# Deformations of Yang-Baxter operators via $n$-Lie algebra cohomology

Mohamed Elhamdadi  University of South Florida, Tampa, FL, USA
<EMAIL_ADDRESS>and  Emanuele Zappala  Yale University, New Haven, CT, USA
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract.

We introduce a cohomology theory of $n$-ary self-distributive objects in the
tensor category of vector spaces that classifies their infinitesimal
deformations. For $n$-ary self-distributive objects obtained from $n$-Lie
algebras we show that ($n$-ary) Lie cohomology naturally injects in the self-
distributive cohomology and we prove, under mild additional assumptions, that
the map is an isomorphism of second cohomology groups. This shows that the
self-distributive deformations are completely classified by the deformations
of the Lie bracket. This theory has important applications in the study of
Yang-Baxter operators as the self-distributive deformations determine
nontrivial deformations of the Yang-Baxter operators derived from $n$-ary
self-distributive structures. In particular, we show that there is a
homomorphism from the second self-distributive cohomology to the second
cohomology of the associated Yang-Baxter operator. Moreover, we prove that
when the self-distributive structure is induced by a Lie algebra with trivial
center, we get a monomorphism. We construct a deformation theory based on
simultaneous deformations, where both the coalgebra and self-distributive
structures are deformed simultaneously. We show that when the Lie algebra has
trivial cohomology (e.g. for semi-simple Lie algebras) the simultaneous
deformations might still be nontrivial, producing corresponding Yang-Baxter
operator deformations. We provide examples and computations in low dimensions,
and we completely characterize $2$-cocycles for the self-distributive objects
obtained from all the nontrivial real Lie algebras of dimension $3$, i.e. the
Bianchi I-IX, and all the nontrivial complex Lie algebras of dimension $3$.

##  1\. Introduction

The Yang-Baxter Equation (YBE) was first introduced in the 1970’s in
statistical mechanics to impose the conservation of momenta in scattering
processes [Jim]. It has ever since outgrown its original field of applications
by becoming a central player in geometric topology, more specifically in knot
theory. In fact, YBE is an algebraic version of Reidemeister move III, which
is one of the necessary and sufficient conditions for two diagrams to
represent isotopic knots/links [Oht, Tur]. The $n$-simplex equation,
originally introduced by Zamolodchikov, is a generalization of the YBE that
has been applied (for $n=3$) to the theory of knotted surfaces embedded in
$4$-space [nsimplex, KTS].

A well known class of algebraic structures that produce solutions to the YBE
are quandles. These are sets endowed with an operation that corresponds to
Reidemeister moves. In particular, they satisfy the self-distributive (SD)
condition, which is a set-theoretic version of Reidemeister move III.
Linearization then gives YB operators. The cohomology of quandles was
introduced in [CJKLS] to construct certain partition functions that are
invariants of links in $3$-space and knotted surfaces in $4$-space.

More recently, the study of homology theories of Yang-Baxter (YB) operators
has led to new knot invariants ([CES]) that generalize the quandle cocycle
invariants given in [CJKLS]. Related homology theories for YB operators have
appeared in [PW, PW_Hom, LV], and a skein-theoretic approach to both homology
and cohomology of YB operators was given in [ESZ_YB]. The systematic study of
solutions of the YBE and the (co)homology theories of YB operators is still a
highly nontrivial problem with applications that span from knot theory to
quantum gravity.

$n$-Lie algebras are $n$-ary counterparts to Lie algebras, where the notion of
Lie bracket and Jacobi identity are replaced by $n$-ary brackets and the
Filippov identity. Interestingly, $n$-Lie algebras were introduced in
theoretical physics [Nambu] by Nambu, with the purpose of generalizing
Hamiltonian dynamics. They were then rediscovered by Filippov in the 1980’s
[Fil], who proved that the structure considered by Nambu satisfied a certain
generalization of the Jacobi identity, that was later dubbed Filippov
identity. A cohomology theory for $n$-Lie algebras was then constructed by
Takhtajan [Tak] based on the Chevalley-Eilenberg complex. This cohomology
theory in particular characterizes deformations of $n$-Lie algebras providing
a parallel with the deformation theory of (binary) Lie algebras due to
Nijenhuis-Richardson [NR], which was itself based on the analogous deformation
theory for associative algebras of Gerstenhaber [Gerstenhaber].

In this article we consider $n$-ary quandle objects in the category of vector
spaces. These are analogues of quandles where the defining axioms are
translated from the symmetric monoidal category of sets with the cartesian
product, to that of vector spaces with tensor product. More specifically, we
are interested in the property of self-distributivity (in the categorical
setting) which directly corresponds to Reidemeister move III. We call these
structures self-distributive objects, or SD objects for short. When we want to
mention explicitly the arity of the SD object, we will write $n$-SD object. We
define a cohomology theory for SD objects in vector spaces that generalizes
the cohomology theory of [CES], and we show that this cohomology characterizes
the deformation theory of SD objects, similarly to the aforementioned cases of
associative and Lie algebras. We give a general construction that derives
$n$-SD objects from $n$-Lie algebras. We relate the deformation theory of
$n$-Lie algebras to the deformation theory of the corresponding $n$-SD object
and show that under mild hypotheses the second cohomology groups are
isomorphic. Then we consider simultaneous deformations of SD objects, where
both coalgebra and SD morphism are deformed together. This provides a more
general class of SD deformations. We use these results to obtain deformations
of YB operators arising from the $n$-SD objects and we obtain a relation with
the second YB cohomology group. Along the way, we consider detailed
computations in low dimensions to show that the theory is nontrivial and give
tables of computatoins of the SD cohomology groups of the Lie algebras
appearing in the Bianchi classification.

The starting point of this article are results found in [CCES], where it was
given the construction of SD objects from Lie algebras in the binary case. We
also point out that in [CCES] the cohomology defined becomes substantially
equivalent to the binary case of the simultaneous deformations proposed here.
We observe that other than considering the extended case to $n$-ary
structures, we also focus on the relations between the deformation theory
(i.e. cohomology) of the $n$-Lie structures and their corresponding $n$-ary
quandles. In addition, we consider in detail the application of this theory to
deformations of YB operators and their second cohomology groups.

The article is structured as follows. In Section 2 we recall some basic
definitions regarding the deformation of Lie algebras, and cochain complexes
of $n$-ary Lie algebras. Section 3 gives a brief review of self-distributive
objects in symmetric monoidal category needed for the rest of the article. In
Section 4 the definition of cohomology of $n$-ary quandles is given, and some
properties of the first and second cohomology groups are proved. In Section 5
we prove that this newly defined cohomology characterizes deformations of
$n$-ary quandle objects. Section 6 is devoted to studying the relation between
the cohomology of Lie algebras and the cohomology of the corresponding SD
($n$-quandle) object. Section 7 is a digression on deformations of coalgebra
structures, which is going to be used in the subsequent Section 8, where
simultaneous deformations of $n$-quandle objects are introduced. In Section 9
we provide detailed computations and examples in low dimensions. Here, tables
containing the cohomology groups for low dimensional Lie algebras are also
provided. In Section 10 we apply the theory introduced and developed in this
article to Yang-Baxter operator deformations, and relate the results to the
second cohomology group of Yang-Baxter operators. Several detailed proofs
which consist of cumbersome computations are deferred to the Appendix.

##  2\. Infinitesimal deformations of $n$-ary Lie algebras

In this section we discuss deformation theory of Lie algebras by means of
cohomology theory. We recall some known traditional facts for the case of
binary Lie algebras, and then pass on considering $n$-Lie algebras, in the
sense of Filippov.

###  2.1. Binary Lie algebras

The material of this subsection is standard, and can be found in [GS, GS2, NR]
for example. Let $\mathfrak{g}$ be a (binary) Lie algebra over the field (or
ring) $\mathbb{k}$ with bracket
$[\bullet,\bullet]:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$.
We extend the coefficients of $\mathfrak{g}$ to $\mathbb{k}[[\hbar]]$, the
power series ring on the variable $\hbar$ with coefficients on $\mathbb{k}$.
Let $\mathfrak{g}^{\prime}$ denote the resulting module, namely
$\mathfrak{g}^{\prime}=\mathbb{k}[[\hbar]]\otimes_{\mathbb{k}}\mathfrak{g}$.
We define a family of maps
$[\bullet,\bullet]^{\prime}:\mathfrak{g}^{\prime}\otimes\mathfrak{g}^{\prime}\longrightarrow\mathfrak{g}^{\prime}$
as a formal power series $\sum_{n}[\bullet,\bullet]_{n}\hbar^{n}$, where we
set $[\bullet,\bullet]_{0}=[\bullet,\bullet]$. If the brackets
$[\bullet,\bullet]_{n}$ are such that $[\bullet,\bullet]^{\prime}$ satisfies
the Jacobi identity, then we say that this is a Lie deformation of the bracket
$[\bullet,\bullet]$, since by definition we have that on $\mathfrak{g}$ the
two brackets coincide. This is equivalent to saying that
$[\bullet,\bullet]^{\prime}$ satisfies the Jacobi identity on elements
$x,y,z\in\mathfrak{g}$ seen as elements of $\mathfrak{g}^{\prime}$ with
coefficients in $\mathbb{k}\hookrightarrow\mathbb{k}[[\hbar]]$. Here we are
interested mainly in infinitesimal deformations. These are those deformations
of $[\bullet,\bullet]$ up to the maximal ideal $(\hbar^{2})$ of
$\mathbb{k}[[\hbar]]$. More specifically, we consider the quotient ring
$R:=\mathbb{k}[[\hbar]]/(\hbar^{2})$ and let
$\tilde{\mathfrak{g}}:=R\otimes_{\mathbb{k}}\mathbb{k}[[\hbar]]$. Then a
deformation on $\tilde{\mathfrak{g}}$ needs to satisfy the Jacobi identity on
any choice of $x,y,z\in\mathfrak{g}$. We set, for short,
$\phi(x,y)=[x,y]_{1}$, the bracket corresponding to the first power of
$\hbar$. Then the Jacobi identity on $x,y,z$ gives

$\displaystyle[[x,y]^{\prime},z]^{\prime}+[[y,z]^{\prime},x]^{\prime}+[[z,x]^{\prime},y]^{\prime}$
$\displaystyle=$
$\displaystyle[[x,y]+\phi(x,y),z]^{\prime}+[[y,z]+\phi(y,z),x]^{\prime}+[[z,x]+\phi(z,x),y]^{\prime}$
$\displaystyle=$
$\displaystyle[[x,y],z]+[[y,z],x]+[[z,x],y]+\hbar[\phi(x,y),z]+\hbar[\phi(y,z),x]$
$\displaystyle+\hbar[\phi(z,x),y]+\hbar\phi([x,y],z)+\hbar\phi([y,z],x)+\hbar\phi([z,x],y)$
$\displaystyle+\hbar^{2}\phi(\phi(x,y),z)+\hbar^{2}\phi(\phi(y,z),x)+\hbar^{2}\phi(\phi(z,x),y)$
$\displaystyle=$ $\displaystyle 0.$

The previous equation is usually interpreted as follows. We set
$\delta\phi(x,y,z)=[\phi(x,y),z]+[\phi(y,z),x]+[\phi(z,x),y]+\phi([x,y],z)+\phi([y,z],x)+\phi([z,x],y)$
and also (assuming ${\rm char}\ \mathbb{k}\neq 2$) set
$1/2[\phi,\phi](x,y,z)=\phi(\phi(x,y),z)+\phi(\phi(y,z),x)+\phi(\phi(z,x),y)$.
Since the coefficient of the term $1/2[\phi,\phi]$ is quadratic in the
“infinitesimal” deformation parameter $\hbar$, it follows that
$[\bullet,\bullet]^{\prime}$ is an infinitesimal deformation if and only if
$\delta^{2}\phi=0$. This is the $2$-cocycle condition for Lie cochains
$\phi:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$ with
coefficients in $\mathfrak{g}$.

###  2.2. $n$-Lie algebras

The case of higher arity Lie algebras was first considered in [Tak]. We use a
slightly modified reasoning which leads to definitions that are more suitable
for our purposes in the rest of the present article. Namely, we do not
consider a homology theory and dualize it, but rather we direcly follow the
deformation theory point of view of Gerstenhaber and Schack and adapt it to
the higher arity case. We explicitly write the formulas for ternary brackets,
since this generalizes directly to the $n$-ary case. The following treatment
is closely related to that of Takhtajan in [Tak].

As before, we consider a Lie algebra with (ternary) bracket
$[\bullet,\bullet,\bullet]:\mathfrak{g}\otimes\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$,
see [Fil]. We extend coefficients to the ring $\mathbb{k}[[\hbar]]$ and
quotient out by the maximal ideal $(\hbar^{2})$, with the intention of finding
deformations that are infinitesimal of order $1$ in $\hbar$. We also set, for
simplicity, $\phi(x,y,z)=[x,y,z]_{1}$, for the ternary bracket of order $1$. A
deformed bracket is now required to satisfy the Filippov idenity (FI) on
$x,y,z,w,u\in\mathfrak{g}$. That is, we have

$\displaystyle[[x,y,z]^{\prime},w,u]^{\prime}=[[x,w,u]^{\prime},y,z]^{\prime}+[x,[y,w,u]^{\prime},z]^{\prime}+[x,y,[z,w,u]^{\prime}]^{\prime}.$

Using the definition of $[\bullet,\bullet,\bullet]^{\prime}$ we obtain, as in
the case of binary deformation, the following. For the left hand side of FI we
have

$\displaystyle[[x,y,z]^{\prime},w,u]^{\prime}$ $\displaystyle=$
$\displaystyle[[x,y,z],w,u]+\hbar[\phi(x,y,z),w,u]+\hbar\phi([x,y,z],w,u)$
$\displaystyle+\hbar^{2}\phi(\phi(x,y,z),w,u).$

For the right hand side of FI we have

$\displaystyle[[x,w,u]^{\prime},y,z]^{\prime}+[x,[y,w,u]^{\prime},z]^{\prime}+[x,y,[z,w,u]^{\prime}]^{\prime}$
$\displaystyle=$
$\displaystyle[[x,w,u],y,z]+[x,[y,w,u],z]+[x,y,[z,w,u]]+\hbar[\phi(x,w,u),y,z]$
$\displaystyle+\hbar[x,\phi(y,w,u),z]+\hbar[x,y,\phi(z,w,u)]+\hbar\phi(x,[y,w,u],z)+\hbar\phi(x,y,[z,w,u])$
$\displaystyle+\hbar\phi([x,w,u],y,z)+\hbar^{2}\phi(\phi(x,w,u),y,z)+\hbar^{2}\phi(x,\phi(y,w,u),z)+\hbar^{2}\phi(x,y,\phi(z,w,u)).$

Neglecting terms of order $2$, FI holds if and only if

$\displaystyle\delta^{2}\phi(x,y,z,w,u)$ $\displaystyle:=$
$\displaystyle[\phi(x,y,z),w,u]+\phi([x,y,z],w,u)-[x,\phi(y,w,u),z]$
$\displaystyle-[x,y,\phi(z,w,u)]-\phi([x,w,u],y,z)-\phi(x,[y,w,y],z)-\phi(x,y,[z,w,u])$
$\displaystyle=$ $\displaystyle 0.$

This is defined to be the $2$-cocycle condition for the $3$-Lie algebra
$\mathfrak{g}$ with coefficients in $\mathfrak{g}$.

