k}A)\rightarrow\mathsf{R}\times_{\bf k}A,\qquad
l_{\mathsf{R}}:(\mathsf{R}\otimes_{\bf k}A)\times(\mathsf{R}\otimes_{\bf
k}M)\rightarrow\mathsf{R}\otimes_{\bf k}M,$

$r_{\mathsf{R}}:(\mathsf{R}\otimes_{\bf k}M)\times(\mathsf{R}\otimes_{\bf
k}A)\rightarrow\mathsf{R}\otimes_{\bf k}M$ and a $\mathsf{R}$-linear map
$P_{\mathsf{R}}:\mathsf{R}\otimes_{\bf k}M\rightarrow\mathsf{R}\otimes_{\bf
k}A$

such that the following conditions hold:

(i) $(\mathsf{R}\otimes_{\bf k}A,\mu_{\mathsf{R}})$ is an
$\mathsf{R}$-associative algebra, $(\mathsf{R}\otimes_{\bf
k}M,l_{\mathsf{R}},r_{\mathsf{R}})$ a bimodule over it and the
$\mathsf{R}$-linear map $P_{R}:\mathsf{R}\otimes_{\bf
k}M\rightarrow\mathsf{R}\otimes_{\bf k}A$ is a relative averaging operator. In
other words, $\mathsf{R}\otimes_{\bf
k}M\xrightarrow{P_{\mathsf{R}}}\mathsf{R}\otimes_{\bf k}A$ is a
$\mathsf{R}$-relative averaging algebra by considering the above structures on
$\mathsf{R}\otimes_{\bf k}A$ and $\mathsf{R}\otimes_{\bf k}M$.

(ii) The pair $(\varepsilon\otimes_{\bf
k}\mathrm{id}_{A},\varepsilon\otimes_{\bf
k}\mathrm{id}_{M}):(\mathsf{R}\otimes_{\bf
k}M\xrightarrow{P_{\mathsf{R}}}\mathsf{R}\otimes_{\bf
k}A)\rightsquigarrow(M\xrightarrow{P}A)$ is a morphism of
$\mathsf{R}$-relative averaging algebras.

######  6.2 Definition.

Let $M\xrightarrow{P}A$ be a relative averaging algebra. Two
$\mathsf{R}$-deformations
$(\mu_{\mathsf{R}},l_{\mathsf{R}},r_{\mathsf{R}},P_{\mathsf{R}})$ and
$(\mu^{\prime}_{\mathsf{R}},l^{\prime}_{\mathsf{R}},r^{\prime}_{\mathsf{R}},P^{\prime}_{\mathsf{R}})$
are said to be equivalent if there exists an isomorphism of
$\mathsf{R}$-relative averaging algebras

$\displaystyle(\Phi,\Psi):(\mathsf{R}\otimes_{\bf
k}M\xrightarrow{P_{\mathsf{R}}}\mathsf{R}\otimes_{\bf
k}A)\rightsquigarrow(\mathsf{R}\otimes_{\bf
k}M\xrightarrow{P^{\prime}_{\mathsf{R}}}\mathsf{R}\otimes_{\bf k}A)$

satisfying $(\varepsilon\otimes_{\bf
k}\mathrm{id}_{A})\circ\Phi=(\varepsilon\otimes_{\bf k}\mathrm{id}_{A})$ and
$(\varepsilon\otimes_{\bf k}\mathrm{id}_{M})\circ\Psi=(\varepsilon\otimes_{\bf
k}\mathrm{id}_{M})$.

We will now consider the cases, when $\mathsf{R}={\bf k}[[t]]$ (the ring of
formal power series) and $\mathsf{R}={\bf k}[[t]]/(t^{2})$ (the local Artinian
ring of dual numbers). In the first case, a $\mathsf{R}$-deformation is called
a formal deformation, and in the second case, a $\mathsf{R}$-deformation is
called an infinitesimal deformation. A more precise description of formal
deformation is given by the following.

######  6.3 Definition.

(i) Let $M_{l_{M},r_{M}}\xrightarrow{P}A_{\mu}$ be a given relative averaging
algebra. A formal deformation of it consists of a quadruple
$(\mu_{t},l_{t},r_{t},P_{t})$ of formal sums

(25) $\displaystyle\mu_{t}=\sum_{i=0}^{\infty}t^{i}\mu_{i},\qquad
l_{t}=\sum_{i=0}^{\infty}t^{i}l_{i},\qquad
r_{t}=\sum_{i=0}^{\infty}t^{i}r_{i}~{}~{}~{}~{}\text{ and
}~{}~{}~{}~{}P_{t}=\sum_{i=0}^{\infty}t^{i}P_{i}$

(where $\mu_{i}:A\times A\rightarrow A$, $l_{i}:A\times M\rightarrow M$,
$r_{i}:M\times A\rightarrow M$ and $P_{i}:M\rightarrow A$ are bilinear/linear
maps, for $i\geq 0$, with $\mu_{0}=\mu$, $l_{0}=l_{M}$, $r_{0}=r_{M}$ and
$P_{0}=P$) such that $A[[t]]=(A[[t]],\mu_{t})$ is an associative algebra over
${\bf k}[[t]]$, and $M[[t]]=(M[[t]],l_{t},r_{t})$ is a bimodule over the
algebra $A[[t]]$, and the ${\bf k}[[t]]$-linear map $P_{t}:M[[t]]\rightarrow
A[[t]]$ is a relative averaging operator. In other words,
$M[[t]]\xrightarrow{P_{t}}A[[t]]$ is a relative averaging algebra over ${\bf
k}[[t]]$.

