}}{\mathcal{K}}_{X/Y}^{-s-1}\stackrel{{\scriptstyle\delta^{-(s+1)}}}{{\longrightarrow}}{\mathcal{K}}_{X/Y}^{-s}\stackrel{{\scriptstyle\delta^{(-s)}}}{{\longrightarrow}}\ldots\stackrel{{\scriptstyle\delta^{(-1)}}}{{\longrightarrow}}{\mathcal{K}}_{X/Y}^{0}\stackrel{{\scriptstyle\delta^{(0)}}}{{\rightarrow}}0,$

for $s\geq 0$, is given by ${\rm
Aut}_{(z_{1},\ldots,z_{n})}{\mathcal{O}}_{{\mathcal{S}}^{s}(X/Y)}|_{\mathcal{H}.W}$.

###### Proof.

The proof is a modification of the proof of Theorem (5.3) of [6] for our case.
Let us set ${\mathcal{K}}^{0}_{0}={\rm
Ker}\;i_{0}^{*}(X/Y)\stackrel{{\scriptstyle j^{*}}}{{\to}}{\rm
Ker}\;i_{0}^{*}(X)\stackrel{{\scriptstyle i^{*}}}{{\to}}{\rm
Ker}\;i_{0}^{*}(Y)$. One can see that the cohomology of the main sequence is
equivalent to the cohomology of the sequence

$\ldots\stackrel{{\scriptstyle\delta^{(-2)}}}{{\to}}{\mathcal{K}}_{X/Y}^{-1}\stackrel{{\scriptstyle\delta^{(-1)}}}{{\to}}{\mathcal{K}}^{0}_{0}\stackrel{{\scriptstyle\delta^{(0)}}}{{\rightarrow}}0,$
(7.1)

defined by $H^{0}_{{\mathcal{K}}_{X/Y}}={\rm Ker}\;\delta^{(-1)}/{\rm
Im}\;\delta^{(0)}$ is ${\rm Aut}_{(z_{1},\ldots,z_{n})}$
${\mathcal{O}}_{X/Y}|_{\mathcal{H}.W}$. By replacing $X/Y$ by
${\mathcal{S}}^{s}(X/Y)$ for $s\geq 1$, we obtain cohomology of an infinite
sequence continuing (7.1). Then by replacing $X/Y$ by $X_{0}/Y_{0}$ where
$X/Y$ is any pair in ${\mathcal{C}}_{c}\times{\mathcal{C}}_{c}$, we get the
cohomology of the infinite sequence in our Proposition. Recall that Lemma 13
gives the cohomology of mappings inside the cell $K_{0}$ of (7.1). To
determine cohomology of other parts of the sequence will apply Lemma 13 to
$Z=X_{C}Y=X\bigcup CY$, $Z/X$ and $Z_{C}X/Z=(Z\bigcup CX)/Z$. Let
$\nu_{1}:Z\to Z/X$ and $\nu_{2}:X\to Z$ bet two inclusions. By considering
$Z/X$ we get the cohomology $H_{K_{0},Z}={\rm Ker}\;\nu_{2}^{*}/{\rm
Im}\;\nu_{1}^{*}$ of the sequence
$K(A(\mathfrak{g}),Z/X)\stackrel{{\scriptstyle\nu_{1}^{*}}}{{\to}}{\rm
Ker}\;i_{0}^{*}(A(\mathfrak{g}),Z)\stackrel{{\scriptstyle\nu_{2}^{*}}}{{\to}}{\rm
Ker}\;i_{0}^{*}(X)$, given by ${\rm
Aut}_{(z_{\xi_{1}},\ldots,z_{\xi_{n}})}\;{\mathcal{O}}_{(\epsilon_{1},\ldots,\epsilon_{n})}$
for $(\epsilon_{1},\ldots,\epsilon_{n})$ beeing base points for $Z/X$. Let
$\widetilde{Y}$ be the neighborhood of $Y$ in $X$ as in Lemma 12. Note that
$CY$ is contractible. Then by Lemma 2 any twisted $A(\mathfrak{g})$-bundle
$\mathcal{E}$ on $Z$ is trivial on $\widetilde{Y}_{C}Y$. Therefore $p^{*}:{\rm
Ker}\;i_{0}^{*}(X/Y)\to{\rm Ker}\;i_{0}^{*}(Z)$ is an isomorphism where
$p:Z\to Z/CY=X/Y$ for the collapsing map. The have the composition
$j^{*}=\nu_{2}^{*}\circ p^{*}$ of maps. We then obtain the sequence
$K(A(\mathfrak{g}),Y)^{-1}\stackrel{{\scriptstyle\delta^{(-1)}}}{{\to}}K(A(\mathfrak{g}),X/Y)\stackrel{{\scriptstyle
j^{*}}}{{\to}}{\rm Ker}\;i_{0}^{*}(X)$ given by ${\rm
Ker}\;i^{*}_{0}({\mathcal{S}}Y)^{-1}\stackrel{{\scriptstyle\delta^{(-1)}}}{{\to}}{\rm
Ker}\;i^{*}_{0}(X/Y)\stackrel{{\scriptstyle j^{*}}}{{\to}}{\rm
Ker}\;i_{0}^{*}(X)$, which cohomology is $H_{K_{0},X/Y}={\rm Ker}\;j^{*}/{\rm
Im}\;\delta^{(-1)}={\rm
Aut}_{(z_{\xi^{\prime}_{1}},\ldots,z_{\xi^{\prime}_{n}})}\;{\mathcal{O}}_{(\epsilon^{\prime}_{1},\ldots,\epsilon^{\prime}_{n})}$.
We apply Lemma 13 to approximations $(X_{C_{1}}Y)$ and
$(X_{C_{1}}Y)_{C_{2}}X/(X_{C_{1}}Y)$ where we have denoted the cones $C_{i}$,
$i=1$, $2$. Thus we obtain the sequence