###  2.3. $2$-cocycles and cohomology

Let $C^{2}_{n}(\mathfrak{g};\mathfrak{g})$, with $n=2,3$ be the group of
$2$-cochains with coefficients in $\mathfrak{g}$. In other words,
$C^{2}_{n}(\mathfrak{g};\mathfrak{g})$ is the space of linear maps
$\mathfrak{g}^{\otimes n}\longrightarrow\mathfrak{g}$. As usual, it is clear
that the $2$-cocycle condition for $n$-Lie algebras, $n=2,3$, is closed under
addition and it determines an additive subgroup of cocycles
$Z^{2}_{n}(\mathfrak{g};\mathfrak{g})\subset
C^{2}_{n}(\mathfrak{g};\mathfrak{g})$ where $n=2,3$ depending on the arity of
$\mathfrak{g}$.

Let us define now the group of $1$-cochains
$C^{1}_{n}(\mathfrak{g};\mathfrak{g})={\rm
Hom}_{\mathbb{k}}(\mathfrak{g},\mathfrak{g})$. We define the first
differential $\delta^{1}:C^{1}_{n}(\mathfrak{g};\mathfrak{g})\longrightarrow
C^{2}_{n}(\mathfrak{g};\mathfrak{g})$ as follows. For $n=2$, we set
$\delta^{1}f(x,y)=f([x,y])-[f(x),y]-[x,f(y)]$, while for $n=3$ we set
$\delta^{1}f(x,y,z)=f([x,y,z])-[f(x),y,z]-[x,f(y),z]-[x,y,f(z)]$.

Clearly, while the $2$-cocycle condition is the obstruction for a bracket to
be infinitesimally deformed, we have that the $1$-cocycle condition is the
obstruction for a $1$-cochain to be a Lie algebra derivation of
$\mathfrak{g}$, where the notion of derivation in the $n$-ary case is a
straightforward generalization of the usual binary definition. We define
$\delta^{1}C^{1}_{n}(\mathfrak{g};\mathfrak{g})=B^{2}_{n}(\mathfrak{g};\mathfrak{g})$,
the subgroup of coboundaries.

Direct computations show that $\delta^{2}\delta^{1}=0$, hence coboundaries are
also $2$-cocycles. Setting
$H^{2}_{n}(\mathfrak{g};\mathfrak{g})=Z^{2}_{n}(\mathfrak{g};\mathfrak{g})/B^{2}_{n}(\mathfrak{g};\mathfrak{g})$
we have the second cohomology group of $\mathfrak{g}$ with coefficients in
$\mathfrak{g}$. The binary case gives the usual cohomology of Lie algebras,
i.e. the dual to the Eilenberg-Chevalley complex. The ternary case closely
parallels the deformation theoretic interpretation of the binary Lie
cohomology and gives a definition of ternary $2$-cohomology that suits well
our purposes.

The definitions for the case $n=3$ extend easily to arbitrary $n$, therefore
providing a notion of second cohomology group for $n$-Lie algebras.

Recall that two deformations of a $2$-Lie bracket are said to be equivalent if
there exists an isomorphism between the two deformed Lie structures that fixes
the zero degree bracket. For higher arity we define equivalent deformations
mutatis mutandis. We have the following standard result.

######  Theorem 2.1.

Let $\mathfrak{g}$ be an $n$-Lie algebra. Then, infinitesimal deformations of
$\mathfrak{g}$ are classified by the second cohomology group
$H^{2}_{n}(\mathfrak{g};\mathfrak{g})$.

###### Proof.

The proof adapts the methods of the binary case to higher arities, and is
therefore only sketched. From the discussion above, it is clear that
infinitesimal deformations arise if and only if
$[\bullet,\ldots,\bullet]^{\prime}=[\bullet,\ldots,\bullet]+\hbar\phi(\bullet,\ldots,\bullet)$
for some $2$-cocycle $\phi$ of appropriate arity. This was seen explicitly for
$n=2,3$, while the general result for arbitrary $n$ is analogous. To complete
the characterization one needs to prove that two extensions of $\mathfrak{g}$
are equivalent if and only if the corresponding $2$-cocycles differ by a term
$\delta^{1}f$, for some $f:\mathfrak{g}\longrightarrow\mathfrak{g}$. In fact,
it is enough to show that $\phi$ defines the trivial deformation if and only
if $\phi=\delta^{1}f$. This is seen directly, for arbitrary $n$. ∎

##  3\. Preliminaries on self-distributive objects

In this section we recall some of the definitions that will be used throughout
the article. This material can be found in [AZ, EZ, CCES, ESZ].

######  Definition 3.1.

An $n$-ary quandle, or $n$-quandle for short, is a set $X$ with an $n$-ary map
$T:X^{\times n}\longrightarrow X$ satisfying the three axioms:

  * •

It holds

$T(T(x_{1},\ldots
x_{n}),x_{n+1},\ldots,x_{2n-1})=T(T(x_{1},x_{n+1},\ldots,x_{2n-1}),\ldots,T(x_{n},x_{n+1},\ldots
x_{2n-1})),$

for all $x_{i}\in X$. This property is called n-ary self-distributivity;

  * •

The map $T_{(x_{2},\ldots,x_{n})}:X\longrightarrow X$ obtained by multiplying
on the right, i.e. $T_{(x_{2},\ldots,x_{n})}(x):=T(x,x_{2},\ldots,x_{n})$, is
bijective for all choices of $(x_{2},\ldots,x_{n})\in X^{\times{(n-1)}}$;

  * •

For all $x\in X$, we have that $T(x,\ldots,x)=x$.

######  Remark 3.2.

The case $n=2$, which is known simply as “quandle” in the literature, has
found profound applications in knot theory (see [Joy, Mat, CJKLS, CEGS]). In
fact, the axioms given above for $n=2$ are seen to be an algebraization of the
Reidemeister moves (III, II and I, respectively). Here we chose to list the
axioms in reversed order than usually found in the literature, because
algebraically, a set with a map $T$ that satisfies only the first axiom is
called a shelf, or self-distributive set, while if $T$ satisfies the first two
axioms is said to be a rack. We will use $n$-shelf and $n$-rack to specify the
arity of the operation $T$, if needed.

The definition of shelf, rack and quandle generalizes to symmetric monoidal
categories as seen in [ESZ, EZ]. The SD property, which is the one of main
interests in the present article, takes the form of the following commutative
diagram

${X^{\boxtimes n^{2}}}$${X^{\boxtimes(2n-1)}}$${X^{\boxtimes
n^{2}}}$${X^{\boxtimes n}}$${X^{\boxtimes
n}}$${X}$$\scriptstyle{\shuffle_{n}}$$\scriptstyle{\mathbb{1}^{\boxtimes
n}\boxtimes\Delta_{n}^{\boxtimes(n-1)}}$$\scriptstyle{W\boxtimes\mathbb{1}^{\boxtimes(n-1)}}$$\scriptstyle{W\boxtimes\cdots\boxtimes
W}$$\scriptstyle{W}$$\scriptstyle{W}$

where $W:X^{\boxtimes n}\longrightarrow X$ is a morphism in the symmetric
monoidal category $(\mathcal{C},\boxtimes)$, $\Delta$ is a comultiplication of
$X$, and the morphism $\shuffle_{n}$ is obtained by a permutation of the
factors (using the symmetry of the category). The specific form of
$\shuffle_{n}$ is given in [ESZ]. Observe that in this case $X$ is endowed
both with an $n$-ary “operation”, and with a (binary) comultiplication. The
comultiplication is used to create $n$-ary comultiplications by compositions.
In fact, the same theory where $\Delta$ is $n$-ary and not decomposable in co-
operations of lower degree can be defined analogously, and it has not yet been
explored, to the authors’ knowledge. The need for a coalgebra structure arises
from the fact that the set-theoretic case uses repetitions of the variables,
which can be thought of as a comultiplication (diagonal) in the symmetric
monoidal category of sets with cartesian product.

The main interest in $n$-ary SD structures (similarly to the binary set-
theoretic case), is that they produce solutions to the Yang-Baxter equation
(YBE). When the structure is a (categorical) rack, then these solutions are
invertible, and they are therefore YB operators. See for instance [EZ] for a
general approach. In [AZ], which constitutes the starting point of the present
article, the ternary approach was considered in the category of vector spaces
and the YB operators studied were derived from Lie algebras, based on work
found in [CCES].

We briefly recall the construction in [AZ] that produces SD objects in the
category of vector spaces from Lie algebras, and associated YB operators. Let
$\mathfrak{g}$ be an $n$-Lie algebra over the ground field $\mathbb{k}$, and
define $X:=\mathbb{k}\oplus\mathfrak{g}$. We define a comultiplication
$\Delta$ (see also [CCES]) as

$(a,x)\mapsto(a,x)\otimes(1,0)+(1,0)\otimes(0,x),$

and counit $\epsilon$ as

$(a,x)\mapsto a.$

With these comultiplication and counit, $X$ is a coalgebra in the category of
vector spaces. Iterating the comultiplication we can define a ternary co-
operation $\Delta_{3}$. Observe that $\Delta$ (hence $\Delta_{3}$) is
cocommutative. We will use this coalgebra structure on the space $X$ defined
above throughout the article. However, we point out that the cohomology that
we are going to define to study deformations of SD structures does not depend
on $\Delta$, but it only uses the fact that we start from an SD object. The SD
structure associated to an $n$-Lie algebra $\mathfrak{g}$ then is then
$(X,\Delta,T)$, where $X$ and $\Delta$ are as above, and for all
$a_{i}\in\mathbb{k}$ and $x_{i}\in\mathfrak{g}$, the map $T$ is defined on
simple tensors as

$(a_{1},x_{1})\otimes\cdots\otimes(a_{n},x_{n})\mapsto(\prod_{i}a_{i},\prod_{i\neq
1}a_{i}x_{1}+[x_{1},\ldots,x_{n}]),$

where $[\bullet,\ldots,\bullet]$ indicates the $n$-ary bracket of
$\mathfrak{g}$. In fact, starting from an $n$-ary structure, one can define a
$(2n-1)$-ary SD structure as it was shown in [ESZ] for symmetric monoidal
categories. The same procedure can be applied in this case. Therefore, we see
that $n$-ary structures in the category of vector spaces are abundant. In [AZ]
it is detailed how to derive a YB operator from a given TSD structure
associated to a ternary Lie algebra. The generalization to the $n$-ary case is
analogous. This will be reconsidered in Section 10, where we will show the
main application of the cohomology theory introduced and studied in this
article.

##  4\. SD cohomology with SD coefficients

We define now a cohomology theory of $n$-SD structues with coefficients in
$n$-SD objects that is suitable for the deformation theory of SD operations.
The fundamental idea is the same as for $n$-Lie algebras above. We explicitly
write the equations for the case $n=3$, since the whole theory generalizes
directly to the case of arbitrary $n$ (including $n=2$). An $n$-ary SD
structure will be denoted by $n$SD, but the binary and ternary cases, due to
their importance in this article will also be called BSD and TSD,
respectively.

Let $(X,T,\Delta_{3})$ be a TSD structure in the symmetric monoidal category
of modules over a unital ring $\mathbb{k}$, where $T\in{\rm
Hom}_{\mathbb{k}}(X^{\otimes 3},X)$ and $\Delta_{3}\in{\rm
Hom}_{\mathbb{k}}(X,X^{\otimes 3})$. For $n=1,2,3$ we define the first and
second cochain modules of $X$ with coefficients in $X$ as follows. Let ${\rm
Hom}_{\mathbb{k}}(X^{\otimes(2n-1)},X)$ denote the spaces of
$\mathbb{k}$-linear maps between $X^{\otimes(2n-1)}$ and $X$. We define
$C^{1}_{\rm TSD}(X;X)$, the first cochain module, to be the submodule of ${\rm
Hom}_{\mathbb{k}}(X,X)$ consisting of ternary coderivations, where a ternary
coderivation is a linear map $f:X\longrightarrow X$ satisfying
$\Delta_{3}\circ f=(\mathbb{1}^{\otimes 2}\otimes f+\mathbb{1}\otimes
f\otimes\mathbb{1}+f\otimes\mathbb{1}^{\otimes 2})\circ\Delta_{3}$. This
naturally generalizes the notion of binary coderivations between coalgebras to
the ternary setting. Next, define $C^{2}_{\rm TSD}(X;X)$, the second cochain
module, to be the submodule of ${\rm Hom}_{\mathbb{k}}(X^{3},X)$ consisting of
maps satisfying the condition $\Delta_{3}\psi=(\psi\otimes T^{\otimes
2}+T\otimes\psi\otimes T+T^{\otimes 2}\otimes\psi)\shuffle\Delta_{3}^{\otimes
3}$, where $\shuffle=\bigl{(}\begin{smallmatrix}1&2&3&4&5&6&7&8&9\\\
1&4&7&2&5&8&3&6&9\end{smallmatrix}\bigr{)}$, i.e. the permutation used to
endow$X^{\otimes 3}$ with the natural ternary coalgebra structure induced by
$\Delta_{3}$. Finally, we let $C^{3}_{\rm TSD}(X;X)={\rm
Hom}_{\mathbb{k}}(X^{\otimes 5},X)$.

We define now differentials between cochain modules $\delta^{1}:C^{1}_{\rm
TSD}(X;X)\longrightarrow C^{2}_{\rm TSD}(X;X)$ and $\delta^{2}:C^{2}_{\rm
TSD}(X;X)\longrightarrow C^{3}_{\rm TSD}(X;X)$ as follows.

For $f:X\longrightarrow X$, ternary coderivation of $X$, we set

$\displaystyle\delta^{1}f(x\otimes y\otimes z)$ $\displaystyle=$
$\displaystyle f(T(x\otimes y\otimes z))-T(f(x)\otimes y\otimes z)-T(x\otimes
f(y)\otimes z)-T(x\otimes y\otimes f(z)).$

Notice that the condition $\delta^{1}f=0$ is the obstruction for $f$ to be a
ternary derivation with respect to the ternary algebraic structure $T$.

For $\psi:X^{\otimes 3}\longrightarrow X$, coalgebra morphism between
$X^{\otimes 3}$ and $X$, we set

$\displaystyle\delta^{2}\psi(x\otimes y\otimes z\otimes w\otimes u)$
$\displaystyle=$ $\displaystyle T(\psi(x\otimes y\otimes z)\otimes w\otimes
u)+\psi(T(x\otimes y\otimes z)\otimes w\otimes u)$
$\displaystyle-\psi(T(x\otimes w^{(1)}\otimes u^{(1)})\otimes T(y\otimes
w^{(2)}\otimes u^{(2)})\otimes T(z\otimes w^{(3)}\otimes u^{(3)}))$
$\displaystyle-T(\psi(x\otimes w^{(1)}\otimes u^{(1)})\otimes T(y\otimes
w^{(2)}\otimes u^{(2)})\otimes T(z\otimes w^{(3)}\otimes u^{(3)}))$
$\displaystyle-T(T(x\otimes w^{(1)}\otimes u^{(1)})\otimes\psi(y\otimes
w^{(2)}\otimes u^{(2)})\otimes T(z\otimes w^{(3)}\otimes u^{(3)}))$
$\displaystyle-T(T(x\otimes w^{(1)}\otimes u^{(1)})\otimes T(y\otimes
w^{(2)}\otimes u^{(2)})\otimes\psi(z\otimes w^{(3)}\otimes u^{(3)})),$

where we have used Sweedler’s notation for the comultiplication of $X$, in the
form $\Delta(x)=x^{(1)}\otimes x^{(2)}\otimes x^{(3)}$. The $2$-cocycle
condition is shown diagrammatically in Figure 1. See also a similar condition
(in multiplicative notation) in [EZ].