(ii) Two formal deformations $(\mu_{t},l_{t},r_{t},P_{t})$ and
$(\mu_{t}^{\prime},l_{t}^{\prime},r_{t}^{\prime},P_{t}^{\prime})$ are
equivalent if there exists a pair $(\varphi_{t},\psi_{t})$ of formal sums

$\displaystyle\varphi_{t}=\sum_{i=0}^{\infty}t^{i}\varphi_{i}\quad\text{ and
}\quad\psi_{t}=\sum_{i=0}^{\infty}t^{i}\psi_{i}$

(where $\varphi_{i}:A\rightarrow A$ and $\psi_{i}:M\rightarrow M$ are linear
maps, for $i\geq 0$, with $\varphi_{0}=\mathrm{id}_{A}$ and
$\psi_{0}=\mathrm{id}_{M}$) such that

$\displaystyle(\varphi_{t},\psi_{t}):(M[[t]]\xrightarrow{P_{t}}A[[t]])\rightsquigarrow(M[[t]]\xrightarrow{P_{t}^{\prime}}A[[t]])$

is an isomorphism of relative averaging algebras over ${\bf k}[[t]]$. Then we
write
$(\mu_{t},l_{t},r_{t},P_{t})\sim(\mu_{t}^{\prime},l_{t}^{\prime},r_{t}^{\prime},P_{t}^{\prime}).$

It follows from the above definition that a quadruple
$(\mu_{t},l_{t},r_{t},P_{t})$ given by (25) is a formal deformation of the
relative averaging algebra $M\xrightarrow{P}A$ if the following system of
equations are hold:

(26) $\displaystyle\sum_{i+j=n}\mu_{i}(\mu_{j}(a,b),c)=~{}$
$\displaystyle\sum_{i+j=n}\mu_{i}(a,\mu_{j}(b,c)),$ (27)
$\displaystyle\sum_{i+j=n}l_{i}(\mu_{j}(a,b),u)=~{}$
$\displaystyle\sum_{i+j=n}l_{i}(a,l_{j}(b,u)),$ (28)
$\displaystyle\sum_{i+j=n}r_{i}(l_{j}(a,u),b)=~{}$
$\displaystyle\sum_{i+j=n}l_{i}(a,r_{j}(u,b)),$ (29)
$\displaystyle\sum_{i+j=n}r_{i}(r_{j}(u,a),b)=~{}$
$\displaystyle\sum_{i+j=n}r_{i}(u,\mu_{j}(a,b)),$ (30)
$\displaystyle\sum_{i+j+k=n}\mu_{i}\big{(}P_{j}(u),P_{k}(v)\big{)}=\sum_{i+j+k=n}P_{i}\big{(}l_{j}(P_{k}(u),v)\big{)}=~{}$
$\displaystyle\sum_{i+j+k=n}P_{i}\big{(}r_{j}(u,P_{k}(v))\big{)},$

for all $a,b,c\in A$, $u,v\in M$ and $n\geq 0$. These are called the
deformation equations. Note that the deformation equations are held for $n=0$
as $M_{l_{M},r_{M}}\xrightarrow{P}A_{\mu}$ is a relative averaging algebra.

For $n=1$. It follows from (26) that

$\displaystyle\mu_{1}(a\cdot b,c)+\mu_{1}(a,b)\cdot c=\mu_{1}(a,b\cdot
c)+a\cdot\mu_{1}(b,c),\text{ for }a,b,c\in A,$

which is equivalent to $\delta_{\mathrm{Hoch}}(\mu_{1})=0$. To summarize the
identities (27), (28), (29) for $n=1$, we define an element
$\beta_{1}\in\mathrm{Hom}(\mathcal{A}^{1,1},M)$ by

(31) $\displaystyle\beta_{1}(a,u)=l_{1}(a,u)~{}~{}\text{ and
}~{}~{}\beta_{1}(u,a)=r_{1}(u,a),\text{ for }a\in A,u\in M.$

Then we get that $\delta_{\mathrm{Hoch}}^{\mu_{1}}(\beta_{1})=0$. Finally, the
identity (30) for $n=1$ is equivalent to

$\displaystyle\big{(}\delta_{\mathrm{Diass}}^{P}(P_{1})+h_{P}(\mu_{1},\beta_{1})\big{)}(y;u,v)=0,\text{
for }y=\leavevmode\hbox to11.78pt{\vbox
to11.78pt{\pgfpicture\makeatletter\hbox{\hskip-16.8727pt\lower-11.58179pt\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{17.0727pt}{0.0pt}\pgfsys@lineto{22.7636pt}{-5.69089pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{22.7636pt}{-5.69089pt}\pgfsys@lineto{28.45448pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{22.7636pt}{-5.69089pt}\pgfsys@lineto{22.7636pt}{-11.38179pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{25.60904pt}{-2.84544pt}\pgfsys@lineto{22.7636pt}{0.0pt}\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
to11.78pt{\vbox to11.78pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-11.58179pt\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{0.0pt}{0.0pt}\pgfsys@lineto{5.69089pt}{-5.69089pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{5.69089pt}{-5.69089pt}\pgfsys@lineto{11.38179pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{5.69089pt}{-5.69089pt}\pgfsys@lineto{5.69089pt}{-11.38179pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{2.84544pt}{-2.84544pt}\pgfsys@lineto{5.69089pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\text{ and }u,v\in M.$

Combining these, we get that
$\delta_{\mathrm{rAvg}}(\mu_{1},\beta_{1},P_{1})=0.$ In other words,
$(\mu_{1},\beta_{1},P_{1})\in Z^{2}_{\mathrm{rAvg}}(M\xrightarrow{P}A)$ is a
$2$-cocycle in the cochain complex of the relative averaging algebra
$M\xrightarrow{P}A$ with coefficients in the adjoint bimodule.