$\displaystyle
K(A(\mathfrak{g}),(X_{C_{1}}Y)_{C_{2}}X/(X_{C_{1}}Y))\stackrel{{\scriptstyle\nu^{*}_{3}}}{{\to}}{\rm
Ker}\;i_{0}^{*}((X_{C_{1}}Y)_{C_{2}}X)\stackrel{{\scriptstyle\nu^{*}_{4}}}{{\to}}{\rm
Ker}\;i_{0}^{*}(X_{C_{1}}Y).$ (7.2)

One shows that this sequence is isomorphic to the sequence in
${\mathcal{K}}^{-1}_{X/Y}$ of (7.1). According to the definition of $\delta$,
it is enough to will show the equivalence of the following two maps

$\displaystyle
K(A(\mathfrak{g}),(X_{C_{1}}Y)_{C_{2}}X/X_{C_{1}}Y)\stackrel{{\scriptstyle\nu^{*}_{3}}}{{\to}}{\rm
Ker}\;i_{0}^{*}((X_{C_{1}}Y)_{C_{2}}X)\stackrel{{\scriptstyle\nu^{*}_{5}}}{{\to}}{\rm
Ker}\;i_{0}^{*}(C_{2}X/X)$ (7.3)
$\displaystyle=K(A(\mathfrak{g}),X)^{-1}\stackrel{{\scriptstyle
i^{*}}}{{\rightarrow}}K(A(\mathfrak{g}),Y)^{-1}={\rm
Ker}\;i_{0}^{*}(C_{1}Y/Y).$

Using Lemma 11, according to definition of $C_{i}$, $i=1$, $2$ we obtain the
sequence ${\rm Ker}\;i_{0}^{*}(\mathcal{S}Y)\to K(A(\mathfrak{g}),C_{1}Y/Y)\to
K(A(\mathfrak{g}),(C_{1}Y)_{C_{2}}Y)$, which is equivalent to
$K(A(\mathfrak{g}),C_{2}Y/Y)\to{\rm Ker}\;i_{0}^{*}(\mathcal{S}Y)$. Finally,
we get