$0=\leavevmode\hbox to85.76pt{\vbox
to57.31pt{\pgfpicture\makeatletter\hbox{\hskip
42.87914pt\lower-28.65276pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{-42.67914pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-21.33957pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{21.33957pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.5pt}{0.0pt}\pgfsys@curveto{2.5pt}{1.38072pt}{1.38072pt}{2.5pt}{0.0pt}{2.5pt}\pgfsys@curveto{-1.38072pt}{2.5pt}{-2.5pt}{1.38072pt}{-2.5pt}{0.0pt}\pgfsys@curveto{-2.5pt}{-1.38072pt}{-1.38072pt}{-2.5pt}{0.0pt}{-2.5pt}\pgfsys@curveto{1.38072pt}{-2.5pt}{2.5pt}{-1.38072pt}{2.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-35.66425pt}{-6.68103pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\delta^{2}(\psi)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\leavevmode\hbox
to71.53pt{\vbox to71.53pt{\pgfpicture\makeatletter\hbox{\hskip
42.87914pt\lower-28.65276pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{-42.67914pt}{42.67914pt}\pgfsys@lineto{-28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-28.45276pt}{42.67914pt}\pgfsys@lineto{-28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-14.22638pt}{42.67914pt}\pgfsys@lineto{-28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-28.45276pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{42.67914pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{28.45276pt}{42.67914pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.5pt}{0.0pt}\pgfsys@curveto{2.5pt}{1.38072pt}{1.38072pt}{2.5pt}{0.0pt}{2.5pt}\pgfsys@curveto{-1.38072pt}{2.5pt}{-2.5pt}{1.38072pt}{-2.5pt}{0.0pt}\pgfsys@curveto{-2.5pt}{-1.38072pt}{-1.38072pt}{-2.5pt}{0.0pt}{-2.5pt}\pgfsys@curveto{1.38072pt}{-2.5pt}{2.5pt}{-1.38072pt}{2.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-28.45276pt}{28.45276pt}\pgfsys@moveto{-25.95276pt}{28.45276pt}\pgfsys@curveto{-25.95276pt}{29.83348pt}{-27.07204pt}{30.95276pt}{-28.45276pt}{30.95276pt}\pgfsys@curveto{-29.83348pt}{30.95276pt}{-30.95276pt}{29.83348pt}{-30.95276pt}{28.45276pt}\pgfsys@curveto{-30.95276pt}{27.07204pt}{-29.83348pt}{25.95276pt}{-28.45276pt}{25.95276pt}\pgfsys@curveto{-27.07204pt}{25.95276pt}{-25.95276pt}{27.07204pt}{-25.95276pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{-28.45276pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.7295pt}{-6.97325pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-38.8229pt}{22.39616pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\psi$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\leavevmode\hbox
to71.53pt{\vbox to71.53pt{\pgfpicture\makeatletter\hbox{\hskip
42.87914pt\lower-28.65276pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{-42.67914pt}{42.67914pt}\pgfsys@lineto{-28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-28.45276pt}{42.67914pt}\pgfsys@lineto{-28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-14.22638pt}{42.67914pt}\pgfsys@lineto{-28.45276pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-28.45276pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{42.67914pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{28.45276pt}{42.67914pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.5pt}{0.0pt}\pgfsys@curveto{2.5pt}{1.38072pt}{1.38072pt}{2.5pt}{0.0pt}{2.5pt}\pgfsys@curveto{-1.38072pt}{2.5pt}{-2.5pt}{1.38072pt}{-2.5pt}{0.0pt}\pgfsys@curveto{-2.5pt}{-1.38072pt}{-1.38072pt}{-2.5pt}{0.0pt}{-2.5pt}\pgfsys@curveto{1.38072pt}{-2.5pt}{2.5pt}{-1.38072pt}{2.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-28.45276pt}{28.45276pt}\pgfsys@moveto{-25.95276pt}{28.45276pt}\pgfsys@curveto{-25.95276pt}{29.83348pt}{-27.07204pt}{30.95276pt}{-28.45276pt}{30.95276pt}\pgfsys@curveto{-29.83348pt}{30.95276pt}{-30.95276pt}{29.83348pt}{-30.95276pt}{28.45276pt}\pgfsys@curveto{-30.95276pt}{27.07204pt}{-29.83348pt}{25.95276pt}{-28.45276pt}{25.95276pt}\pgfsys@curveto{-27.07204pt}{25.95276pt}{-25.95276pt}{27.07204pt}{-25.95276pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{-28.45276pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.37015pt}{-6.0566pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\psi$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-39.18227pt}{21.4795pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$-\leavevmode\hbox to114.21pt{\vbox
to114.21pt{\pgfpicture\makeatletter\hbox{\hskip
57.10551pt\lower-28.65276pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{-56.90552pt}{42.67914pt}\pgfsys@lineto{-42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-42.67914pt}{42.67914pt}\pgfsys@lineto{-42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-28.45276pt}{42.67914pt}\pgfsys@lineto{-42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-42.67914pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-14.22638pt}{42.67914pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{42.67914pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{42.67914pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{56.90552pt}{42.67914pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{42.67914pt}{42.67914pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{28.45276pt}{42.67914pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.5pt}{0.0pt}\pgfsys@curveto{2.5pt}{1.38072pt}{1.38072pt}{2.5pt}{0.0pt}{2.5pt}\pgfsys@curveto{-1.38072pt}{2.5pt}{-2.5pt}{1.38072pt}{-2.5pt}{0.0pt}\pgfsys@curveto{-2.5pt}{-1.38072pt}{-1.38072pt}{-2.5pt}{0.0pt}{-2.5pt}\pgfsys@curveto{1.38072pt}{-2.5pt}{2.5pt}{-1.38072pt}{2.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-42.67914pt}{28.45276pt}\pgfsys@moveto{-40.17914pt}{28.45276pt}\pgfsys@curveto{-40.17914pt}{29.83348pt}{-41.29842pt}{30.95276pt}{-42.67914pt}{30.95276pt}\pgfsys@curveto{-44.05986pt}{30.95276pt}{-45.17914pt}{29.83348pt}{-45.17914pt}{28.45276pt}\pgfsys@curveto{-45.17914pt}{27.07204pt}{-44.05986pt}{25.95276pt}{-42.67914pt}{25.95276pt}\pgfsys@curveto{-41.29842pt}{25.95276pt}{-40.17914pt}{27.07204pt}{-40.17914pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{-42.67914pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@moveto{45.17914pt}{28.45276pt}\pgfsys@curveto{45.17914pt}{29.83348pt}{44.05986pt}{30.95276pt}{42.67914pt}{30.95276pt}\pgfsys@curveto{41.29842pt}{30.95276pt}{40.17914pt}{29.83348pt}{40.17914pt}{28.45276pt}\pgfsys@curveto{40.17914pt}{27.07204pt}{41.29842pt}{25.95276pt}{42.67914pt}{25.95276pt}\pgfsys@curveto{44.05986pt}{25.95276pt}{45.17914pt}{27.07204pt}{45.17914pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{2.5pt}{28.45276pt}\pgfsys@curveto{2.5pt}{29.83348pt}{1.38072pt}{30.95276pt}{0.0pt}{30.95276pt}\pgfsys@curveto{-1.38072pt}{30.95276pt}{-2.5pt}{29.83348pt}{-2.5pt}{28.45276pt}\pgfsys@curveto{-2.5pt}{27.07204pt}{-1.38072pt}{25.95276pt}{0.0pt}{25.95276pt}\pgfsys@curveto{1.38072pt}{25.95276pt}{2.5pt}{27.07204pt}{2.5pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.37015pt}{-6.0566pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\psi$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-53.40865pt}{21.4795pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.7295pt}{21.4795pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{46.17601pt}{21.4795pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{85.35828pt}\pgfsys@lineto{0.0pt}{71.1319pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@moveto{2.5pt}{71.1319pt}\pgfsys@curveto{2.5pt}{72.51262pt}{1.38072pt}{73.6319pt}{0.0pt}{73.6319pt}\pgfsys@curveto{-1.38072pt}{73.6319pt}{-2.5pt}{72.51262pt}{-2.5pt}{71.1319pt}\pgfsys@curveto{-2.5pt}{69.75117pt}{-1.38072pt}{68.6319pt}{0.0pt}{68.6319pt}\pgfsys@curveto{1.38072pt}{68.6319pt}{2.5pt}{69.75117pt}{2.5pt}{71.1319pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-40.00865pt}{72.17406pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\shuffle_{3}\circ\Delta_{3}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{0.0pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{-14.22638pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{-28.45276pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{-42.67914pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{-56.90552pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{14.22638pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{28.45276pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{42.67914pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{56.90552pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to114.21pt{\vbox to114.21pt{\pgfpicture\makeatletter\hbox{\hskip
57.10551pt\lower-28.65276pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{-56.90552pt}{42.67914pt}\pgfsys@lineto{-42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-42.67914pt}{42.67914pt}\pgfsys@lineto{-42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-28.45276pt}{42.67914pt}\pgfsys@lineto{-42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-42.67914pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-14.22638pt}{42.67914pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{42.67914pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{42.67914pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{56.90552pt}{42.67914pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{42.67914pt}{42.67914pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{28.45276pt}{42.67914pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.5pt}{0.0pt}\pgfsys@curveto{2.5pt}{1.38072pt}{1.38072pt}{2.5pt}{0.0pt}{2.5pt}\pgfsys@curveto{-1.38072pt}{2.5pt}{-2.5pt}{1.38072pt}{-2.5pt}{0.0pt}\pgfsys@curveto{-2.5pt}{-1.38072pt}{-1.38072pt}{-2.5pt}{0.0pt}{-2.5pt}\pgfsys@curveto{1.38072pt}{-2.5pt}{2.5pt}{-1.38072pt}{2.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-42.67914pt}{28.45276pt}\pgfsys@moveto{-40.17914pt}{28.45276pt}\pgfsys@curveto{-40.17914pt}{29.83348pt}{-41.29842pt}{30.95276pt}{-42.67914pt}{30.95276pt}\pgfsys@curveto{-44.05986pt}{30.95276pt}{-45.17914pt}{29.83348pt}{-45.17914pt}{28.45276pt}\pgfsys@curveto{-45.17914pt}{27.07204pt}{-44.05986pt}{25.95276pt}{-42.67914pt}{25.95276pt}\pgfsys@curveto{-41.29842pt}{25.95276pt}{-40.17914pt}{27.07204pt}{-40.17914pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{-42.67914pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@moveto{45.17914pt}{28.45276pt}\pgfsys@curveto{45.17914pt}{29.83348pt}{44.05986pt}{30.95276pt}{42.67914pt}{30.95276pt}\pgfsys@curveto{41.29842pt}{30.95276pt}{40.17914pt}{29.83348pt}{40.17914pt}{28.45276pt}\pgfsys@curveto{40.17914pt}{27.07204pt}{41.29842pt}{25.95276pt}{42.67914pt}{25.95276pt}\pgfsys@curveto{44.05986pt}{25.95276pt}{45.17914pt}{27.07204pt}{45.17914pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{2.5pt}{28.45276pt}\pgfsys@curveto{2.5pt}{29.83348pt}{1.38072pt}{30.95276pt}{0.0pt}{30.95276pt}\pgfsys@curveto{-1.38072pt}{30.95276pt}{-2.5pt}{29.83348pt}{-2.5pt}{28.45276pt}\pgfsys@curveto{-2.5pt}{27.07204pt}{-1.38072pt}{25.95276pt}{0.0pt}{25.95276pt}\pgfsys@curveto{1.38072pt}{25.95276pt}{2.5pt}{27.07204pt}{2.5pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.7295pt}{-6.97325pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-53.04929pt}{22.39616pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\psi$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.7295pt}{21.4795pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{46.17601pt}{21.4795pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{85.35828pt}\pgfsys@lineto{0.0pt}{71.1319pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@moveto{2.5pt}{71.1319pt}\pgfsys@curveto{2.5pt}{72.51262pt}{1.38072pt}{73.6319pt}{0.0pt}{73.6319pt}\pgfsys@curveto{-1.38072pt}{73.6319pt}{-2.5pt}{72.51262pt}{-2.5pt}{71.1319pt}\pgfsys@curveto{-2.5pt}{69.75117pt}{-1.38072pt}{68.6319pt}{0.0pt}{68.6319pt}\pgfsys@curveto{1.38072pt}{68.6319pt}{2.5pt}{69.75117pt}{2.5pt}{71.1319pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-40.00865pt}{72.17406pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\shuffle_{3}\circ\Delta_{3}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{0.0pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{-14.22638pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{-28.45276pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{-42.67914pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{-56.90552pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{14.22638pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{28.45276pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{42.67914pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{56.90552pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$
$-\leavevmode\hbox to114.21pt{\vbox
to114.21pt{\pgfpicture\makeatletter\hbox{\hskip
57.10551pt\lower-28.65276pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{-56.90552pt}{42.67914pt}\pgfsys@lineto{-42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-42.67914pt}{42.67914pt}\pgfsys@lineto{-42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-28.45276pt}{42.67914pt}\pgfsys@lineto{-42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-42.67914pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-14.22638pt}{42.67914pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{42.67914pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{42.67914pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{56.90552pt}{42.67914pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{42.67914pt}{42.67914pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{28.45276pt}{42.67914pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.5pt}{0.0pt}\pgfsys@curveto{2.5pt}{1.38072pt}{1.38072pt}{2.5pt}{0.0pt}{2.5pt}\pgfsys@curveto{-1.38072pt}{2.5pt}{-2.5pt}{1.38072pt}{-2.5pt}{0.0pt}\pgfsys@curveto{-2.5pt}{-1.38072pt}{-1.38072pt}{-2.5pt}{0.0pt}{-2.5pt}\pgfsys@curveto{1.38072pt}{-2.5pt}{2.5pt}{-1.38072pt}{2.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-42.67914pt}{28.45276pt}\pgfsys@moveto{-40.17914pt}{28.45276pt}\pgfsys@curveto{-40.17914pt}{29.83348pt}{-41.29842pt}{30.95276pt}{-42.67914pt}{30.95276pt}\pgfsys@curveto{-44.05986pt}{30.95276pt}{-45.17914pt}{29.83348pt}{-45.17914pt}{28.45276pt}\pgfsys@curveto{-45.17914pt}{27.07204pt}{-44.05986pt}{25.95276pt}{-42.67914pt}{25.95276pt}\pgfsys@curveto{-41.29842pt}{25.95276pt}{-40.17914pt}{27.07204pt}{-40.17914pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{-42.67914pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@moveto{45.17914pt}{28.45276pt}\pgfsys@curveto{45.17914pt}{29.83348pt}{44.05986pt}{30.95276pt}{42.67914pt}{30.95276pt}\pgfsys@curveto{41.29842pt}{30.95276pt}{40.17914pt}{29.83348pt}{40.17914pt}{28.45276pt}\pgfsys@curveto{40.17914pt}{27.07204pt}{41.29842pt}{25.95276pt}{42.67914pt}{25.95276pt}\pgfsys@curveto{44.05986pt}{25.95276pt}{45.17914pt}{27.07204pt}{45.17914pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{2.5pt}{28.45276pt}\pgfsys@curveto{2.5pt}{29.83348pt}{1.38072pt}{30.95276pt}{0.0pt}{30.95276pt}\pgfsys@curveto{-1.38072pt}{30.95276pt}{-2.5pt}{29.83348pt}{-2.5pt}{28.45276pt}\pgfsys@curveto{-2.5pt}{27.07204pt}{-1.38072pt}{25.95276pt}{0.0pt}{25.95276pt}\pgfsys@curveto{1.38072pt}{25.95276pt}{2.5pt}{27.07204pt}{2.5pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.7295pt}{-6.97325pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-53.40865pt}{21.4795pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.37015pt}{22.39616pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\psi$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{46.17601pt}{21.4795pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{85.35828pt}\pgfsys@lineto{0.0pt}{71.1319pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@moveto{2.5pt}{71.1319pt}\pgfsys@curveto{2.5pt}{72.51262pt}{1.38072pt}{73.6319pt}{0.0pt}{73.6319pt}\pgfsys@curveto{-1.38072pt}{73.6319pt}{-2.5pt}{72.51262pt}{-2.5pt}{71.1319pt}\pgfsys@curveto{-2.5pt}{69.75117pt}{-1.38072pt}{68.6319pt}{0.0pt}{68.6319pt}\pgfsys@curveto{1.38072pt}{68.6319pt}{2.5pt}{69.75117pt}{2.5pt}{71.1319pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-40.00865pt}{72.17406pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\shuffle_{3}\circ\Delta_{3}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{0.0pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{-14.22638pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{-28.45276pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{-42.67914pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{-56.90552pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{14.22638pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{28.45276pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{42.67914pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{56.90552pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}-\leavevmode\hbox
to114.21pt{\vbox to114.21pt{\pgfpicture\makeatletter\hbox{\hskip
57.10551pt\lower-28.65276pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{}{}\pgfsys@moveto{-56.90552pt}{42.67914pt}\pgfsys@lineto{-42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-42.67914pt}{42.67914pt}\pgfsys@lineto{-42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-28.45276pt}{42.67914pt}\pgfsys@lineto{-42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-42.67914pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{-14.22638pt}{42.67914pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{42.67914pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{14.22638pt}{42.67914pt}\pgfsys@lineto{0.0pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{56.90552pt}{42.67914pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{42.67914pt}{42.67914pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{28.45276pt}{42.67914pt}\pgfsys@lineto{42.67914pt}{28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-28.45276pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.5pt}{0.0pt}\pgfsys@curveto{2.5pt}{1.38072pt}{1.38072pt}{2.5pt}{0.0pt}{2.5pt}\pgfsys@curveto{-1.38072pt}{2.5pt}{-2.5pt}{1.38072pt}{-2.5pt}{0.0pt}\pgfsys@curveto{-2.5pt}{-1.38072pt}{-1.38072pt}{-2.5pt}{0.0pt}{-2.5pt}\pgfsys@curveto{1.38072pt}{-2.5pt}{2.5pt}{-1.38072pt}{2.5pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{-42.67914pt}{28.45276pt}\pgfsys@moveto{-40.17914pt}{28.45276pt}\pgfsys@curveto{-40.17914pt}{29.83348pt}{-41.29842pt}{30.95276pt}{-42.67914pt}{30.95276pt}\pgfsys@curveto{-44.05986pt}{30.95276pt}{-45.17914pt}{29.83348pt}{-45.17914pt}{28.45276pt}\pgfsys@curveto{-45.17914pt}{27.07204pt}{-44.05986pt}{25.95276pt}{-42.67914pt}{25.95276pt}\pgfsys@curveto{-41.29842pt}{25.95276pt}{-40.17914pt}{27.07204pt}{-40.17914pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{-42.67914pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@moveto{45.17914pt}{28.45276pt}\pgfsys@curveto{45.17914pt}{29.83348pt}{44.05986pt}{30.95276pt}{42.67914pt}{30.95276pt}\pgfsys@curveto{41.29842pt}{30.95276pt}{40.17914pt}{29.83348pt}{40.17914pt}{28.45276pt}\pgfsys@curveto{40.17914pt}{27.07204pt}{41.29842pt}{25.95276pt}{42.67914pt}{25.95276pt}\pgfsys@curveto{44.05986pt}{25.95276pt}{45.17914pt}{27.07204pt}{45.17914pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{42.67914pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@moveto{2.5pt}{28.45276pt}\pgfsys@curveto{2.5pt}{29.83348pt}{1.38072pt}{30.95276pt}{0.0pt}{30.95276pt}\pgfsys@curveto{-1.38072pt}{30.95276pt}{-2.5pt}{29.83348pt}{-2.5pt}{28.45276pt}\pgfsys@curveto{-2.5pt}{27.07204pt}{-1.38072pt}{25.95276pt}{0.0pt}{25.95276pt}\pgfsys@curveto{1.38072pt}{25.95276pt}{2.5pt}{27.07204pt}{2.5pt}{28.45276pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{28.45276pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.7295pt}{-6.97325pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-53.40865pt}{21.4795pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.7295pt}{21.4795pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{46.53537pt}{22.39616pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\psi$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{85.35828pt}\pgfsys@lineto{0.0pt}{71.1319pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@moveto{2.5pt}{71.1319pt}\pgfsys@curveto{2.5pt}{72.51262pt}{1.38072pt}{73.6319pt}{0.0pt}{73.6319pt}\pgfsys@curveto{-1.38072pt}{73.6319pt}{-2.5pt}{72.51262pt}{-2.5pt}{71.1319pt}\pgfsys@curveto{-2.5pt}{69.75117pt}{-1.38072pt}{68.6319pt}{0.0pt}{68.6319pt}\pgfsys@curveto{1.38072pt}{68.6319pt}{2.5pt}{69.75117pt}{2.5pt}{71.1319pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-40.00865pt}{72.17406pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\shuffle_{3}\circ\Delta_{3}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{0.0pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{-14.22638pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{-28.45276pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{-42.67914pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{-56.90552pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{14.22638pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{28.45276pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{42.67914pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{71.1319pt}\pgfsys@lineto{56.90552pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