Next, let $(\mu_{t},l_{t},r_{t},P_{t})$ and
$(\mu^{\prime}_{t},l^{\prime}_{t},r^{\prime}_{t},P^{\prime}_{t})$ be two
equivalent formal deformations of a relative averaging algebra
$M_{l_{M},r_{M}}\xrightarrow{P}A_{\mu}$. Then it follows from Definition
6.3(ii) that there exists a pair $(\varphi_{t},\psi_{t})$ of formal sums such
that the followings are hold:

$\displaystyle\sum_{i+j=n}\varphi_{i}(\mu_{j}(a,b))=~{}$
$\displaystyle\sum_{i+j+k=n}\mu_{i}^{\prime}\big{(}\varphi_{j}(a),\varphi_{k}(b)\big{)},$
$\displaystyle\sum_{i+j=n}\psi_{i}(l_{j}(a,u))=~{}$
$\displaystyle\sum_{i+j+k=n}l_{i}^{\prime}\big{(}\varphi_{j}(a),\psi_{k}(u)\big{)},$
$\displaystyle\sum_{i+j=n}\psi_{i}(r_{j}(u,a))=~{}$
$\displaystyle\sum_{i+j+k=n}r_{i}^{\prime}\big{(}\psi_{j}(u),\varphi_{k}(a)\big{)},$
$\displaystyle\sum_{i+j=n}\varphi_{i}\circ P_{j}=~{}$
$\displaystyle\sum_{i+j=n}P_{i}^{\prime}\circ\psi_{j},$

for $a,b\in A$ and $u\in M$. As before, all these relations are hold for $n=0$
as $\varphi_{0}=\mathrm{id}_{A}$ and $\psi_{0}=\mathrm{id}_{M}$. However, for
$n=1$, we get that

$\displaystyle\quad\mu_{1}(a,b)-\mu_{1}^{\prime}(a,b)=~{}$ $\displaystyle
a\cdot\varphi_{1}(b)+\varphi_{1}(a)\cdot b-\varphi_{1}(a\cdot b),$
$\displaystyle\beta_{1}(a,u)-\beta_{1}^{\prime}(a,u)=~{}$ $\displaystyle
a\cdot_{M}\psi_{1}(u)+\varphi_{1}(a)\cdot_{M}u-\psi_{1}(a\cdot_{M}u),$
$\displaystyle\beta_{1}(u,a)-\beta_{1}^{\prime}(u,a)=~{}$ $\displaystyle
u\cdot_{M}\varphi_{1}(a)+\psi_{1}(u)\cdot_{M}a-\psi_{1}(u\cdot_{M}a),$
$\displaystyle\quad P_{1}-P_{1}^{\prime}=~{}$ $\displaystyle
P\circ\psi_{1}-\varphi_{1}\circ P.$

Combining these relations, we get that

$\displaystyle(\mu_{1},\beta_{1},P_{1})-(\mu_{1}^{\prime},\beta_{1}^{\prime},P_{1})=\big{(}\delta_{\mathrm{Hoch}}(\varphi_{1}),\delta_{\mathrm{Hoch}}^{\varphi_{1}}(\psi_{1}),h_{P}(\varphi_{1},\psi_{1})\big{)}=\delta_{\mathrm{rAvg}}((\varphi_{1},\psi_{1})).$

Thus, equivalent formal deformations give rise to cohomologous $2$-cocycles.

As mentioned earlier that an infinitesimal deformation of a relative averaging
algebra is a $\mathsf{R}$-deformation, where $\mathsf{R}={\bf
k}[[t]]/(t^{2})$. Thus, an infinitesimal deformation can be regarded as a
truncated version (modulo $t^{2}$) of formal deformation. Equivalences between
infinitesimal deformations can be defined similarly.

######  6.4 Theorem.

Let $M\xrightarrow{P}A$ be a relative averaging algebra. Then there is a
bijection

$\displaystyle(\text{infinitesimal deformations of
}M\xrightarrow{P}A)/\sim\quad\leftrightsquigarrow\quad
H^{2}_{\mathrm{rAvg}}(M\xrightarrow{P}A).$

###### Proof.

Let $(\mu_{t}=\mu+t\mu_{1},l_{t}=l+tl_{1},r_{t}=r+tr_{1},P_{t}=P+tP_{1})$ be
an infinitesimal deformation of the relative averaging algebra
$M\xrightarrow{P}A$. Then one can show that (similar to formal deformation)
the triple $(\mu_{1},\beta_{1},P_{1})\in
Z^{2}_{\mathrm{rAvg}}(M\xrightarrow{P}A)$ is a $2$-cocycle, where
$\beta_{1}\in\mathrm{Hom}(\mathcal{A}^{1,1},M)$ is given by (31). Moreover,
equivalent infinitesimal deformations give rise to cohomologous $2$-cocycles.
Hence, there is a well-defined map

$\displaystyle\Gamma:(\text{infinitesimal deformations of
}M\xrightarrow{P}A)/\sim~{}~{}\rightarrow~{}~{}H^{2}_{\mathrm{rAvg}}(M\xrightarrow{P}A).$

To obtain a map in the other direction, we first consider a $2$-cocycle
$(\mu_{1},\beta_{1},P_{1})\in Z^{2}_{\mathrm{rAvg}}(M\xrightarrow{P}A)$. Then
it is easy to see that the $2$-cocycle $(\mu_{1},\beta_{1},P_{1})$ induces an
infinitesimal deformation

$\displaystyle(\mu_{t}=\mu+t\mu_{1},l_{t}=l+tl_{1},r_{t}=r+tr_{1},P_{t}=P+tP_{1})$

of the relative averaging algebra $M\xrightarrow{P}A$, where the maps
$l_{1},r_{1}$ are defined from $\beta_{1}$ by (31). Let
$(\mu_{1}^{\prime},\beta_{1}^{\prime},P_{1}^{\prime})$ be another $2$-cocycle
cohomologous to $(\mu_{1},\beta_{1},P_{1})$, i.e.
$(\mu_{1},\beta_{1},P_{1})-(\mu_{1}^{\prime},\beta_{1}^{\prime},P_{1}^{\prime})=\delta_{\mathrm{rAvg}}((\varphi_{1},\psi_{1}))$,
for some $(\varphi_{1},\psi_{1})\in C^{1}_{\mathrm{rAvg}}(M\xrightarrow{P}A)$.
Then it is easy to verify that the corresponding infinitesimal deformations
$(\mu_{t},l_{t},r_{t},P_{t})$ and
$(\mu^{\prime}_{t},l^{\prime}_{t},r^{\prime}_{t},P^{\prime}_{t})$ are
equivalent via the pair
$(\varphi_{t}=\mathrm{id}_{A}+t\varphi_{1},\psi_{t}=\mathrm{id}_{M}+t\psi_{1})$.
As a consequence, we obtain a map