$\displaystyle H^{0}_{{\mathcal{K}}_{X/Y}}={\rm Ker}\;\delta^{(0)}/{\rm
Im}\;\delta^{(-1)}=H_{{\mathcal{K}}^{0}_{0}}={\rm
Aut}_{(z_{1},\ldots,z_{n})}\;{\mathcal{O}}_{X/Y}|_{\mathcal{H}.W}.$

$\square$

## References

  * [1] M. F. Atiyah, K-theory. W. A. Benjamin, Inc., New York-Amsterdam 1967
  * [2] Frenkel, E.; Ben-Zvi, D. Vertex algebras and algebraic curves. Mathematical Surveys and Monographs, 88\. American Mathematical Society, Providence, RI, 2001. xii+348 pp.
  * [3] R. Bott, Lectures on characteristic classes and foliations. Springer LNM 279 (1972), 1–94.
  * [4] R. Bott, G.Segal, The cohomology of the vector fields on a manifold, Topology Volume 16, Issue 4, 1977, Pages 285–298.
  * [5] Dong C., Li H. and Mason G., Twisted representations of vertex operator algebras, Math. Ann. 310 (1998), 571–600.
  * [6] Ch. Dong, K. Liu, X. Ma, J. Zhou, K-theory associated to vertex operator algebras. Mathematical Research Letters 11, 629–647 (2004)
  * [7] Feigin, B. L.: Conformal field theory and Cohomologies of the Lie algebra of holomorphic vector fields on a complex curve. Proc. ICM, Kyoto, Japan, 71-85 (1990)
  * [8] Frenkel I.B., Huang Y. and Lepowsky J., On axiomatic approach to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104, 1993.
  * [9] Fuks, D. B.:Cohomology of Infinite Dimensional Lie algebras. New York and London: Consultant Bureau 1986
  * [10] I.M. Gelfand, D.A. Kazhdan and D.B. Fuchs, The actions of infinite-dimensional Lie algebras, Funct. Anal. Appl. 6 (1972) 9–13.
  * [11] Gerstenhaber, M., Schack, S. D.: Algebraic Cohomology and Deformation Theory, in: Deformation Theory of Algebras and Structures and Applications, NATO Adv. Sci. Inst. Ser. C 247, Kluwer Dodrecht 11-264 (1988)
  * [12] Hatcher A. Vector bundles and K-Theory, Cornell Lecture Notes.
  * [13] Huang Y.-Zh. A cohomology theory of grading-restricted vertex algebras. Comm. Math. Phys. 327 (2014), no. 1, 279–307.
  * [14] Kac V., Infinite-dimensional Lie algebras, Cambridge Univ. Press, London, 1991.
  * [15] Kodaira, K.: Complex Manifolds and Deformation of Complex Structures. Springer Grundlehren 283 Berlin Heidelberg New York 1986
  * [16] Manetti M. Lectures on deformations of complex manifolds (deformations from differential graded viewpoint). Rend. Mat. Appl. (7) 24 (2004), no. 1, 1–183.
  * [17] Patras, F., Thomas, J.-C. Cochain algebras of mapping spaces and finite group actions. Topology Appl. 128 (2003), no. 2-3, 189–207.
  * [18] A. Tsuchiya, K. Ueno and Y. Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, in: Advanced Studies in Pure Math., Vol. 19, Kinokuniya Company Ltd., Tokyo, 1989, 459–566.
  * [19] Wagemann, F.: Differential graded cohomology and Lie algebras of holomorphic vector fields. Comm. Math. Phys. 208 (1999), no. 2, 521–540
  * [20] E. Witten, Quantum field theory, Grassmannians and algebraic curves, Comm. Math. Phys. 113 (1988) 529–600.
  * [21] Witten E., The index of the Dirac operator in loop space, in Elliptic Curves and Modular forms in Algebraic Topology, Landweber P.S., SLNM 1326, Springer, Berlin, 161-186.
  * [22] Zhu Y., Modular invariance of characters of vertex operator algebras, J. AMS 9, (1996), 237-301.
  * [23] Zuevsky, A. Associative algebra twisted bundles over compact topological spaces. J. Geom. Phys. 170 (2021), Paper No. 104393.