Figure 1. String diagram for the $2$-cocycle condition of a $2$-cochain
$\psi$. The strings are permuted using the natural shuffling morphism for the
category of modules.

######  Lemma 4.1.

The differential $\delta^{1}$ maps $1$-cochains into $C^{2}_{\rm TSD}(X;X)$.
Moreover, the composition of differentials $\delta^{2}\circ\delta^{1}$ is
trivial.

###### Proof.

This is a long direct computation which we postpone to the Appendix. ∎

As a consequence of Lemma 4.1, the following is well posed.

######  Definition 4.2.

Let $X$ be an $n$-SD structure over the unital ring $\mathbb{k}$. Following
traditional notation, we set $Z^{2}_{n\rm SD}(X;X)={\rm ker}\ \delta^{2}$, and
$B^{2}_{n\rm SD}(X;X)={\rm im}\ \delta^{1}$. The second cohomology group of
$X$ with coefficients in $X$ is then the quotient $H^{2}_{n\rm
SD}(X;X)=Z^{2}_{n\rm SD}(X;X)/B^{2}_{n\rm SD}(X;X)$. For the ternary case we
will use the notation $H^{2}_{\rm TSD}(X;X)$, and similarly for cocycles and
coboundaries.

Straightforward modifications generalize the previous definitions to the case
of cohomology of $X$ with coefficients in a TSD object $A$ different from $X$
itself. The proof of Lemma 4.1 is easily adapted to this case.

######  Remark 4.3.

We observe that Definition 4.2 does not explicitly require $\Delta$ to be the
comultiplication given above, as long as $(T,\Delta,\epsilon)$ is an SD
object. In fact, Lemma 4.1 does not need any special consideration regarding
the actual form of $\Delta$ to hold, but only uses the SD property (see
Appendix). This fact is going to play a role in the simultaneous deformations
considered in Section 8. Until then, we will consider the special form for
$\Delta$ described above.

######  Lemma 4.4.

Let $V$ be a vector space and let $X=\mathbb{k}\oplus V$ be the coalgebra with
comultiplication $\Delta(a,x)=(a,x)\otimes(1,0)+(1,0)\otimes(0,x)$. Then, a
linear map $f:X\longrightarrow X$ is an $n$-ary coderivation if and only if it
is of type $f(a,x)=(0,g(x))$ for some linear map $g:V\longrightarrow V$.

###### Proof.

We consider the binary case, since the $n$-ary one is substantially similar.
This is a direct inspection of the coderivation condition $\Delta
f(a,x)=(f\otimes\mathbb{1}+\mathbb{1}\otimes f)\Delta(a,x)$, where
$f(a,x)=f_{0}(a,x)+f_{1}(a,x)$ for maps $f_{0}:X\longrightarrow\mathbb{k}$ and
$f_{1}:X\longrightarrow V$. From the definition of $\Delta$, we have that the
LHS of the coderivation condition is

$\Delta f(a,x)=f(a,x)\otimes(1,0)+(1,0)\otimes(0,f_{1}(a,x)),$

while for the RHS we have

$\displaystyle(f\otimes\mathbb{1}+\mathbb{1}\otimes f)\Delta(a,x)$
$\displaystyle=$ $\displaystyle(f\otimes\mathbb{1}+\mathbb{1}\otimes
f)[(a,x)\otimes(1,0)+(1,0)\otimes(0,x)]$ $\displaystyle=$ $\displaystyle
f(a,x)\otimes(1,0)+f(1,0)\otimes(0,x)+(a,x)\otimes f(1,0)+(1,0)\otimes
f(0,x).$

Therefore, the coderivation condition becomes

$(1,0)\otimes(0,f_{1}(a,x))=f(1,0)\otimes(0,x)+(a,x)\otimes
f(1,0)+(1,0)\otimes f(0,x).$

Since this equation should hold for all $x\in V$, it follows that $f(1,0)=0$,
since otherwise the RHS woud have a component in $V\otimes X$ which could not
vanish. It follows that the coderivation condition gives

$(1,0)\otimes(0,f_{1}(a,x))=(1,0)\otimes f(0,x)$

which, writing $(a,x)$ as a sum in $X$, holds if and only if
$af_{1}(1,0)+f_{1}(0,x)=f_{0}(0,x)+f_{1}(0,x)$. Therefore, we have found that
it must hold $af(1,0)=f_{0}(0,x)$ which forces $f_{0}(0,x)=0$, completing the
proof. ∎

Applying Lemma 4.4, it follows that the space of $1$-cochains for the SD
cohomology consists of linear maps $f:X\longrightarrow X$ that factors through
the Lie algebra $\mathfrak{g}$ as

${X}$${X}$${\mathfrak{g}}$${\mathfrak{g}}$$\scriptstyle{f}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\tilde{f}}$$\scriptstyle{\iota_{2}}$

for some $\tilde{f}$, where $\pi_{2}$ is the projection on the second summand
of $X=\mathbb{k}\oplus\mathfrak{g}$, and $\iota_{2}$ is the obvious injection
$\mathfrak{g}\hookrightarrow X=\mathbb{k}\oplus\mathfrak{g}$. In particular,
this provides a direct criterion for an SD $2$-cocycle to be nontrivial as
follows. Lemma 4.4 applied to a $1$-cochain $f$, along with the definition of
differential $\delta_{SD}^{1}f$, shows that $\delta_{SD}^{1}f$ is trivial on
$\mathbb{k}\otimes X$, and morover it has image on
$\mathfrak{g}\otimes\mathfrak{g}$. Therefore, any $2$-cocycle $\phi$ that is
nontrivial on a simple tensor of type $(1,0)\otimes(a,x)$, or such that $\phi$
has nontrivial image in $\mathbb{k}\otimes X+X\otimes\mathbb{k}$, is not
cobounded. It therefore represents a nontrivial cohomological class in
$H^{2}_{\rm SD}(X;X)$. In fact, it turns out that there are examples of
$2$-cocycles of this sort. For instance, the Lie algebra Bianchi IV over the
complex field has two nontrivial cohomology classes of this type, cf. Section
9.

######  Remark 4.5.

We point out that a formulation of higher order cochains (along with their
corresponding cohomology groups) might be attempted along the same lines,
although we have not explicitly tried so. However, we are not aware of current
potential applications of cohomology groups of degree larger than $3$. For
degree $3$, J. Stasheff has pointed to us the possible interesting application
of Gerstenhaber bracket type obstructions (elements of the third cohomology
groups) to extending the deformations to higher orders. This is a particularly
interesting perspective when considering the Yang-Baxter operator deformations
discussed in Section 10 below, especially with the target of producing link
invariants, from arbitrarily deformed YB operators. We have not yet explored
this promising line of research.

##  5\. Infinitesimal deformations of SD structures

Once again, let us consider the ternary case for simplicity, even though the
results of this section generalize to the $n$-ary case following an analogous
reasoning. We show that the second cohomology group $H^{2}_{\rm TSD}(X;X)$ of
a TSD object with coefficients in itself classifies the infinitesimal
deformations of $X$, as for associative and Lie algebras.

The notion of infinitesimal deformation for a TSD object follows the same
lines as for $n$-Lie algebras above. Namely, given a TSD object $X$ over a
field $\mathbb{k}$, we extend coefficients of $X$ to
$\mathbb{k}[[\hbar]]/(\hbar^{2})$, and let us denote it as $X_{\hbar}$. We
define a new ternary operation on $X_{\hbar}$ as a linear map
$T^{\prime}=T+\hbar\psi_{1}$, where $\psi_{1}:X^{\otimes 3}\longrightarrow X$.
We say that $T^{\prime}$ is an infinitesimal deformation of $T$ if
$T^{\prime}$ turns $X_{\hbar}$ into a TSD object, where the comultiplication
of $X_{\hbar}$ is that of $X$, $\Delta_{3}$, extended to $X_{\hbar}$ by
linearity. Two infinitesimal deformations $T^{\prime}$ and $T^{\prime\prime}$
of $T$ are said to be equivalent if there exists an ismorphism between
$(X_{\hbar},T^{\prime})$ and $(X_{\hbar},T^{\prime\prime})$ fixing the
underlying zero degree structure.

######  Theorem 5.1.

Let $X$ be a TSD object over $\mathbb{k}$. Then, the cohomology group
$H^{2}_{\rm TSD}(X;X)$ classifies the infinitesimal deformations of $X$.