$\displaystyle\Theta:H^{2}_{\mathrm{rAvg}}(M\xrightarrow{P}A)~{}~{}\rightarrow~{}~{}(\text{infinitesimal
deformations of }M\xrightarrow{P}A)/\sim.$

Finally, it is routine task to check that the maps $\Gamma$ and $\Theta$ are
inverses to each other. This completes the proof. ∎

##  7\. Abelian extensions of relative averaging algebras

Our aim in this section is to study abelian extensions of a relative averaging
algebra $M\xrightarrow{P}A$ by a bimodule $(N\xrightarrow{Q}B,l,r)$ of it. We
show that the isomorphism classes of such abelian extensions are in bijective
correspondence with the second cohomology group
$H^{2}_{\mathrm{rAvg}}(M\xrightarrow{P}A;N\xrightarrow{Q}B)$.

Let $M\xrightarrow{P}A$ be a relative averaging algebra and
$N\xrightarrow{Q}B$ be a $2$-term chain complex (not necessarily a bimodule).
Note that $N\xrightarrow{Q}B$ can be regarded as a relative averaging algebra
with the trivial associative multiplication on $B$ and the trivial
$B$-bimodule structure on $N$. With this consideration, we have the following
definition.

######  7.1 Definition.

An abelian extension of a relative averaging algebra $M\xrightarrow{P}A$ by a
$2$-term chain complex $N\xrightarrow{Q}B$ is a relative averaging algebra
$\widehat{M}\xrightarrow{\widehat{P}}\widehat{A}$ with a short exact sequence
of relative averaging algebras of the form

(36)

Sometimes, we denote an abelian extension as above by the relative averaging
algebra $\widehat{M}\xrightarrow{\widehat{P}}\widehat{A}$ when the exact
sequence is understood.

A section of the abelian extension (36) is given by a pair $(s,\overline{s})$
of linear maps $s:A\rightarrow\widehat{A}$ and
$\overline{s}:M\rightarrow\widehat{M}$ satisfying $p\circ s=\mathrm{id}_{A}$
and $\overline{p}\circ\overline{s}=\mathrm{id}_{M}$. Given any section
$(s,\overline{s})$, we define two bilinear maps (both denoted by the same
notation) $\cdot_{B}:A\times B\rightarrow B$ and $\cdot_{B}:B\times
A\rightarrow B$ by

$\displaystyle a\cdot_{B}b=s(a)\cdot_{\widehat{A}}i(b)~{}~{}~{}~{}\text{ and
}~{}~{}~{}~{}b\cdot_{B}a=i(b)\cdot_{\widehat{A}}s(a),\text{ for }a\in A,b\in
B.$

These two maps make $B$ into an $A$-bimodule. Similarly, there are two
bilinear maps (both denoted by the same notation) $\cdot_{N}:A\times
N\rightarrow N$ and $\cdot_{N}:N\times A\rightarrow N$ given by

$\displaystyle
a\cdot_{N}n=s(a)\cdot_{\widehat{M}}\overline{i}(n)~{}~{}~{}~{}\text{ and
}~{}~{}~{}~{}n\cdot_{N}a=\overline{i}(n)\cdot_{\widehat{M}}s(a),\text{ for
}a\in A,n\in N.$

Here $\cdot_{\widehat{M}}$ denotes both the left and right
$\widehat{A}$-actions on $\widehat{M}$. These two maps make $N$ into an
$A$-bimodule. Finally, we define bilinear maps $l:M\times B\rightarrow N$ and
$r:B\times M\rightarrow N$ by

$\displaystyle l(u,b)=\overline{s}(u)\cdot_{\widehat{M}}i(b)~{}~{}~{}~{}\text{
and }~{}~{}~{}~{}r(b,u)=i(b)\cdot_{\widehat{M}}\overline{s}(u),\text{ for
}u\in M,b\in B.$

It is straightforward to see that the maps $l,r$ satisfy the identities (10)
and (11). Finally, for any $u\in M$ and $n\in N$,

$\displaystyle
P(u)\cdot_{B}Q(n)=sP(u)\cdot_{\widehat{A}}iQ(n)=\widehat{P}\overline{s}(u)\cdot_{\widehat{A}}\widehat{P}\overline{i}(n)=~{}$
$\displaystyle\begin{cases}=\widehat{P}\big{(}\widehat{P}(\overline{s}(u))\cdot_{\widehat{M}}\overline{i}(n)\big{)}\\\
=\widehat{P}\big{(}\overline{s}(u)\cdot_{\widehat{M}}\widehat{P}\overline{i}(n)\big{)}\end{cases}$
$\displaystyle=~{}$
$\displaystyle\begin{cases}=\widehat{P}\big{(}{s}P(u)\cdot_{\widehat{M}}\overline{i}(n)\big{)}~{}=~{}Q\big{(}P(u)\cdot_{N}n\big{)},\\\
=\widehat{P}\big{(}\overline{s}(u)\cdot_{\widehat{M}}{i}Q(n)\big{)}~{}=~{}Q\big{(}l(u,Q(n))\big{)}.\end{cases}$

Similarly, one can show that

$\displaystyle
Q(n)\cdot_{B}P(u)=iQ(n)\cdot_{\widehat{A}}sP(u)=\widehat{P}\overline{i}(n)\cdot_{\widehat{A}}\widehat{P}\overline{s}(u)=~{}$
$\displaystyle\begin{cases}=\widehat{P}\big{(}\widehat{P}\overline{i}(n)\cdot_{\widehat{M}}\overline{s}(u)\big{)}\\\
=\widehat{P}\big{(}\overline{i}(n)\cdot_{\widehat{M}}\widehat{P}\overline{s}(u)\big{)}\end{cases}$
$\displaystyle=~{}$
$\displaystyle\begin{cases}=\widehat{P}\big{(}iQ(n)\cdot_{\widehat{M}}\overline{s}(u)\big{)}~{}=~{}Q\big{(}r(Q(n),u)\big{)},\\\
=\widehat{P}\big{(}\overline{i}(n)\cdot_{\widehat{M}}sP(u)\big{)}~{}=~{}Q(n\cdot_{N}P(u)).\end{cases}$