# Emulating AC OPF solvers for Obtaining Sub-second Feasible, Near-Optimal
Solutions

Kyri Baker
University of Colorado Boulder
Renewable and Sustainable Energy Institute
<EMAIL_ADDRESS>
###### Abstract

Using machine learning to obtain solutions to AC optimal power flow has
recently been a very active area of research due to the astounding speedups
that result from bypassing traditional optimization techniques. However,
generally ensuring feasibility of the resulting predictions while maintaining
these speedups is a challenging, unsolved problem. In this paper, we train a
neural network to emulate an iterative solver in order to cheaply and
approximately iterate towards the optimum. Once we are close to convergence,
we then solve a power flow to obtain an overall AC-feasible solution. Results
shown for networks up to 1,354 buses indicate the proposed method is capable
of finding feasible, near-optimal solutions to AC OPF in milliseconds on a
laptop computer. In addition, it is shown that the proposed method can find
“difficult” AC OPF solutions that cause flat-start or DC-warm started
algorithms to diverge. Lastly, we show that for larger networks, the learning-
based solution finds approximate solutions to AC OPF faster than it takes to
find a solution to DC OPF, with a smaller optimality gap than DC OPF provides,
and without the AC infeasibility of DC OPF.

##  I Introduction

AC optimal power flow (OPF) is a canonical power systems operation problem
that is at the heart of optimizing large-scale power networks. Solving this
problem quickly and efficiently has been the subject of decades of research.
One particularly interesting development in this area is the use of machine
learning (ML), in particular deep learning, to obtain solutions to AC OPF [1,
2, 3]. Within this area, ensuring feasibility of the resulting solution has
been a challenge. In this paper, we propose a deep learning model which aims
to emulate an AC OPF solver (in particular, the Matpower Interior Point
Solver, or MIPS). The benefit of using an ML model instead of the MIPS solver
directly is that no matrix inverses or factorizations are needed, and
inference is extremely fast, resulting in an overall faster convergence. While
we do not claim that feasibility can be guaranteed for every single output of
the learning-based model, empirically, we have observed very positive results
on the chosen networks in terms of optimality gap, speed, and convergence
success.

The model proposed in this paper is comprised of a fully connected three-layer
neural network (NN) $F_{R}$ with feedback, where input ${\bf x}^{k}$ is the
candidate optimal solution vector at iteration $k$. Reminiscent of a simple
recurrent neural network, the model iteratively uses feedback from the output
layer as inputs until convergence ($||{\bf x}^{k+1}-{\bf
x}^{k}||\leq\epsilon$). The model thus bypasses any construction of a Jacobian
matrix or associated inverse, for example. This has benefits when trying to
solve challenging AC OPF problems that may encounter near-singular power flow
Jacobians while solving. In order to obtain a final AC feasible solution, a
subset of the learned variables are sent to a power flow solver. See Fig. 1
for an overview of the testing phase of the algorithm.

Some works have looked at penalizing constraint violations during training,
which can help preserve AC feasibility, but cannot guarantee it [2, 3]. In
addition, these techniques can result in very cumbersome-to-train loss
functions. Other work using ML for AC OPF has recovered AC feasibility by
using a post-processing procedure with the AC power flow equations, but
requires a restricted training set generated from a modified AC OPF problem,
sometimes requires PV/PQ switching, and was only tested on small networks [1].
The approach presented in this paper utilizes a similar concept to the latter
technique, but offers advantages over all previous techniques to ensure
feasibility:

  * •

No restriction on the training set is required; previous AC OPF runs can be
used to train the ML model.

  * •

The model _emulates_ an iterative algorithm, meaning that each model run is a
small step towards the optimum, instead of directly predicting the AC OPF
solution.

  * •

While only a subset of variables is sent to the power flow solver, the ML
model utilizes information about the entire OPF solution, better informing the
model as it iterates towards the optimum.

Figure 1: Using a neural network to approximate fast iterations towards the
optimum, then solving a power flow to recover feasibility.