###### Proof.

First, we show that $T+\hbar\psi$ defines an infinitesimal deformation of $T$
if and only if $\psi\in Z^{2}_{\rm TSD}(X;X)$. Let $x\otimes y\otimes z\otimes
w\otimes u\in X^{\otimes 5}$ be a simple tensor. The left hand side of TSD
condition on $T^{\prime}=T+\hbar\psi$ gives

$\displaystyle T^{\prime}(T^{\prime}(x\otimes y\otimes z)\otimes w\otimes u)$
$\displaystyle=$ $\displaystyle T(T(x\otimes y\otimes z)\otimes w\otimes
u)+\hbar T(\psi(x\otimes y\otimes z)\otimes w\otimes u)$
$\displaystyle+\hbar\psi(T(x\otimes y\otimes z)\otimes w\otimes
u)+\hbar^{2}\psi(\psi(x\otimes y\otimes z)\otimes w\otimes u).$

The right hand side gives

$\displaystyle T^{\prime}(T^{\prime}(x\otimes w^{(1)}\otimes u^{(1)})\otimes
T^{\prime}(y\otimes w^{(2)}\otimes u^{(2)})\otimes T^{\prime}(z\otimes
w^{(3)}\otimes u^{(3)}))$ $\displaystyle=$ $\displaystyle T(T(x\otimes
w^{(1)}\otimes u^{(1)})\otimes T(y\otimes w^{(2)}\otimes u^{(2)})\otimes
T(z\otimes w^{(3)}\otimes u^{(3)}))$ $\displaystyle+\hbar\psi(T(x\otimes
w^{(1)}\otimes u^{(1)})\otimes T(y\otimes w^{(2)}\otimes u^{(2)})\otimes
T(z\otimes w^{(3)}\otimes u^{(3)}))$ $\displaystyle+\hbar T(\psi(x\otimes
w^{(1)}\otimes u^{(1)})\otimes T(y\otimes w^{(2)}\otimes u^{(2)})\otimes
T(z\otimes w^{(3)}\otimes u^{(3)}))$ $\displaystyle+\hbar T(T(x\otimes
w^{(1)}\otimes u^{(1)})\otimes\psi(y\otimes w^{(2)}\otimes u^{(2)})\otimes
T(z\otimes w^{(3)}\otimes u^{(3)}))$ $\displaystyle+T(T(x\otimes
w^{(1)}\otimes u^{(1)})\otimes T(y\otimes w^{(2)}\otimes
u^{(2)})\otimes\psi(z\otimes w^{(3)}\otimes u^{(3)}))$
$\displaystyle+\hbar^{2}(\cdots)+\hbar^{3}(\cdots),$

where the terms in brackets correspond to terms where $\psi$ appears twice and
thrice. Since higher powers of $\hbar$ are zero, equating terms of degree one
we find that $T^{\prime}$ satisfies the TSD condition if and only if $\psi$
satisfies the $2$-cocycle condition. A similar direct computation also shows
that $T^{\prime}$ is a coalgebra morphism if and only if $\psi$ satisfies the
condition $\Delta_{1}\psi=(\psi\otimes T^{\otimes 2}+T\otimes\psi\otimes
T+T^{\otimes 2}\otimes\psi)\shuffle\Delta_{3}^{\otimes 3}$.

To complete the proof, we need to show that a deformation is equivalent to the
trivial deformation if and only if the corresponding $2$-cocycle $\psi$ is
cobounded by some coderivation $f$. In fact, suppose that
$T^{\prime}=T+\hbar\psi$ is equivalent to the trivial deformation. Then, by
definition, there exists a $\mathbb{k}[[\hbar]]/(\hbar^{2})$-linear map $g$ of
the form $g=\mathbb{1}+\hbar f$ such that $g$ is a coalgebra morphism and a
TSD morphism. The latter condition enforces that
$g(T+\hbar\psi)(g^{-1}(x)\otimes g^{-1}(y)\otimes g^{-1}(z))=T(x\otimes
y\otimes z)$. Expanding $g^{-1}$ as a geometric series in $\hbar$ and grouping
terms of order one, we see that $\psi=\delta^{1}f$. Imposing that $g$ is a
coalgebra morphism and, again, taking terms of degree $1$ shows that $f$ is a
coderivation. It follows that $[\psi]=0$ in $H^{2}_{\rm TSD}(X;X)$. A similar
reasoning also shows that coboundaries give rise to trivial deformations,
completing the proof. ∎

##  6\. From Lie algebra cohomology to SD cohomology

In this section we construct homomorphisms between the second cohomology
groups of a Lie algebra and the corresponding SD structures. Observe here that
we consider maps from binary and ternary Lie algebras to both binary and
ternary SD structures, along with maps from ternary Lie algebras to ternary SD
structures. In fact, as seen in [ESZ], composing $n$-SD structures produces
$(2n-1)$-SD structures, therefore from binary Lie algebras one gets $3$-SD
structures as well. The results of this section can be generalized to maps (in
cohomology) between $n$-Lie algebras and $n$-SD as well as $(2n-1)$-SD
structures. In fact, it is reasonable to assume that the same procedure can be
iterated deriving maps to higher order SD structures, even though we have not
explicitly verified that this is the case. This is a relevant question which
we leave to future work.

###  6.1. Binary Lie algebras

Let $\mathfrak{g}$ be a Lie algebra, and let $X=\mathbb{C}\oplus\mathfrak{g}$
denote the associated BSD structure. Then, we define a map
$\Psi^{2}:C^{2}_{\rm Lie}(\mathfrak{g};\mathfrak{g})\longrightarrow C^{2}_{\rm
BSD}(X;X)$ as follows. On $2$-chains $(a,x)\otimes(b,y)$ we set
$\Psi^{2}(\phi)(a,x)\otimes(b,y)=(0,\phi(x,y))$, where $\phi$ denotes a Lie
algebra $2$-cocycle with coefficients in $\mathfrak{g}$.

######  Proposition 6.1.

Let $\mathfrak{g}$, $X$ and $\Psi^{2}$ be as above. Then $\Psi^{2}$ induces a
well defined map between cohomology groups $H^{2}_{\rm
Lie}(\mathfrak{g};\mathfrak{g})$ and $H^{2}_{\rm TSD}(X;X)$.

In what follows we use the notation
$\phi((a,x)\otimes(b,y))=(\phi_{0}((a,x)\otimes(b,y)),\phi_{1}((a,x)\otimes(b,y)))$,
for $2$-cochains, and when no confusion arises, we simply write $a+x$ instead
of $(a,x)$ to indicate the direct sum $\mathbb{k}\oplus\mathfrak{g}$. So,
$\phi_{1}((a,x)\otimes(b,y))$ will denote $(0,\phi_{1}((a,x)\otimes(b,y)))$,
for instance.

######  Definition 6.2.

Let $X$ denote a BSD structure corresponding to a Lie algebra $\mathfrak{g}$.
Let $\phi\in C^{2}_{\rm BSD}(X;X)$. Then, if
$\hat{\psi}(a,x)\otimes(b,y)=(0,\psi(x,y))$ for all $(a,x)$ and $(b,y)$, we
say that the cohomology class $[\hat{\psi}]$ is special, and we simply denote
it by the map $\psi$.

The following lemma will be key in proving the main result of the section.

######  Lemma 6.3.

Let $\mathfrak{g}$ be a Lie algebra and $X$ the associated BSD object. Then,
any $2$-cochain $\phi\in C^{2}(X;X)$ is such that
$\phi((a,x)\otimes(b,y))=\phi_{1}((a,x)\otimes(b,y))$, and moreover
$\phi_{1}(\mathbb{k}\otimes X)=0$. If in addition $\mathfrak{g}$ has trivial
center, any $2$-cocycle $\phi\in Z^{2}_{\rm BSD}(X;X)$ is special.

###### Proof.

Let $\phi$ denote a $2$-cochain representing a cohomology class $[\phi]\in
H^{2}_{\rm BSD}(X;X)$. Then, by definition of $C^{2}_{\rm BSD}(X;X)$ we have
that $\phi$ satisfies the equation

$\Delta\phi((a,x)\otimes(b,y))=[\phi\otimes
q+q\otimes\phi](\mathbb{1}\otimes\tau\otimes\mathbb{1})(\Delta\otimes\Delta)((a,x)\otimes(b,y)),$

where $q:X\times X\rightarrow X$ is given by
$q((a,x)\otimes(b,y))=(ab,bx+[x,y])$. Using the definitions, the $2$-cochain
condition for $\phi$ becomes

$\displaystyle\phi((a,x)\otimes(b,y))\otimes(1,0)+(1,0)\otimes\phi_{1}((a,x)\otimes(b,y))$
$\displaystyle=$
$\displaystyle\phi((a,x)\otimes(b,y))\otimes(1,0)+\phi((a,x)\otimes(1,0))\otimes
q((1,0)\otimes(0,y))$
$\displaystyle+\phi((1,0)\otimes(b,y))\otimes(0,x)+\phi((1,0)\otimes(1,0))\otimes(0,[x,y])$
$\displaystyle+(ab,bx+[x,y])\otimes\phi((1,0)\otimes(1,0))+(a,x)\otimes\phi((1,0)\otimes(0,y))$
$\displaystyle+b(1,0)\otimes\phi((0,x)\otimes(1,0))+(1,0)\otimes\phi((0,x)\otimes(0,y)),$

and this is readily rewritten as

$\displaystyle(1,0)\otimes\phi_{1}((a,x)\otimes(b,y))$ $\displaystyle=$
$\displaystyle+\phi((1,0)\otimes(b,y))\otimes(0,x)+\phi((1,0)\otimes(1,0))\otimes(0,[x,y])$
$\displaystyle+(ab,bx+[x,y])\otimes\phi((1,0)\otimes(1,0))+(a,x)\otimes\phi((1,0)\otimes(0,y))$
$\displaystyle+b(1,0)\otimes\phi((0,x)\otimes(1,0))+(1,0)\otimes\phi((0,x)\otimes(0,y)).$

Let us take $y=0$ in the previous equation. Then, the RHS has a component of
type $(0,x)\otimes(c,z)$, coming from
$(ab,bx+[x,y])\otimes\phi((1,0)\otimes(1,0))$, which is not present on the
LHS. This forces $\phi((1,0)\otimes(1,0))=0$. The $2$-cochain condition,
therefore, becomes

$\displaystyle(1,0)\otimes\phi_{1}((a,x)\otimes(b,y))$ $\displaystyle=$
$\displaystyle\phi((1,0)\otimes(0,y))\otimes(0,x)+(a,x)\otimes\phi((1,0)\otimes(0,y))$
$\displaystyle+b(1,0)\otimes\phi((0,x)\otimes(1,0))+(1,0)\otimes\phi((0,x)\otimes(0,y)),$

where we have also simplified the term $\phi((1,0)\otimes(b,y))\otimes(0,x)$
to $\phi((1,0)\otimes(0,y))\otimes(0,x)$ by virtue of the vanishing of
$\phi((1,0)\otimes(1,0))$. But then this forces $\phi((1,0)\otimes(0,y))$ to
be trivial, since otherwise $(a,x)\otimes\phi((1,0)\otimes(0,y))$ would have a
component of type $(a,x)\otimes(c,z)$ which has no counterpart in the LHS. We
have obtained

$\displaystyle(1,0)\otimes\phi_{1}((a,x)\otimes(b,y))$ $\displaystyle=$
$\displaystyle
b(1,0)\otimes\phi((0,x)\otimes(1,0))+(1,0)\otimes\phi((0,x)\otimes(0,y)).$

Now, let us consider the case $y=0$. From this we obtain that

$b(1,0)\otimes(0,\phi_{1}((0,x)\otimes(1,0)))=b(1,0)\otimes\phi((0,x)\otimes(1,0)),$

which shows that $\phi_{0}(X\otimes\mathbb{k})=0$. Similarly, setting $b=0$
gives that $\phi((0,x)\otimes(0,y))=\phi_{1}((0,x)\otimes(0,y))$, showing that
$\phi_{0}=0$ on $\mathfrak{g}\otimes\mathfrak{g}$ as well. This completes the
first part of the statement.

Let us now suppose that $\mathfrak{g}$ has trivial center, and that $\phi$ is
a $2$-cocycle. Applying the $2$-cocycle condition on simple tensors of type
$(0,x)\otimes(1,0)\otimes(0,z)$ we find

$\displaystyle\phi(q((0,x)\otimes(1,0))\otimes(0,z))+q(\phi((0,x)\otimes(1,0))\otimes(0,z))$
$\displaystyle=$ $\displaystyle
q(\phi((0,x)\otimes(0,z))\otimes(1,0))+q(\phi((0,x)\otimes(1,0))\otimes
q((1,0)\otimes(0,z)))$
$\displaystyle+q(q((0,x)\otimes(0,z))\otimes\phi((1,0)\otimes(1,0)))+q((0,x)\otimes\phi((1,0)\otimes(1,0)))$
$\displaystyle+\phi(q((0,x)\otimes(0,z))\otimes(1,0))+\phi((0,x)\otimes
q((1,0)\otimes(0,z)))$ $\displaystyle=$
$\displaystyle\phi((0,x)\otimes(0,z))+\phi((0,[x,z])\otimes(1,0)),$

where we have used the conditions on $\phi$ proved in the first part of the
proof, and the definition of $q$, in particular the fact that
$q((1,0)\otimes(0,x))=0$ for all $x\in\mathfrak{g}$. The previous equality
then is seen to reduce to

$[\phi_{1}((0,x)\otimes(1,0)),z]=\phi_{1}((0,[x,z])\otimes(1,0)),$

which holds for arbitrary $x,z\in\mathfrak{g}$. We proceed similarly now on
simple tensors of type $(0,x)\otimes(0,y)\otimes(1,0)$, for which the
$2$-cocycle condition takes the form

$\displaystyle\phi(q((0,x)\otimes(0,x))\otimes(1,0))+q(\phi((0,x)\otimes(0,y))\otimes(1,0))$
$\displaystyle=$ $\displaystyle q(\phi((0,x)\otimes(1,0))\otimes(0,y))$
$\displaystyle+q((0,x)\otimes\phi((0,y)\otimes(1,0))$
$\displaystyle+\phi((0,x)\otimes(0,y)).$

Applying the condition obtained from the previous case, this equation is seen
to reduce to $0=(0,[x,\phi_{1}((0,y)\otimes(1,0))])$, which holds for all
$x,y\in\mathfrak{g}$. Therefore, $\phi_{1}$ maps
$\mathfrak{g}\otimes\mathbb{k}$ in the center of $\mathfrak{g}$, which is
trivial by hypothesis. Consequently,
$\phi_{1}(\mathfrak{g}\otimes\mathbb{k})=0$, and $\phi$ is special, completing
the proof. ∎

######  Theorem 6.4.

Let $\mathfrak{g}$ be a Lie algebra and let $X$ be its associated BSD object.
Then $\Psi^{2}$ induces a monomorphism $H^{2}_{\rm
Lie}(\mathfrak{g};\mathfrak{g})\longrightarrow H^{2}_{\rm BSD}(X;X)$ in
cohomology. Suppose, in addition, that $\mathfrak{g}$ has trivial center (e.g.
it is semisimple), then $H^{2}_{\rm Lie}(\mathfrak{g};\mathfrak{g})\cong
H^{2}_{\rm BSD}(X;X)$.