Combining all these, we get that $(N\xrightarrow{Q}B,l,r)$ is a bimodule over
the relative averaging algebra $M\xrightarrow{P}A$. This is called the induced
bimodule structure starting from the abelian extension $(\ref{abl-fig})$. Note
that this bimodule structure is independent of the choice of section. To see
this, let $(s^{\prime},\overline{s}^{\prime})$ be any other section of
$(\ref{abl-fig})$. Then we observe that
$s(a)-s^{\prime}(a)\in\mathrm{ker}(p)=\mathrm{im}(i)$ and
$\overline{s}(u)-\overline{s}^{\prime}(u)\in\mathrm{ker}(\overline{p})=\mathrm{im}(\overline{i})$,
for $a\in A$ and $u\in M$. Let $\cdot_{B}^{\prime}$, $\cdot_{N}^{\prime}$ and
$l^{\prime},r^{\prime}$ be the maps induced by the section
$(s^{\prime},\overline{s}^{\prime})$. Then we have

$\displaystyle
a\cdot_{B}b-a\cdot_{B}^{\prime}b=\big{(}s(a)-s^{\prime}(a)\big{)}\cdot_{\widehat{A}}i(b)=0~{}~{}\text{
and
}~{}~{}b\cdot_{B}a-b\cdot_{B}^{\prime}a=i(b)\cdot_{\widehat{A}}\big{(}s(a)-s^{\prime}(a)\big{)}=0,$
$\displaystyle
a\cdot_{N}n-a\cdot_{N}^{\prime}n=\big{(}s(a)-s^{\prime}(a)\big{)}\cdot_{\widehat{M}}\overline{i}(n)=0~{}~{}\text{
and
}~{}~{}n\cdot_{N}a-n\cdot_{N}^{\prime}a=\overline{i}(n)\cdot_{\widehat{M}}\big{(}s(a)-s^{\prime}(a)\big{)}=0,$
$\displaystyle
l(u,b)-l^{\prime}(u,b)=\big{(}\overline{s}(u)-\overline{s}^{\prime}(u)\big{)}\cdot_{\widehat{M}}i(b)=0~{}~{}\text{
and
}~{}~{}r(b,u)-r^{\prime}(b,u)=i(b)\cdot_{\widehat{M}}\big{(}\overline{s}(u)-\overline{s}^{\prime}(u)\big{)}=0.$

Hence our claim follows.

######  7.2 Definition.

Let $M\xrightarrow{P}A$ be a relative averaging algebra and
$N\xrightarrow{Q}B$ be a $2$-term chain complex. Two abelian extensions
$\widehat{M}\xrightarrow{\widehat{P}}\widehat{A}$ and
$\widehat{M}^{\prime}\xrightarrow{\widehat{P}^{\prime}}\widehat{A}^{\prime}$
are said to be isomorphic if there is an isomorphism
$(\varphi,\psi):(\widehat{M}\xrightarrow{\widehat{P}}\widehat{A})\rightsquigarrow(\widehat{M}^{\prime}\xrightarrow{\widehat{P}^{\prime}}\widehat{A}^{\prime})$
of relative averaging algebras that makes the following diagram commutative

(45)

Let $\widehat{M}\xrightarrow{\widehat{p}}\widehat{A}$ and
$\widehat{M}^{\prime}\xrightarrow{\widehat{p}^{\prime}}\widehat{A}^{\prime}$
be two isomorphic abelian extensions as in the above definition. Then it is
easy to see that the corresponding induced bimodules on the $2$-term chain
complex $N\xrightarrow{Q}B$ are the same.

######  7.3 Notation.

Let $M\xrightarrow{P}A$ be a relative averaging algebra and
$(N\xrightarrow{Q}B.l,r)$ be a given bimodule over it. We denote by
$\mathrm{Ext}(M\xrightarrow{{P}}{A};N\xrightarrow{Q}B)$ the set of all
isomorphism classes of abelian extensions of $M\xrightarrow{P}A$ by the
$2$-term complex $N\xrightarrow{Q}B$ so that the induced bimodule coincides
with the prescribed one.

In the following result, we parametrize the space
$\mathrm{Ext}(M\xrightarrow{{P}}{A};N\xrightarrow{Q}B)$ by the second
cohomology group of the relative averaging algebra.

######  7.4 Theorem.

Let $M\xrightarrow{P}A$ be a relative averaging algebra and
$(N\xrightarrow{Q}B,l,r)$ be a given bimodule over it. Then there is a
bijective correspondence between
$\mathrm{Ext}(M\xrightarrow{{P}}{A};N\xrightarrow{Q}B)$ and the second
cohomology group
$H^{2}_{\mathrm{rAvg}}(M\xrightarrow{{P}}{A};N\xrightarrow{Q}B).$

###### Proof.

Let $\widehat{M}\xrightarrow{\widehat{P}}\widehat{A}$ be an abelian extension
of the relative averaging algebra $M\xrightarrow{P}A$ by the $2$-term complex
$N\xrightarrow{Q}B$ representing an element in
$\mathrm{Ext}(M\xrightarrow{P}A;N\xrightarrow{Q}B)$. Let $(s,\overline{s})$ be
a section. Then we define a triple $(\alpha,\beta,\gamma)$ of maps