These prior works [1, 2, 3] additionally only perturb the given loads by 10%
or 20% in both training and testing, which results in a much smaller set of
optimal solutions, making it easier for the ML model to map these conditions
onto optimal values. Here, we generate data than spans a larger region of the
feasible space, including solutions which represent the state of the system
near voltage collapse. These situations pose challenging situations for
traditional AC OPF solvers, which will be discussed in more detail.

Promising initial results are shown for 30, 500, and 1,354-bus networks
suggesting that the learning-based model may provide a desirable tradeoff
between speed and optimality gap while maintaining feasibility.

##  II Iterating using inference

A general nonconvex optimization problem with $n$-dimensional optimization
variable vector ${\bf x}$, cost function
$f(\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}$, $M$ equality constraints
$g_{i}(x)=0$, $g_{i}(\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}$, and $P$
inequality constraints $h_{j}(x)\leq 0$,
$h_{j}(\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}$ can be written as

$\displaystyle\min_{\begin{subarray}{c}{\bf x}\end{subarray}}~{}~{}f({\bf x})$
(1a) $\displaystyle\mathrm{s.t:~{}}g_{i}({\bf x})$
$\displaystyle=0,~{}~{}i=1,...M$ (1b) $\displaystyle h_{j}({\bf x})$
$\displaystyle\leq 0,~{}~{}j=1,...,P$ (1c)

Many iterative optimization solvers use Hessians of the Lagrangian function to
iterate towards the optimal solution of constrained nonconvex problems,
including the Matpower Interior Point Solver (MIPS) [4], which leverages a
primal-dual interior point algorithm to update candidate solution ${\bf
x}^{k}$ at iteration $k$. Instead of using Lagrangian functions or forming
Hessian matrices, the learning-based method uses a deep learning model
$F_{R}(\cdot):\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ that takes in ${\bf
x}^{k}$ as an input and provides ${\bf x}^{k+1}$ as an output; e.g.

${\bf x}^{k+1}=F_{R}({\bf x}^{k}).$ (2)

A fully connected three-layer NN is used here. The variable vector ${\bf
x}^{k}=[{\bf v}^{k},~{}{\mbox{\boldmath$\theta$}}^{k},~{}{\bf
P}^{k}_{g},~{}{\bf Q}^{k}_{g}]^{T}$, where the iteration index is $k$, ${\bf
v}$ contains the complex voltage magnitudes at each bus, $\theta$ contains the
complex voltage angles, and ${\bf P}_{g}$ and ${\bf Q}_{g}$ are the real and
reactive power outputs at each generator, respectively.

###  II-A Network architecture

A rectified linear unit (ReLU) was used as the activation function on the
input layer; a hyperbolic tangent (tanh) activation function was used in the
hidden layer, and a linear function was used on the output. Bounds on
generation and voltages were enforced with a threshold on the output layer of
the NN. Normalization of the inputs was also performed such that all inputs
were in the range $[0,1]$, which improved NN performance. In addition to ${\bf
x}^{k}$, the network loading (constant throughout inference) was given as an
input to the NN. The chosen number of nodes in the hidden layer and training
dataset sizes are shown in Table I. The data generation, training and testing
of the network, and simulations were performed on a 2017 MacBook Pro laptop
with 16 GB of memory. Keras with the Tensorflow backend was used to train the
neural network using the Adam optimizer.

TABLE I: Number of nodes and training samples Case |  | Input
---
Nodes
| Output
---
Nodes
| Hidden
---
Nodes
| Training
---
Samples
30-bus | 112 | 72 | 100 | 72,111
300-bus | 1,120 | 738 | 800 | 91,432
500-bus | 1,512 | 1,112 | 2,300 | 111,674
1,354-bus | 4,470 | 3,228 | 6,000 | 126,724