###### Proof.

Let us indicate the map $H^{2}_{\rm
Lie}(\mathfrak{g};\mathfrak{g})\longrightarrow H^{2}_{\rm BSD}(X;X)$ induced
by $\Psi^{2}$ in cohomology (from Proposition 6.1) by the same symbol
$\Psi^{2}$. We start by showing that $\Psi^{2}$ is injective. Let $\phi$ be a
representative of a cohomology class in $H^{2}_{\rm
Lie}(\mathfrak{g};\mathfrak{g})$ and suppose that $[\Psi^{2}(\phi)]=[0]$.
Then, there exists a $1$-cochain $f:X\longrightarrow X$ such that
$\Psi^{2}(\phi)=\delta^{1}f$. Observe that, by definition of $\Psi^{2}$,
$\Psi^{2}(\phi)$ is special. Therefore, it follows that $\delta_{SD}^{1}$ is
special as well, since it would not otherwise cobound $\Psi^{2}(\phi)$.
Consequently, we can write
$\delta^{1}f(a,x)\otimes(b,y)=(0,g(a,x)\otimes(b,y))$. A direct computation
shows that $g(a,x)\otimes(b,y)=\delta^{1}_{\rm Lie}f_{1}x\otimes y$. Then,
this means that $\phi=\delta^{1}_{\rm Lie}f_{1}$, which in turn implies that
$[\phi]=[0]$ in the Lie cohomology group. This shows that $\psi^{2}$ is
injective.

Let us now consider the second part of the statement. Surjectivity is derived
from the fact that cohomology classes of $X$ are special, i.e. Lemma 6.3 where
$\mathfrak{g}$ has trivial center. Let $[\psi]$ be a cohomology class in
$H^{2}_{\rm BSD}(X;X)$. Then, the fact that $\psi$ is special implies that
$\psi((a,x)\otimes(b,y))=\psi_{1}((0,x)\otimes(0,y))$. Then, we can define
$\sigma:\mathfrak{g}^{\otimes 2}\longrightarrow\mathfrak{g}$ by the assignment
$g(x\otimes y):=\psi_{1}((0,x)\otimes(0,y))$. It follows that
$\Psi^{2}(\sigma)=\psi$, showing that $\Psi^{2}$ is surjective. ∎

######  Remark 6.5.

We observe that Theorem 6.4 has a deep meaning in terms of deformation theory.
It substantially states, in fact, that the deformation theory of SD structures
coincides with the deformation theory of the Lie algebra that generates the SD
object.

We consider now the case of binary Lie algebras and TSD structure associated
to it via composition of binary self-disrtibutive operations. Let
$\mathfrak{g}$ denote a (binary) Lie algebra, and let $X$ denote the
corresponding TSD structure arising from $\mathfrak{g}$. Then, for a
$2$-cocycle $\phi\in Z^{2}_{\rm Lie}(\mathfrak{g},\mathfrak{g})$, we define a
map $\Theta(\phi):X^{\otimes 3}\longrightarrow X$ according to the assignment

$\Theta^{2}(\phi)((a,x)\otimes(b,y)\otimes(c,z))=(0,b\phi(x\otimes
z)+c\phi(x\otimes y)+[\phi(x,y),z]+\phi([x,y],z)).$

######  Proposition 6.6.

Let $\mathfrak{g}$, $X$ and $\Theta^{2}$ be as above. Then $\Theta^{2}$
induces a map between cohomology groups $H^{2}_{\rm
Lie}(\mathfrak{g};\mathfrak{g})$ and $H^{2}_{\rm TSD}(X;X)$.

###### Proof.

We need to show that $\Theta^{2}$ maps a $2$-cocycle $\phi$ to a $2$-cochain
of $X$, and that it satisfies the $2$-cocycle condition. This means that
$\Theta^{2}(\phi)$ must satisfy the property
$\Delta_{3}\Theta^{2}(\phi)=[\Theta^{2}(\phi)\otimes T^{\otimes
2}+T\otimes\Theta^{2}(\phi)\otimes T+T^{\otimes
2}\otimes\Theta^{2}(\phi)]\shuffle\Delta_{3}^{\otimes 3}$, and moreover
$\delta^{2}\Theta^{2}(\phi)=0$, where $\delta^{2}$ is the differential of
Section 5. Both properties are easily verified as follows. Let
$[\bullet,\bullet]_{\psi}$ denote the Lie alegbra structure obtained as a
deformation of the bracket $[\bullet,\bullet]$ of $\mathfrak{g}$, as an
application of Theorem 2.1. From Theorem 8.1 in [ESZ], the map
$T_{\phi}:X_{\hbar}^{\otimes 3}\longrightarrow X_{\hbar}$ associated to
$[\bullet,\bullet]_{\phi}$ turns $X_{\hbar}$ into a TSD object. Direct
inspection shows that $T_{\phi}=T+\hbar\Theta^{2}(\phi)$. Then, applying
Theorem 5.1 we conclude that $\Theta^{2}(\phi)$ satisfies the required
properties. To complete the proof, we need to show that
$\Theta^{2}(\delta^{1}f)$ is a coboundary for any
$f:\mathfrak{g}\longrightarrow\mathfrak{g}$. In fact, given such a $1$-cochain
$f$, we define $g:X\longrightarrow X$ as $g(a,x)=(0,f(x))$. Then, we see that
$\delta^{1}_{\rm
TSD}g((a,x)\otimes(b,y)\otimes(c,z)=(0,bf([x,y])+cf([x,y])+f([[x,y],z])-b[f(x),z]-c[f(x),y]-[[f(x),y],z]-c[x,f(y)]-[[x,f(y)],z]-b[x,f(z)]-[[x,y],f(z)])$.
It is a direct verification to see that the previous term coincides with
$\Theta^{2}(\delta^{1}_{\rm Lie}f)$ evaluated on
$(a,x)\otimes(b,y)\otimes(c,z)$, up to an overall negative sign. It follows
that setting $\Theta^{1}(f)(a,x)=(0,-f(x))$ gives a cochain map at level $1$
and $2$ between Lie cohomology and TSD cohomology, and $\Theta^{2}$ induces a
well defined map in cohomology. ∎

We now show that the morphism of Proposition 6.6 is injective.

######  Proposition 6.7.

The map $\Theta^{2}$ is an injection.

###### Proof.

Let $\phi$ be a Lie $2$-cocycle and suppose that $[\Theta^{2}(\phi)]=0$ in
$H^{2}_{\rm TSD}(X;X)$. Let $f$ be a $1$-cochain that cobounds
$\Theta^{2}(\phi)$. As in the proof of Theorem 6.4, since $\Theta^{2}$ maps
$2$-cochains to special ones, $\delta^{1}f$ needs to be special as well, which
in turn implies that $f(a,x)=(0,g(x))$ for some $g$. Then, the equation
$\delta^{1}_{\rm
TSD}f((a,x)\otimes(b,y)\otimes(c,z))=\Theta^{2}(\phi)((a,x)\otimes(b,y)\otimes(c,z))$
gives

$\displaystyle
bg([x,z])+cg([x,y])+g([[x,y],z])-b[g(x),z]-c[g(x),y]-[[g(x),y],z]$
$\displaystyle-b[x,z]-c[x,g(y)]-[[x,g(y)],z]-[x,g(z)]-c[x,y]-[[x,y],g(z)]$
$\displaystyle=$ $\displaystyle b\phi(x\otimes z)+c\phi(x\otimes
y)+[\phi(x\otimes y),z]+\phi([x,y]\otimes z).$

Considering the case $(1,x)\otimes(1,y)\otimes(1,0)$ reduces to the equation

$g([x,y])-[g(x),y]-[x,g(y)]=\phi(x\otimes y),$

which shows that $\phi=\delta^{1}_{\rm Lie}g$, therefore completing the proof.
∎

###  6.2. $3$-Lie algebras

A construction similar to that of $2$-Lie algebras is applicable in the case
of $3$-Lie algebras and their corresponding TSD objects. The main difference
is that due to the fact that the $3$-bracket is not obtained as the
composition of lower degree ones, the chain map involved is simpler than that
of Proposition 6.6.

Let $\mathfrak{g}$ be a $3$-Lie algebra and let $X$ denote the corresponding
TSD object. Given a $2$-cocycle $\phi:\mathfrak{g}^{\otimes
3}\longrightarrow\mathfrak{g}$ of $\mathfrak{g}$, we define the map
$\Lambda^{2}(\phi):X^{\otimes 3}\longrightarrow X$ according to

$\Lambda^{2}(\phi)((a,x)\otimes(b,y)\otimes(c,z))=(0,\phi(x\otimes y\otimes
z)).$

######  Proposition 6.8.

Let $\mathfrak{g}$, $X$ and $\Lambda^{2}$ be as above. Then $\Lambda^{2}$
induces a map between cohomology groups $H^{2}_{3{\rm
Lie}}(\mathfrak{g};\mathfrak{g})$ and $H^{2}_{\rm TSD}(X;X)$.

###### Proof.

The proof is analogous to that of Proposition 6.6. First one shows that the
TSD structure induced by a deformed $3$-Lie bracket can be written as
$T+\hbar\Lambda^{2}(\phi)$, and then one sees that $\Lambda^{2}$ can be
extended to a cochain map at $n=1,2$, for some $\Lambda^{1}$. The
corresponding computations are easier than those in Proposition 6.6. ∎

Recall ([Fil]) that for a ternary Lie algebra $\mathfrak{g}$, we say that the
$3$-bracket $[\bullet,\bullet,\bullet]$ has trivial center if $[x,y,z]=0$ for
all $x$ implies either $y=0$, or $z=0$. The following result is completely
analogous to Theorem 6.4 and it is proved following similar techniques adapted
to ternary operations.

######  Theorem 6.9.

Let $\mathfrak{g}$ be a ternary Lie algebra and let $X$ denote the associated
TSD structure. Then, there is a monomorphism $\Lambda^{2}:H^{2}_{3\rm
Lie}(\mathfrak{g};\mathfrak{g})\longrightarrow H^{2}_{\rm TSD}(X;X)$.
Moreover, if $\mathfrak{g}$ has trivial center, $\Lambda^{2}$ gives an
isomorphism $H^{2}_{3\rm Lie}(\mathfrak{g};\mathfrak{g})\cong H^{2}_{\rm
TSD}(X;X)$.

##  7\. Coalgebra deformations

We consider now the problem of deforming the coalgebra structure of a
coalgebra $(X,\Delta)$. In the present section, the map $T$ of a TSD object
does not appear, and the deformations only refer to the comultiplication and
counit structures. We will consider the simultaneous deformation of coalgebra
and SD structures afterwards.

Let $\Delta^{\prime}=\Delta+\hbar\psi$ denote a deformed comultiplication, and
let $\epsilon^{\prime}=\epsilon+\hbar\iota$ be a deformed counit. Then, we
have the following.

######  Lemma 7.1.

Let $(X,\Delta,\epsilon)$ be a coalgebra over $\mathbb{k}$, and let
$\Delta^{\prime}$, $\epsilon^{\prime}$ be deformations over
$\mathbb{k}^{\prime}$ as above. Then,
$(X^{\prime},\Delta^{\prime},\epsilon^{\prime})$ is a coalgebra if and only if

(1)
$\displaystyle(\mathbb{1}\otimes\Delta)\psi+(\mathbb{1}\otimes\psi)\Delta=(\Delta\otimes\mathbb{1})\psi+(\psi\otimes\mathbb{1})\Delta,$

and

(2)
$\displaystyle(\epsilon\otimes\mathbb{1})\psi+(\iota\otimes\mathbb{1})\Delta=0$
(3)
$\displaystyle(\mathbb{1}\otimes\epsilon)\psi+(\mathbb{1}\otimes\iota)\Delta=0.$

###### Proof.

We need to verify the coassociativity property for $\Delta^{\prime}$. We have

$\displaystyle(\mathbb{1}\otimes\Delta^{\prime})\Delta^{\prime}=(\mathbb{1}\otimes\Delta)\Delta+\hbar[(\mathbb{1}\otimes\Delta)\psi+(\mathbb{1}\otimes\psi)\Delta]+\hbar^{2}(\mathbb{1}\otimes\psi)\psi$
$\displaystyle(\Delta^{\prime}\otimes\mathbb{1})\Delta^{\prime}=(\Delta\otimes\mathbb{1})\Delta+\hbar[(\Delta\otimes\mathbb{1})\psi+(\psi\otimes\mathbb{1})\Delta]+\hbar^{2}(\psi\otimes\mathbb{1})\psi.$

Equating the two results, using the fact that $\Delta$ is coassociative and
considering that $\hbar^{2}=0$, we obtain that $\Delta^{\prime}$ is
coassociative if and only if equation (1) holds.

Let us now consider the counit axiom, i.e.
$(\epsilon^{\prime}\otimes\mathbb{1})\Delta^{\prime}=(\mathbb{1}\otimes\epsilon^{\prime})\Delta^{\prime}=\mathbb{1}$.
We have

$\displaystyle(\epsilon^{\prime}\otimes\mathbb{1})\Delta^{\prime}=(\epsilon\otimes\mathbb{1})\Delta+\hbar[(\epsilon\otimes\mathbb{1})\psi+(\iota\otimes\mathbb{1})\Delta]+\hbar^{2}(\iota\otimes\mathbb{1})\psi$

which, as before, is equal to $\mathbb{1}$ if and only if the terms of degree
$1$ in $\hbar$ vanish, i.e. if and only if Equation (2) holds. Similar
considerations apply for Equation (3). ∎

######  Example 7.2.

Let $X=\mathbb{k}\oplus V$, where $V$ is a vector space. Let
$\Delta(1)=1\otimes 1$ and $\Delta(x)=x\otimes 1+1\otimes x$, extended by
linearity, and let $\epsilon(1)=1$, $\epsilon(x)=0$ for all $x\in V$. Then
$(X,\Delta,\epsilon)$ is a coalgebra. Let $\psi(x):=x\otimes x$ and let
$\iota=0$. One verifies by direct computation that
$\Delta^{\prime}=\Delta+\hbar\psi$ and $\epsilon^{\prime}=\epsilon$ satisfy
Equations (1–3) and therefore provide a coalgebra deformation of
$(X,\Delta,\epsilon)$.

With the result of Lemma 7.1 in mind, we now want to construct a cohomology
group for coalgebras that classifies infinitesimal deformations of the
coalgebra structure. We suppose that the counit $\epsilon$ is not deformed
here. Moreover, we assume that $\Delta$ is cocommutative.

Let us define a cochain complex in low dimensions as follows. For $n=1,2,3$ we
set $C^{n}_{\rm CA}(X;X)={\rm Hom}_{\mathbb{k}}(X,X^{\otimes n})$. The, we
define $\delta^{1}:C^{1}_{\rm CA}(X;X)\longrightarrow C^{2}_{\rm CA}(X;X)$
according to $f\mapsto\Delta f-[f\otimes\mathbb{1}+\mathbb{1}\otimes
f]\Delta$. Observe that $\delta^{1}f$ entails how far the homomorphism $f$ is
from being a coderivation with respect to $\Delta$. In other words,
$\delta^{1}f=0$ if and only if $f$ is a coderivation. Next, we define
$\delta^{2}:C^{2}_{\rm CA}(X;X)\longrightarrow C^{3}_{\rm CA}(X;X)$ as
$\psi\mapsto(\mathbb{1}\otimes\Delta)\psi+(\mathbb{1}\otimes\psi)\Delta-(\Delta\otimes\mathbb{1})\psi-(\psi\otimes\mathbb{1})\Delta$.
As expected, the composition $\delta^{2}\delta^{1}$ vanishes.

######  Lemma 7.3.

With the notation of the previous paragraph, we have $\delta^{2}\delta^{1}=0$.