$\displaystyle\alpha\in\mathrm{Hom}(A^{\otimes
2},B),\quad\alpha(a,b)=s(a)\cdot_{\widehat{A}}s(b)-s(a\cdot b),$
$\displaystyle\beta\in\mathrm{Hom}(\mathcal{A}^{1,1},N),\quad\begin{cases}\beta(a,u)=s(a)\cdot_{\widehat{M}}\overline{s}(u)-\overline{s}(a\cdot_{M}u),\\\
\beta(u,a)=\overline{s}(u)\cdot_{\widehat{M}}{s}(a)-\overline{s}(u\cdot_{M}a),\end{cases}$
$\displaystyle\gamma\in\mathrm{Hom}(M,B),\quad\gamma(u)=(\widehat{P}\circ\widehat{s}-s\circ
P)(u),$

for $a,b\in A$ and $u\in M$. Then it is easy to verify that
$(\alpha,\beta,\gamma)\in
Z^{2}_{\mathrm{rAvg}}(M\xrightarrow{P}A;N\xrightarrow{Q}B)$ is a $2$-cocycle
in the cohomology complex of the relative averaging algebra
$M\xrightarrow{P}A$ with coefficients in the bimodule
$(N\xrightarrow{Q}B,l,r)$. Moreover, the corresponding cohomology class in
$H^{2}_{\mathrm{rAvg}}(M\xrightarrow{P}A;N\xrightarrow{Q}B)$ doesn’t depend on
the choice of the section.

Let $\widehat{M}\xrightarrow{\widehat{P}}\widehat{A}$ and
$\widehat{M}^{\prime}\xrightarrow{\widehat{P}^{\prime}}\widehat{A}^{\prime}$
be two isomorphic abelian extensions. For any section $(s,\overline{s})$ of
the first abelian extension, we have

$\displaystyle p^{\prime}\circ(\varphi\circ s)=p\circ
s=\mathrm{id}_{A}\quad\text{ and
}\quad\overline{p}^{\prime}\circ(\psi\circ\overline{s})=\overline{p}\circ\overline{s}=\mathrm{id}_{M}.$

Thus $(\varphi\circ s,\psi\circ\overline{s})$ is a section of the second
abelian extension. If $(\alpha^{\prime},\beta^{\prime},\gamma^{\prime})\in
Z^{2}_{\mathrm{rAvg}}(M\xrightarrow{P}A;N\xrightarrow{Q}B)$ is the $2$-cocycle
corresponding to the second abelian extension and its section $(\varphi\circ
s,\psi\circ\overline{s})$, then

$\displaystyle\alpha^{\prime}(a,b)=~{}$ $\displaystyle(\varphi\circ
s)(a)\cdot_{\widehat{A}^{\prime}}(\varphi\circ s)(b)-(\varphi\circ s)(a\cdot
b)$ $\displaystyle=~{}$
$\displaystyle\varphi\big{(}s(a)\cdot_{\widehat{A}}s(b)-s(a\cdot
b)\big{)}=\varphi(\alpha(a,b))=\alpha(a,b)\quad(\because\varphi|_{B}=\mathrm{id}_{B}).$

Similarly, one can show that $\beta^{\prime}=\beta$ and
$\gamma^{\prime}=\gamma$. Thus, we obtain
$(\alpha,\beta,\gamma)=(\alpha^{\prime},\beta^{\prime},\gamma^{\prime})$. As a
consequence, we obtain a well-defined map

$\displaystyle\Lambda:\mathrm{Ext}(M\xrightarrow{P}A;N\xrightarrow{Q}B)~{}\rightarrow~{}H^{2}_{\mathrm{rAvg}}(M\xrightarrow{P}A;N\xrightarrow{Q}B).$

To obtain a map in the other direction, we take a $2$-cocycle
$(\alpha,\beta,\gamma)\in
Z^{2}_{\mathrm{rAvg}}(M\xrightarrow{P}A;N\xrightarrow{Q}B)$. Take
$\widehat{A}=A\oplus B$ and $\widehat{M}=M\oplus N$, and consider bilinear
maps

$\displaystyle\mu_{\widehat{A}}:\widehat{A}\times\widehat{A}\rightarrow\widehat{A},\quad$
$\displaystyle\mu_{\widehat{A}}\big{(}(a,b),(a^{\prime},b^{\prime})\big{)}=\big{(}a\cdot
a^{\prime},a\cdot_{B}b^{\prime}+b\cdot_{B}a^{\prime}+\alpha(a,a^{\prime})\big{)},$
$\displaystyle
l_{\widehat{M}}:\widehat{A}\times\widehat{M}\rightarrow\widehat{M},\quad$
$\displaystyle
l_{\widehat{M}}\big{(}(a,b),(u,n)\big{)}=\big{(}a\cdot_{M}u,a\cdot_{N}n+r(b,u)+\beta(a,u)\big{)},$
$\displaystyle
r_{\widehat{M}}:\widehat{M}\times\widehat{A}\rightarrow\widehat{M},\quad$
$\displaystyle
r_{\widehat{M}}\big{(}(u,n),(a,b)\big{)}=\big{(}u\cdot_{M}a,l(u,b)+n\cdot_{N}a+\beta(u,a)\big{)},$

for $(a,b),(a^{\prime},b^{\prime})\in\widehat{A}$ and $(u,n)\in\widehat{M}$.
Then it is easy to see that $(\widehat{A},\mu_{\widehat{A}})$ is an
associative algebra and $(\widehat{M},l_{\widehat{M}},r_{\widehat{M}})$ is a
bimodule over it. Finally, we define a map
$\widehat{P}:\widehat{M}\rightarrow\widehat{A}$ by

$\displaystyle\widehat{P}((u,n))=\big{(}P(u),Q(n)+\gamma(u)\big{)},\text{ for
}(u,n)\in\widehat{M}.$