###  II-B Data Generation

The MATPOWER Interior Point Solver (MIPS) [4] was used to generate the data
and was used as the baseline for comparison with the NN model. A single
training sample consists of the pair $[{\bf x}^{k},{\bf x}^{k+1}]$ obtained
from the solver. The termination tolerance of the MIPS solver was set to
$10^{-9}$ for data generation and $10^{-4}$ for testing. The tolerance of the
learning-based solver was set to $10^{-4}$, where convergence is reached when
$||{\bf x}^{k+1}-{\bf x}^{k}||\leq\epsilon$. A smaller tolerance was used for
data generation to promote smoother convergence and “basins of attraction"
within the ML model. For a fair comparison, the same convergence criteria was
used for the NN model and MIPS during testing. 500 different loading scenarios
were randomly generated at each load bus from a uniform distribution of $\pm
40\%$ around the given base loading scenario in MATPOWER. Table I shows the
number of training samples generated for each scenario. Each generated set of
loads was used to solve a standard power flow, and if the power flow could not
find a solution, the data was not included in the training set. It is
recognized that generating a diverse and representative dataset is an
important and essential thrust of research within learning-based OPF methods.
This is an important direction of future work.

TABLE II: The ML solution finds feasible AC OPF solutions faster than traditional methods and solves in a comparable time to DC OPF. Network | OPF Type |  | Average Solve
---
Time (s)
| Percent Solve
---
Success (%)
|  | AC OPF
---
w/Flat Start
0.069 s | 100%
|  | AC OPF
---
w/DC Start
0.079 s | 100%
30-bus |  | AC OPF
---
w/PF Start
0.068 s | 100%
|  | NN with PF
---
0.050 s | 100%
| DC OPF | 0.010 s | 100%
|  | AC OPF
---
w/Flat Start
3.26 s | 100%
|  | AC OPF
---
w/DC Start
3.34 s | 100%
500-bus |  | AC OPF
---
w/PF Start
3.15 s | 100%
|  | NN with PF
---
0.12 s | 100%
| DC OPF | 0.12 s | 100%
|  | AC OPF
---
w/Flat Start
20.12 s | 59.4%
|  | AC OPF
---
w/DC Start
11.63 s | 59.4%
1354-bus |  | AC OPF
---
w/PF Start
10.64 s | 59.8%
|  | NN with PF
---
0.25 s | 98.00%
| DC OPF | 0.33 s | 100%

##  III Simulation Results

Here, three networks were considered: The IEEE 30 bus, IEEE 300 bus, and
1,354-bus PEGASE networks [5]. Line flow constraints were neglected (although
some networks did not have them to begin with). In Table II, the learning-
based method (“NN”) was compared with 4 other cases across the 500-sample
training set: AC OPF flat-started, AC OPF warm-started with both a DC OPF
solution and a power flow solution, and a DC OPF. While the DC OPF never
produces an AC feasible solution, it is often used as an approximation for AC
OPF and used in many LMP-based markets to calculate prices and thus provides
an interesting comparison. Note that in these simulations, no “extra”
feasibility steps were performed (like PV/PQ switching in [1]); the output of
the NN was directly sent to a power flow solver. Performance could perhaps be
further improved in the future with additional steps.

###  III-A Speedups

As the table shows, especially for the 500 and 1,354-bus networks, the
learning-based solution can provide solutions with speedups of over twenty
times faster than using a traditional solver, even one that has been warm-
started. A DC OPF, in comparison, takes about the same amount of time to solve
a much simpler, convex problem, but whose solution does not satisfy the AC
power flow equations. For smaller networks, the benefit of using a ML-based
method is negligible; sub-second solution times are already achieved by
standard AC OPF methods. Another interesting idea is to use a ML model to
warm-start the AC OPF as in [6] or the AC PF as in [7], but we only compare
with traditional warm-start methods here.

###  III-B Challenging OPF scenarios

In the 1,354-bus test case, load profiles were generated that challenged the
MIPS solver. Typically for these difficult cases, continuation methods can be
used to robustly solve the AC OPF by solving a series of simpler OPF problems
[8]. While robust, these methods can be very time consuming and may not be
suitable for real-time operation. The learning-based method may provide an
alternative to these methods, as issues with singular Jacobian matrices and
solutions close to voltage instability do not affect inference as much. These
solutions may still prove challenging for the post-processing power flow step,
however, which is perhaps why a few failures were still encountered when using
the learning-based method.