###### Proof.

By direct computation, for $f\in Hom_{\mathbb{k}}(X,X)$, one obtains

$\displaystyle\delta^{2}\delta^{1}(f)$ $\displaystyle=$
$\displaystyle(\mathbb{1}\otimes\Delta)\Delta
f-(\mathbb{1}\otimes\Delta)(f\otimes\mathbb{1})\Delta-(\mathbb{1}\otimes
f)\Delta+[\mathbb{1}\otimes(\Delta f)]\Delta$
$\displaystyle-[\mathbb{1}\otimes([f\otimes
1]\Delta)]\Delta-[\mathbb{1}\otimes([\mathbb{1}\otimes
f])\Delta]\Delta-(\Delta\otimes\mathbb{1})\Delta
f+(\Delta\otimes\mathbb{1})[(f\otimes\mathbb{1})\Delta]$
$\displaystyle+(\Delta\otimes\mathbb{1})[(\mathbb{1}\otimes f)\Delta]-[(\Delta
f)\otimes\mathbb{1}]\Delta+[([f\otimes\mathbb{1}]\Delta)\otimes\mathbb{1}]\Delta+[([\mathbb{1}\otimes
f]\Delta)\otimes\mathbb{1}]\Delta.$

Using cocommutativity of $\Delta$, the previous quantity vanishes, and it
therefore follows that $\delta^{2}\delta^{1}=0$. ∎

Applyig Lemma 7.3 it follows that we can define $H^{2}_{\rm CA}(X;X)={\rm
ker}(\delta^{2})/{\rm im}(\delta^{1})$ as usual. This is the second cohomology
group of the coalgebra $(X,\Delta)$.

######  Theorem 7.4.

Let $(X,\Delta,\epsilon)$ be a coalgebra. Then, the infinitesimal deformations
of $\Delta$ ($\epsilon$ is not deformed here) are classified by the coalgebra
cohomology group $H^{2}_{\rm CA}(X;X)$.

###### Proof.

The proof is similar to the TSD case considered before. Applying Lemma 7.1
with $\iota=0$ one sees that $2$-cocycles are equivalent to deformations,
while to show that trivial deformations correspond to $2$-cocycles that
cobound, one expands the isomorphisms between the trivial deformation and the
given deformation in a geometric series with respect to $\hbar$. ∎

##  8\. Simultaneous deformations

As it was shown in [AZ], it is a problem of relevance to produce deformations
of the TSD structure $(X,T)$ by simultaneously deforming both $T$ and
$\Delta$. This requires that the new $T^{\prime}$ satisfies the TSD property
with respect to the deformed $\Delta^{\prime}$, rather than $\Delta$, as
considered above. Moreover, $T^{\prime}$ needs to be a morphism of coalgebras
with respect to $(X^{\prime},\Delta^{\prime})$. These considerations require a
modification of the equations used above, although the base philosophy is
identical. Once again, the passage to $n$-ary SD structures is deduced without
excessive issues from the ternary one shown here, although the equations do
become convoluted.

We proceed as follows. First, we consider deformations of the coalgebra
structure of $(X,\Delta,\epsilon)$ as in Section 7, and then we determine the
conditions that a deformation of $T$ needs to satisfy in order to produce a
TSD object. This requires substantially two conditions. First, $T^{\prime}$
needs to satisfy the SD condition, then $T^{\prime}$ needs to be a coalgebra
morphism with respect to $\Delta^{\prime}$. Observe that in both conditions,
one needs to use $\Delta^{\prime}$ and therefore the considerations derived in
Section 5 need to be modified. We start with binary deformations, as these
involve much simpler conditions, and composing deformed binary operations we
can obtain deformed ternary operations as well.

######  Lemma 8.1.

Let $(X,\Delta,q)$ be a binary SD object, and let $\Delta^{\prime}$ be such
that $(X^{\prime},\Delta^{\prime})$ is a coalgebra. Let $\shuffle$ indicate
the shuffle map corresponding to binary self-distributivity. Then
$q^{\prime}=q+\hbar\phi$ satisfies the SD condition with respect to
$(X^{\prime},\Delta^{\prime})$ if and only if

$\displaystyle q(\phi\otimes\mathbb{1})+\phi(q\otimes\mathbb{1})$
$\displaystyle=$ $\displaystyle q[\phi\otimes
q+q\otimes\phi]\shuffle(\mathbb{1}^{\otimes 2}\otimes\Delta)+\phi(q\otimes
q)\shuffle(\mathbb{1}^{\otimes 2}\otimes\Delta)+q(q\otimes
q)\shuffle(\mathbb{1}^{\otimes 2}\otimes\psi).$

Moreover, $q^{\prime}$ is a morphism of coaglebras if and only if

$\displaystyle\Delta\phi+\psi q$ $\displaystyle=$ $\displaystyle[\phi\otimes
q+q\otimes\phi](\mathbb{1}\otimes\tau\otimes\mathbb{1})(\Delta\otimes\Delta)$
$\displaystyle+(q\otimes
q)(\mathbb{1}\otimes\tau\otimes\mathbb{1})(\psi\otimes\Delta+\Delta\otimes\psi),$

where $\tau$ indicates the switching map.

We provide, next, the conditions for deformations of ternary structures. A
similar statement for $n$-ary SD structures is obtained by appropriately
changing the number of tensorands and the arities of $T$ and $\Delta$. We just
provide the ternary case for simplicity of notation.

######  Lemma 8.2.

Let $(X,\Delta,T)$ be a TSD object and let $\Delta^{\prime}$ denote a
deformation of the coalgebra structure. Then, $T^{\prime}=T+\hbar\phi$ is a
deformatioin of $T$ if and only if the following two conditions hold

$\displaystyle T(\phi\otimes\mathbb{1}^{\otimes
2})+\phi(T\otimes\mathbb{1}^{\otimes 2})$ $\displaystyle=$
$\displaystyle\phi[T^{\otimes 3}\shuffle(\mathbb{1}^{\otimes
3}\otimes\Delta_{3}^{\otimes 2})]$ $\displaystyle+T[(\phi\otimes T^{\otimes
2}+T\otimes\phi\otimes T+T^{\otimes 2}\otimes\phi)\shuffle(\mathbb{1}^{\otimes
3}\otimes\Delta_{3}^{\otimes 2})]$ $\displaystyle T[T^{\otimes
3}\shuffle(\mathbb{1}^{\otimes
3}\otimes[(\mathbb{1}\otimes\Delta)\psi+(\mathbb{1}\otimes\psi)\Delta]^{\otimes
2})],$

$\displaystyle(\mathbb{1}\otimes\psi)\Delta T+(\mathbb{1}\otimes\Delta)\psi
T+\Delta_{3}\phi$ $\displaystyle=$ $\displaystyle[\phi\otimes T^{\otimes
2}+T\otimes\phi\otimes T+T^{\otimes
2}\otimes\phi]\tilde{\shuffle}\Delta_{3}^{\otimes 3}$
$\displaystyle+T^{\otimes
3}\tilde{\shuffle}[(\mathbb{1}\otimes\Delta)\psi+(\mathbb{1}\otimes\psi)\Delta]^{\otimes
3},$

where $\shuffle$ is the permutation associated to TSD property, and
$\tilde{\shuffle}$ is the permutation that gives the natural coalgebra
structure on $X^{\otimes 3}$.

######  Remark 8.3.

We observe that simultaneous deformations of $q$ or $T$, i.e. a binary or a
ternary SD operation, can be interpreted as cohomology classes of $H^{2}_{\rm
SD}(X_{\psi};X_{\psi})$, where $X_{\psi}$ is considered with the
comultiplication on $X$ obtained by deforming $\Delta$ by $\psi$. So, the
coalgebra cohomology parametrizes the SD second cohomology group of $X$. The
structure so obtained is an SD object in the category of vector spaces in the
sense of [ESZ, AZ].

We point out that the simultaneous deformations of this section naturally
generalize to higher arity SD operations the cohomology introduced in [CCES],
based on the deformation theory approach of [MS]. In fact, in [CCES], the
cochain complexes contain direct sums corresponding to the coalgebra
deformation and the SD deformation. In the present work, such an approach to
put together the two deformations in a single complex would work as well, and
the proofs of the well definedness of the second cohomology group
substantially reduce the Lemma 8.2. We therefore pose the following.

######  Definition 8.4.

Let $(X,\Delta,q)$ denote a SD object. Then the second simultaneous cohomology
group of $X$, denoted by the symbol $H^{2}_{\rm\Sigma SD}(X;X)$ is defined to
be the group

$H^{2}_{\rm\Sigma SD}(X;X):=\bigoplus_{[\psi]\in H^{2}_{\rm
CA}(X;X)}H^{2}_{SD}(X_{\psi},X_{\psi}),$

where $\psi$ varies over the $2$-cocycles representatives of the coalgebra
cohomology of $(X,\Delta)$.

######  Remark 8.5.

We note that when $\psi=0$ is the trivial deformation, one recovers the
deformation of SD structures of Section 4, where the coalgebra structure is
not deformed.

##  9\. Examples and Computations

In this section we consider examples of nontrivial SD cohomology classes, and
provide some computations for certain low-dimensional Lie algebras.

###  9.1. SD deformations

First, we consider deformations of SD objects where the comultiplication is
fixed. For simplicity of computation we focus on the binary case and we
consider the SD structures arising from all the $3$-dimensional Lie algebras
over the real and complex fields.

We now provide the computations for the SD second cohomology groups of the SD
objects associated to the binary Lie algebras in the Bianchi classification
for the real and complex ground fields, where the comultiplication is not
deformed. This accounts to having $\psi=\iota=0$ in the previous computations.
The results are summarised in Table 1 and Table 2, for the real and complex
cases, respectively.

Real Lie algebra | Second Cohomology Group
---|---
Bianchi II (Heisenberg) | $\mathbb{R}^{\oplus 7}$
Bianchi III | $\mathbb{R}$
Bianchi IV | $\mathbb{R}^{\oplus 6}$
Bianchi V | $\mathbb{R}$
Bianchi VI | $\mathbb{R}$
Bianchi VII | $\mathbb{R}^{\oplus 4}$
Bianchi VIII ($\mathfrak{sl}_{2}(\mathbb{R})$) | 0
Bianchi IX ($\mathfrak{so}(3)$) | 0
Table 1. Cohomology of SD obsjects associated to real $3$D Lie algebras Complex Lie algebra | Second Cohomology Group
---|---
Bianchi II (Heisenberg) | $\mathbb{C}^{\oplus 6}$
Bianchi III | $\mathbb{C}$
Bianchi IV | $\mathbb{C}^{\oplus 6}$
Bianchi V | $\mathbb{C}$
Bianchi VI | $\mathbb{C}^{4}$
Bianchi VIII ($\mathfrak{sl}_{2}(\mathbb{C})$) | 0
Table 2. Cohomology of SD obsjects associated to complex $3$D Lie algebras

We observe that, as expected, the computation of the cohomology groups for the
simple algebras (i.e. $\mathfrak{sl_{2}}$ and $\mathfrak{so}(3)$) are trivial,
since they are simple, and consequently have trivial center. In fact, known
results (see for instance [Jacob]) in Lie theory show that semisimple Lie
algebras have trivial cohomology and, in addition, since they have trivial
center it follows that the the corresponding SD cohomology is trivial as well
from Theorem 6.4. However, niplotent and solvable algebras present nontrivial
cohomology, which means that they have nontrivial deformations of the
associated SD objects.

###  9.2. Simultaneous deformations

We now consider the case of simultaneous deformations, where again we focus on
the binary case for computational simplicity. We first give examples of
simultaneously deformed structures for $\mathfrak{sl}_{2}(\mathbb{C})$ and the
Heisenberg Lie algebra. We recall that the main objective of introducing
simultaneous deformations is, in practice, to produce nontrivial deformations
of the SD objects associated to Lie algebras with trivial center, so that our
main interest here lies in the deformations of
$\mathfrak{sl}_{2}(\mathbb{C})$, for which we explicitly compute the
simultaneous cohomology $H^{2}_{\rm\Sigma SD}$.

######  Example 9.1.

Let $\mathfrak{sl}_{2}(\mathbb{C})$ denote the special linear algebra of order
$2$ over the complex numbers. Recall that this is a $3$-dimensional Lie
algebra whose generators we indicate as $x,y,z$ and Lie bracket specified by
$[x,y]=z$, $[z,y]=-2y$ and $[z,x]=2x$. Let $q$ denote the corresponding BSD
structure on $X=\mathbb{C}\oplus\mathfrak{sl}_{2}(\mathbb{C})$. We consider
now simultaneous deformations of $\Delta$ and $q$. Following Section 8 this
accounts to finding $\phi$ and $\psi$ satisfying the conditions such that
$\Delta^{\prime}=\Delta+\hbar\psi$ and $q^{\prime}=q+\hbar\phi$ define an SD
object structure on $X^{\prime}$, which is $X$ with coefficients in
$\mathbb{C}[[\hbar]]/(\hbar^{2})$. Let us set, for clarity,
$e^{1}=(1,0),e^{2}=(0,x),e^{3}=(0,y)$ and $e^{4}=(0,z)$, so that $2$-cochains
$\phi:X\otimes X\longrightarrow X$ are given by the expression
$\phi(e^{i}\otimes e^{j})=\alpha^{ij}_{k}e^{k}$, where Einstein summation
convention is understood. Similarly, for $\psi:X\longrightarrow X\otimes X$ we
set $\psi(e^{j})=\beta^{j}_{kl}e^{k}\otimes e^{l}$. Then, a direct computer
aided calculation shows that the set of maps $\phi$ correspond to matrices