Then $\widehat{P}$ is a relative averaging operator. In other words,
$\widehat{M}\xrightarrow{\widehat{P}}\widehat{A}$ is a relative averaging
algebra. This is an abelian extension of the relative averaging algebra
$M\xrightarrow{P}A$ by the $2$-term chain complex $N\xrightarrow{Q}B$, and
defines an element in $\mathrm{Ext}(M\xrightarrow{P}A;N\xrightarrow{Q}B)$.
Finally, let $(\alpha,\beta,\gamma)$ and
$(\alpha^{\prime},\beta^{\prime},\gamma^{\prime})$ be two cohomologous
$2$-cocycles, say
$(\alpha,\beta,\gamma)-(\alpha^{\prime},\beta^{\prime},\gamma^{\prime})=\delta_{\mathrm{rAvg}}((\kappa,\eta))$,
for some $(\kappa,\eta)\in
C^{1}_{\mathrm{rAvg}}(M\xrightarrow{P}A;N\xrightarrow{Q}B)$. If
$\widehat{M}^{\prime}\xrightarrow{\widehat{P}^{\prime}}\widehat{A}^{\prime}$
is the relative averaging algebra induced by the $2$-cocycle
$(\alpha^{\prime},\beta^{\prime},\gamma^{\prime})$, then the pair of maps

$\displaystyle(\varphi,\psi):(\widehat{M}\xrightarrow{\widehat{P}}\widehat{A})\rightsquigarrow(\widehat{M}^{\prime}\xrightarrow{\widehat{P}^{\prime}}\widehat{A}^{\prime})$

is an isomorphism of abelian extensions, where
$\varphi:\widehat{A}\rightarrow\widehat{A}^{\prime}$,
$\varphi(a,b)=(a,b+\kappa(a))$ and
$\psi:\widehat{M}\rightarrow\widehat{M}^{\prime}$, $\psi(u,n)=(u,n+\eta(u))$.
This shows that there is a well-defined map

$\displaystyle\Upsilon:H^{2}_{\mathrm{rAvg}}(M\xrightarrow{P}A;N\xrightarrow{Q}B)~{}\rightarrow~{}\mathrm{Ext}(M\xrightarrow{P}A;N\xrightarrow{Q}B).$

Finally, the maps $\Lambda$ and $\Upsilon$ are inverses to each other. This
shows the required bijection. ∎

##  8\. Homotopy relative averaging algebras and homotopy diassociative
algebras

In this section, we first consider $Diass_{\infty}$-algebras introduced by
Loday. However, our definition is more simple to use. Next, we introduce the
homotopy relative averaging operators and homotopy relative averaging
algebras. We show that a homotopy relative averaging algebra naturally induces
a $Diass_{\infty}$-algebra structure. We first recall some basic definitions
related to $A_{\infty}$-algebras [17].

######  8.1 Definition.

An $A_{\infty}$-algebra is a pair $({A},\\{\mu_{k}\\}_{k=1}^{\infty})$
consisting of a graded vector space ${A}=\oplus_{i\in\mathbb{Z}}{A}_{i}$
together with a collection $\\{\mu_{k}\\}_{k=1}^{\infty}$ of degree $1$ graded
linear maps $\mu_{k}:{A}^{\otimes k}\rightarrow{A}$, for $k\geq 1$, satisfying
the following identities (called higher associativities)

(46)
$\displaystyle\sum_{k+l=n+1}\sum_{i=1}^{n-l+1}(-1)^{|a_{1}|+\cdots+|a_{i-1}|}~{}\mu_{k}\big{(}a_{1},\ldots,a_{i-1},\mu_{l}(a_{i},\ldots,a_{i+l-1}),a_{i+l},\ldots,a_{n})=0,$

for all $n\geq 1$ and homogeneous elements $a_{1},\ldots,a_{n}\in{A}.$

Let ${A}=\oplus_{i\in\mathbb{Z}}{A}_{i}$ be a graded vector space. Let
$\overline{T}(A)=\oplus_{n=1}^{\infty}A^{\otimes n}$ be the free tensor
algebra over the graded vector space $A$. For each $n\in\mathbb{Z}$, let
$C^{n}(A,A):=\mathrm{Hom}_{n}(\overline{T}(A),A)$ be the space of all degree
$n$ graded linear maps from the graded vector space $\overline{T}(A)$ to $A$.
Thus, an element $\mu\in C^{n}(A,A)$ is given by a sum
$\mu=\sum_{k=1}^{\infty}\mu_{k}$, where $\mu_{k}:A^{\otimes k}\rightarrow A$
is a degree $n$ linear map, for $k\geq 1$. For
$\mu=\sum_{k=1}^{\infty}\mu_{k}\in C^{m}(A,A)$ and
$\nu=\sum_{l=1}^{\infty}\nu_{l}\in C^{n}(A,A)$, we define a bracket
$[\mu,\nu]\in C^{m+n}(A,A)$ by

$\displaystyle[\mu,\nu]:=\sum_{s=1}^{\infty}\sum_{k+l=s+1}\big{(}\mu_{k}\circ\nu_{l}-(-1)^{mn}~{}\nu_{l}\circ\mu_{k}\big{)},\text{
where }$
$\displaystyle(\mu_{k}\circ\nu_{l})(a_{1},\ldots,a_{s})=\sum_{i=1}^{s-l+1}(-1)^{|a_{1}|+\cdots+|a_{i-1}|}~{}\mu_{k}\big{(}a_{1},\ldots,a_{i-1},\nu_{l}(a_{i},\ldots,a_{i+l-1}),a_{i+l},\ldots,a_{s}\big{)}.$

The graded vector space $\oplus_{n\in\mathbb{Z}}C^{n}(A,A)$ with the above
bracket is a graded Lie algebra. An element $\mu=\sum_{k=1}^{\infty}\mu_{k}\in
C^{1}(A,A)$ is a Maurer-Cartan element of the graded Lie algebra
$(\oplus_{n\in\mathbb{Z}}C^{n}(A,A),[~{},~{}])$ if and only if the pair
$(A,\\{\mu_{k}\\}_{k=1}^{\infty})$ is an $A_{\infty}$-algebra.