###  III-C Optimality gaps

Further, Table III shows the optimality gap (difference in cost function
value) for the learning-based solution (“NN”) and DC OPF. This is a key
comparison because in many markets, DC OPF is used to calculate prices; thus,
it is currently deemed an acceptable way of approximating the cost of network
operation. As the table shows, however, the learning-based method produces
even lower optimality gaps than DC OPF, on average and in the worst case
throughout the testing set. A larger neural network, more training data, and
more hyperparameter tuning may improve these results even further.

TABLE III: Percent optimality gap for the proposed method and for DC OPF across the 500-sample testing sets. Network |  | Average
---
Gap: NN
| Average
---
Gap: DC
| Worst
---
Gap: NN
| Worst
---
Gap: DC
30-bus | 0.20% | 1.46% | 0.31% | 1.82%
500-bus | 2.70% | 5.30% | 10.12% | 16.22%
1354-bus | 0.09% | 1.40% | 0.20% | 1.73%

##  IV Conclusion and Discussion

This paper provided a learning-based approximation for solving AC optimal
power flow. Promising initial results indicate that the method can achieve
very fast convergence speeds with minimal optimality gaps, even converging
faster and with more accuracy than DC OPF on large networks. Directions of
future work include development of datasets or dataset generation methods for
learning-based OPF, inclusion of additional constraints such as line limits,
and speed/accuracy comparisons with other relaxations or convexifications.

There are multiple drawbacks of this learning-based method that should be
discussed here; although these results show an effective speed gain,
traditional optimization has multiple upsides that are not yet covered by the
“learning-for-OPF” models in literature (which also provide promising future
directions of research). First, the model is trained on one static
configuration of the network. That means that for any transformer tap-
changing, any line switching, or switchable shunt control for example, a
different model would have to be trained. Second, the performance of the model
is limited by the dataset from which it was trained on. If the data is not
representative of the entire AC OPF space, the model may perform poorly under
certain conditions. Lastly, black-box models such as neural networks do not
offer grid operators as much insight or confidence in the resulting decision-
making. A combination of greater understanding of these models and increased
failsafes can help expedite the potential use of these models in actual
operation.

## References

  * [1] A. Zamzam and K. Baker, “Learning optimal solutions for extremely fast AC optimal power flow,” in _IEEE SmartGridComm_ , Dec. 2020, available at: https://arxiv.org/abs/2861719.
  * [2] M. Chatzos, F. Fioretto, T. W. K. Mak, and P. V. Hentenryck, “High-fidelity machine learning approximations of large-scale optimal power flow,” _arXiv preprint arXiv:2006.16356_ , 2020.
  * [3] X. Pan, M. Chen, T. Zhao, and S. Low, “DeepOPF: A feasibility- optimized deep neural network approach for AC optimal power flow problems,” _arXiv preprint arXiv:2007.0100_ , Jul. 2020.
  * [4] R. D. Zimmerman, C. E. Murillo-Sanchez, and R. J. Thomas, “MATPOWER: Steady-state operations, planning, and analysis tools for power systems research and education,” _IEEE Trans. on Power Systems_ , vol. 26, no. 1, pp. 12–19, Feb 2011.
  * [5] S. Fliscounakis, P. Panciatici, F. Capitanescu, and L. Wehenkel, “Contingency ranking with respect to overloads in very large power systems taking into account uncertainty, preventive, and corrective actions,” _IEEE Trans. on Power Systems_ , vol. 28, no. 4, pp. 4909–4917, 2013.
  * [6] K. Baker, “Learning warm-start points for AC optimal power flow,” in _IEEE Machine Learning for Signal Proc. Conf. (MLSP)_ , Oct. 2019.
  * [7] L. Chen and J. E. Tate, “Hot-starting the Ac power flow with convolutional neural networks,” _arXiv preprint arXiv:2004.09342_ , 2020.
  * [8] S. Cvijic, P. Feldmann, and M. Ilic, “Applications of homotopy for solving AC power flow and AC optimal power flow,” in _2012 IEEE Power and Energy Society General Meeting_ , 2012, pp. 1–8.