$A^{T}_{\phi}=\left(\begin{matrix}2\beta^{3}_{34}+\alpha_{3}^{31}&&&\\\
\alpha_{4}^{24}+\alpha_{4}^{42}&-2\beta^{3}_{34}&&\frac{\alpha_{3}^{41}}{2}\\\
\alpha_{4}^{34}+\alpha_{4}^{43}&&2\beta^{3}_{34}&\frac{\alpha_{2}^{41}}{2}\\\
\alpha_{4}^{44}&-\alpha_{2}^{41}&-\alpha_{3}^{41}&\\\
&4\beta^{3}_{34}+\alpha_{3}^{31}&&-\frac{\alpha_{3}^{41}}{2}\\\
&\alpha_{4}^{24}+\alpha_{4}^{42}&&\\\
-\alpha_{4}^{44}&2\alpha_{4}^{34}+\alpha_{4}^{34}&-\alpha_{4}^{24}&-\alpha_{4}^{32}\\\
2\alpha_{4}^{24}+2\alpha_{4}^{42}&-2\alpha_{2}^{42}+\alpha_{4}^{44}&-\alpha_{3}^{42}&\alpha_{4}^{24}\\\
&&\alpha_{3}^{31}&-\frac{\alpha_{2}^{41}}{2}\\\
\alpha_{4}^{44}&-\alpha_{4}^{34}&2\alpha_{4}^{24}+\alpha_{4}^{42}&\alpha_{4}^{32}\\\
&&\alpha_{4}^{34}+\alpha_{4}^{34}&\\\
-2\alpha_{4}^{34}-2\alpha_{4}^{34}&-\alpha_{2}^{43}&\alpha_{2}^{42}+\alpha_{4}^{44}&\alpha_{4}^{34}\\\
&\alpha_{2}^{41}&\alpha_{3}^{41}&2\beta^{3}_{34}+\alpha_{3}^{31}\\\
-2\alpha_{4}^{24}-2\alpha_{4}^{42}&\alpha_{2}^{42}&\alpha_{3}^{42}&\alpha_{4}^{42}\\\
2\alpha_{4}^{34}+2\alpha_{4}^{34}&\alpha_{2}^{43}&-\alpha_{2}^{42}&\alpha_{4}^{34}\\\
&&&\alpha_{4}^{44}\end{matrix}\right)$

while the maps $\psi$ are characterized as matrices

$B_{\psi}=\left(\begin{matrix}-2\beta^{3}_{34}-\alpha_{3}^{31}&-\alpha_{4}^{24}-\alpha_{4}^{42}&-\alpha_{4}^{34}-\alpha_{4}^{34}&-\alpha_{4}^{44}\\\
&-2\beta^{3}_{34}-\alpha_{3}^{31}&&\\\ &&-2\beta^{3}_{34}-\alpha_{3}^{31}&\\\
&&&-2\beta^{3}_{34}-\alpha_{3}^{31}\\\ &-2\beta^{3}_{34}-\alpha_{3}^{31}&&\\\
&-\alpha_{2}^{41}&&\\\
&\frac{\alpha_{3}^{41}}{2}&-\frac{\alpha_{2}^{41}}{2}&\\\
&\beta^{3}_{34}&&-\frac{\alpha_{2}^{41}}{2}\\\
&&-2\beta^{3}_{34}-\alpha_{3}^{31}&\\\
&\frac{\alpha_{3}^{41}}{2}&-\frac{\alpha_{2}^{41}}{2}&\\\
&&\alpha_{3}^{41}&\\\ &&\beta^{3}_{34}&\frac{\alpha_{3}^{41}}{2}\\\
&&&-2\beta^{3}_{34}-\alpha_{3}^{31}\\\
&\beta^{3}_{34}&&-\frac{\alpha_{2}^{41}}{2}\\\
&&\beta^{3}_{34}&\frac{\alpha_{3}^{41}}{2}\\\
&&&2\beta^{3}_{34}\end{matrix}\right),$

where we have omitted the zeros in both matrices, and $A_{\phi}$ has been
transposed to better fit the page. The total number of free complex parameters
is $14$, as it can be seen directly by the matrices of $\phi$ and $\psi$. In
this example, the counit deformation, i.e. the map $\iota$ appearing in
Section 8, is non-trivial as well.

######  Example 9.2.

Let us now consider the $3$-dimensional Heisenberg algebra over the complex
numbers, which we here denote as $\mathcal{H}_{3}(\mathbb{C})$. Recall that
this is generated by three elements $x,y,z$ and its Lie bracket is defined by
$[x,y]=z$, $[x,z]=[y,z]=0$. Basis vectors for
$X=\mathbb{C}\oplus\mathcal{H}_{3}$ are again
$e^{1}=(1,0),e^{2}=(0,x),e^{3}=(0,y)$ and $e^{4}=(0,z)$ as before, and we will
indicate $\phi$ and $\psi$ as in Example 9.1, while
$\iota(e^{k})=\iota_{k}\in\mathbb{R}$ for all $k=1,2,3,4$. Once again omitting
the zeros, for simplicity, we have

$A^{T}_{\phi}=\left(\begin{matrix}\alpha_{4}^{31}&&&\\\
\alpha_{1}^{12}&&&\beta^{4}_{43}\\\ \alpha_{1}^{13}&&&-\beta^{4}_{24}\\\
\alpha_{4}^{44}&&&\\\ &\alpha_{4}^{31}&&\alpha_{4}^{21}\\\
&\alpha_{1}^{12}&&\alpha_{4}^{22}\\\
-\alpha_{4}^{44}&\alpha_{1}^{13}-\alpha_{2}^{32}&\alpha_{1}^{12}-\alpha_{3}^{32}&\alpha_{4}^{23}\\\
&\alpha_{3}^{43}+\alpha_{4}^{44}&-\alpha_{3}^{42}&\alpha_{1}^{12}-\alpha_{4}^{42}\\\
&&\alpha_{4}^{41}&\alpha_{4}^{31}\\\
\alpha_{4}^{44}&-\alpha_{2}^{43}&\alpha_{2}^{33}&\alpha_{4}^{32}\\\
&&\alpha_{1}^{13}&\alpha_{4}^{33}\\\
&-\alpha_{3}^{43}&-\alpha_{3}^{43}+\alpha_{4}^{44}&\alpha_{1}^{13}-\alpha_{4}^{43}\\\
&\alpha_{2}^{43}&&\alpha_{4}^{41}\\\ &&\alpha_{3}^{42}&\alpha_{4}^{42}\\\
&&\alpha_{3}^{43}&\alpha_{4}^{43}\\\ &&&\alpha_{4}^{44}\end{matrix}\right)$

as well as

$B_{\psi}=\left(\begin{matrix}-\alpha_{4}^{41}&-\alpha_{1}^{12}&-\alpha_{1}^{13}&-\alpha_{4}^{44}\\\
&-\alpha_{4}^{41}&&\\\ &&-\alpha_{4}^{41}&\\\ &&&-\alpha_{4}^{41}\\\
&-\alpha_{4}^{41}&&\\\ &2\beta^{4}_{24}&&\\\
&\beta^{4}_{43}&\beta^{4}_{24}&\\\
&\beta^{4}_{44}-\beta^{2}_{24}&-\beta^{3}_{24}&\beta^{4}_{24}\\\
&&-\alpha_{4}^{41}&\\\ &\beta^{4}_{43}&\beta^{4}_{24}&\\\
&&2\beta^{4}_{43}&\\\
&-\beta^{2}_{34}&\beta^{4}_{44}-\beta^{4}_{34}&\beta^{4}_{43}\\\
&&&-\alpha_{4}^{41}\\\ &\beta^{2}_{24}&\beta^{3}_{24}&\beta^{4}_{24}\\\
&\beta^{2}_{34}&\beta^{3}_{34}&\beta^{4}_{43}\\\
&\beta^{2}_{44}&\beta^{3}_{44}&\beta^{4}_{44}\end{matrix}\right).$

The counit deformation is nontrivial too, and it takes the form

$\iota=(\alpha_{4}^{41},\alpha_{1}^{12},\alpha_{1}^{13},\alpha_{4}^{44}).$

The previous matrices completely characterize the deformations of the SD
struture obtained from the Heisenberg algebra $\mathcal{H}_{3}$, as well as
the coalgebra structure $(X,\Delta,\epsilon)$. We have a total of $26$ free
parameters.

######  Remark 9.3.

It is well known that semisimple Lie algebras have trivial cohomology group
$H^{2}_{\rm Lie}(\mathfrak{g};\mathfrak{g})$, see [Jacob] for example. Since
they also have trivial center, from Theorem 6.4 it follows that the BSD
cohomology group of semisimple Lie algebras, e.g. $\mathfrak{sl}_{2}$, is
trivial. Then, the $2$-cocycles computed above are all cobounded when $\psi=0$
(i.e. the coalgebra structure is not deformed). However, in the case of
simultaneous deformations where $\psi\neq 0$, nontrivial $2$-cocycles can
arise, as the following example shows.

######  Example 9.4.

Acomputer aided calculation for the second cohomology group of
$\mathfrak{sl}_{2}(\mathbb{C})$ of Definition 8.4 shows that this is
$8$-dimensional, i.e. $H^{2}_{\rm\Sigma SD}(X;X)=\mathbb{C}^{\oplus 8}$. This
example, in particular shows that when simultaneous deformations are
considered, SD structures that do not admit deformations (with fixed
comultiplication) can be deformed. Interestingly, the counit admits nontrivial
deformations as well, showing the the coalgebra structure can be deformed in
its entirety, not just through its comultiplication.

##  10\. Deformations of Yang-Baxter operators

We consider now the main application of the cohomology theories introduced and
studied in this article. It was shown in [EZ] that ternary SD objects in
symmetric monoidal categories can be used to construct braided monoidal
caategories. In particular, it was shown that from a ternary SD object $(X,T)$
one can obtain a Yang-Baxter operator on the tensor product $X^{\otimes
2}\otimes X^{\otimes 2}$. The same paradigm is applied to obtain a Yang-Baxter
operator on $X^{\otimes 2(n-1)}$, starting from an $n$-ary SD object. In [AZ]
it was given a construction that associates an $n$-ary SD object, in the
category of vector spaces, and a Yang-Baxter operator starting to an $n$-Lie
algebra. Since simultaneous deformations of a given binary (resp. ternary) SD
object provide a new $(X^{\prime},q^{\prime})$ (resp.
$(X^{\prime},T^{\prime})$) which is a deformation of $(X,q)$ (resp. $(X,T)$),
it follows that the construction of [AZ] gives a corresponding deformation of
the associated Yang-Baxter operator. Before investigating this correspondence,
we pose the following.

######  Definition 10.1.

Let $\mathfrak{g}$ denote an $n$-Lie algebra. Then, let $X$ denote the
associated $n$-ary SD object. In this situation, the Yang-Baxter operator
corresponding to $X$ is denoted by $R_{\mathfrak{g}}$.

As before, we will consider explicitly binary and ternary cases, although the
ternary case can be extended to $n$-ary in a straightforward manner. However,
our assumption simplifies notation significantly.

######  Definition 10.2 ([Eis, Eis2]).

Let $R:V^{\otimes 2}\longrightarrow V^{\otimes 2}$ be a Yang-Baxter operator
over the vector space $V$. An operator $\tilde{R}:V^{\otimes 2}\longrightarrow
V^{\otimes 2}$ is said to be a Yang-Baxter deformation of $R$, or YB
deformation for short, is a Yang-Baxter operator such that
$[R-\tilde{R}](V^{\otimes 2})\subset\hbar V^{\otimes 2}$.

We formalize the previous discussion regarding deformations of Yang-Baxter
operators arising from (binary) Lie algebras in the following result.

######  Lemma 10.3.

Let $\mathfrak{g}$ be a binary Lie algebra and let $(\phi,\psi,\iota)$ denote
a simultaneous deformation of the associated SD object $(X,\Delta)$. Then,
$(\phi,\psi,\iota)$ induces a Yang-Baxter deformation of the operator
$R_{\mathfrak{g}}$ given by

$\tilde{R}_{\mathfrak{g}}=R_{\mathfrak{g}}+\hbar[(\mathbb{1}\otimes\phi)\shuffle(\mathbb{1}\otimes\Delta)+(\mathbb{1}\otimes
q)\shuffle(\mathbb{1}\otimes\psi)],$

where $\shuffle=(\mathbb{1}\otimes\tau\otimes\mathbb{1})$.

###### Proof.

The deformation of $R_{\mathfrak{g}}$ is the Yang-Baxter operator $\tilde{R}$
corresponding to the SD object $(X^{\prime},\Delta^{\prime})$ (see Theorem 3.4
and Theorem 3.11 in [AZ]), where $X^{\prime}$ is endowed with the SD map
$q^{\prime}=q+\hbar\phi$, and the coalgebra structure of $X^{\prime}$ is given
by $\Delta^{\prime}=\Delta+\hbar\psi$,
$\epsilon^{\prime}=\epsilon+\hbar\iota$. To see this explicitly, we write the
YB operator $\tilde{R}_{\mathfrak{g}}=(\mathbb{1}\otimes
q^{\prime})\shuffle(\mathbb{1}\otimes\Delta^{\prime})$ and then, discarding
terms with higher powers in $\hbar$ than $1$, we obtain the operator in the
statement. The fact that $R-\tilde{R}$ maps the space $X^{\prime\otimes 2}$
into $\hbar X^{\prime\otimes 2}$ i.e. $\tilde{R}$ is a YB deformation, follows
readily as well, since $\tilde{R}_{\mathfrak{g}}$ differs from
$R_{\mathfrak{g}}$ by a term proportional to $\hbar$. ∎

The same constructions hold when one considers a ternary Lie algebra (cf.
results in Section 4 of [AZ]). The only difference is that starting with a
ternary bracket the construction is “doubled”, and the corresopnding YB
operators are given on the tensor products of the TSD objects. Therefore, an
analogous version of Proposition 10.3 for ternary Lie algebras is obtained
simply by changing binary with ternary, and YB operator over $X^{\prime}$ with
YB operator over $X^{\prime\otimes 2}$. The proof is substantially identical.

######  Lemma 10.4.

Let $\mathfrak{g}$ be a ternary Lie algebra and let $(\phi,\psi,\iota)$ denote
a simultaneous deformation of the associated SD object $(X,\Delta)$. Then,
$(\phi,\psi,\iota)$ induces a Yang-Baxter deformation of the operator
$R_{\mathfrak{g}}$ given by

$\tilde{R}_{\mathfrak{g}}=R_{\mathfrak{g}}+\hbar[(\mathbb{1}^{\otimes
2}\otimes\phi)\shuffle(\mathbb{1}^{\otimes 2}\otimes\Delta^{\otimes
2})+(\mathbb{1}^{\otimes 2}\otimes T)\shuffle(\mathbb{1}^{\otimes
2}\otimes\psi^{\otimes 2})],$

where $\shuffle$ is the linearization of the permuation associated to the TSD
property.

Let us now restrict our attention to the case of deformations of the SD
structure where the coalgebra is fixed. These are simply simultaneous
deformations where $\psi$ and $\iota$ are trivial. According to the
computations given in Section 9, only three Lie algebras over the reals, and
two over the complex, admit deformations where $\psi$ can be nontrivial. In
what follows, we use a notation to indicate YB cohomology that is similar to
Eisermann’s in [Eis, Eis2], where the YB operator and an ideal $\mathfrak{m}$
appear explicitly. Since we are dealing with the case where the ring is
$\mathbb{k}[[\hbar]]$ taken along its unique maximal idea $(\hbar)$, we write
$H^{2}_{\rm YB}(R,\hbar)$, or $H^{2}(R,\hbar)$ for short. We want to show now
that SD cohomology classes give rise to YB cohomology classes. Specifically,
we have the following.

######  Theorem 10.5.

Let notation be as above. Then there is a homomorphism $\Xi^{2}$ from SD
cohomology to Yang-Baxter cohomology $\Xi^{2}:H^{2}_{\rm
SD}(X;X)\longrightarrow H^{2}_{\rm YB}(R,\hbar)$. If, moreover, $\mathfrak{g}$
has trivial center, $\Xi^{2}$ is a monomorphism.

###### Proof.

Throughout this proof, we use the same notation as in Lemma 6.3, where maps
with image in $X$ have been written as consisting of a sum of a term with
image in $\mathbb{k}$, and one with image in $\mathfrak{g}$. For $\phi\in
C^{2}_{\rm SD}(X;X)$, we define the following map $c_{\phi}$, by providing its
image on simple tensors as

$(a,x)\otimes(b,y)\mapsto(b,y)^{(1)}\otimes\phi((a,x)\otimes(b,y)^{(2)}).$

First, we want to show that $\Xi^{2}$ is well defined in cohomology, which
accounts to the first part of the statement of the theorem. From Lemma 10.3
(with $\psi,\iota=0$) it follows that $\tilde{R}=R+\hbar c_{\phi}$ is a YB
operator, provided that $\phi$ is an SD $2$-cocycle. But we can write
$\tilde{R}=R\circ(\mathbb{1}^{\otimes 2}+\hbar\Xi^{2}(\phi))$. Applying