Let $(A,\\{\mu_{k}\\}_{k=1}^{\infty})$ be an $A_{\infty}$-algebra. A
representation of this $A_{\infty}$-algebra is given by a pair
$(M,\\{\eta_{k}\\}_{k=1}^{\infty})$ that consists of a graded vector space
$M=\oplus_{i\in\mathbb{Z}}M_{i}$ with a collection
$\\{\eta_{k}:\mathcal{A}^{k-1,1}\rightarrow M\\}_{k=1}^{\infty}$ of degree $1$
linear maps satisfying the identities (46) when exactly one of the variables
$a_{1},\ldots,a_{n}$ comes from $M$ and the corresponding linear operation
$\mu_{i}$ or $\mu_{j}$ replaced by $\eta_{i}$ or $\eta_{j}$. Like the ungraded
case, here $\mathcal{A}^{k-1,1}$ denotes the direct sum of all possible tensor
powers of $A$ and $M$ in which $A$ appears $k-1$ times (and hence $M$ appears
exactly once). Note that any $A_{\infty}$-algebra
$(A,\\{\mu_{k}\\}_{k=1}^{\infty})$ can be realized as a representation of
itself, where $\eta_{k}=\mu_{k}$, for $k\geq 1$.

######  8.2 Definition.

A $Diass_{\infty}$-algebra (also called a strongly homotopy diassociative
algebra) is a pair $(D,\\{\pi_{k}\\}_{k=1}^{\infty})$ consisting of a graded
vector space $D=\oplus_{i\in\mathbb{Z}}D_{i}$ equipped with a collection of
degree $1$ graded linear maps $\\{\pi_{k}:{\bf k}[Y_{k}]\otimes D^{\otimes
k}\rightarrow D\\}_{k=1}^{\infty}$ satisfying the following set of identities

(47)
$\displaystyle\sum_{k+l=n+1}\sum_{i=1}^{n-l+1}(-1)^{|a_{1}|+\cdots+|a_{i-1}|}\pi_{k}\big{(}R_{0}^{k;i,l}(y);a_{1},\ldots,a_{i-1},\pi_{l}\big{(}R_{i}^{k;i,l}(y);a_{i},\ldots,a_{i+l-1}\big{)},a_{i+l},\ldots,a_{n}\big{)}=0,$

for all $n\geq 1$, $y\in Y_{n}$ and homogeneous elements
$a_{1},\ldots,a_{n}\in D$. The maps $R_{0}^{k;i,l}$ and $R_{i}^{k;i,l}$ are
described in (3), (4).

Note that any diassociative algebra can be realized as a
$Diass_{\infty}$-algebra concentrated in degree $-1$. More precisely, if
$(D,\dashv,\vdash)$ is a diassociative algebra then $s^{-1}D$ (considered as a
graded vector space with $(s^{-1}D)_{-1}=D$ and $(s^{-1}D)_{i}=0$ for
$i\neq-1$) can be given a $Diass_{\infty}$-algebra structure with the
operations $\\{\pi_{k}:{\bf k}[Y_{k}]\otimes(s^{-1}D)^{\otimes k}\rightarrow
s^{-1}D\\}_{k=1}^{\infty}$ given by

$\displaystyle\pi_{2}(\leavevmode\hbox to17.47pt{\vbox
to17.47pt{\pgfpicture\makeatletter\hbox{\hskip-25.40643pt\lower-17.27095pt\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{25.60643pt}{0.0pt}\pgfsys@lineto{34.1419pt}{-8.53548pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{34.1419pt}{-8.53548pt}\pgfsys@lineto{42.6774pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{34.1419pt}{-8.53548pt}\pgfsys@lineto{34.1419pt}{-17.07095pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{38.40965pt}{-4.26773pt}\pgfsys@lineto{34.1419pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}};s^{-1}a,s^{-1}b)=s^{-1}(a\dashv
b),\quad\pi_{2}(\leavevmode\hbox to17.47pt{\vbox
to17.47pt{\pgfpicture\makeatletter\hbox{\hskip-25.40643pt\lower-17.27095pt\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{25.60643pt}{0.0pt}\pgfsys@lineto{34.1419pt}{-8.53548pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{34.1419pt}{-8.53548pt}\pgfsys@lineto{42.6774pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{34.1419pt}{-8.53548pt}\pgfsys@lineto{34.1419pt}{-17.07095pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}
{}{}{}\pgfsys@moveto{29.87418pt}{-4.26773pt}\pgfsys@lineto{34.1419pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}};s^{-1}a,s^{-1}b)=s^{-1}(a\dashv
b)~{}~{}\text{ and }\pi_{k}=0\text{ for }k\neq 2.$

######  8.3 Remark.

Let $(D,\\{\pi_{k}\\}_{k=1}^{\infty})$ be any $Diass_{\infty}$-algebra. Using
the higher diassociative identities (47) and the mathematical induction on
$k$, we can show that

$\displaystyle\pi_{k}(y;a_{1},\ldots,a_{k})=\pi_{k}(y^{\prime};a_{1},\ldots,a_{k}),\text{
for }a_{1},\ldots,a_{k}\in D,$

when both of $y,y^{\prime}\in Y_{k}$ can be written as the grafting of a
$(i-1)$-tree and $(k-i)$-tree.

######  8.4 Proposition.

Let $(A,\\{\mu_{k}\\}_{k=1}^{\infty})$ be an $A_{\infty}$-algebra and
$(M,\\{\eta_{k}\\}_{k=1}^{\infty})$ be a representation of it. Then the graded
vector space $A\oplus M$ can be equipped with a $Diass_{\infty}$-algebra
structure with the operations $\\{\pi_{k}:{\bf k}[Y_{k}]\otimes(A\oplus
M)^{\otimes k}\rightarrow A\oplus M\\}_{k=1}^{\infty}$ given by

(48)
$\displaystyle\pi_{k}\big{(}y;(a_{1},u_{1}),\ldots,(a_{k},u_{k})\big{)}=\big{(}\mu_{k}(a_{1},\ldots,a_{k}),\eta_{k}(a_{1},\ldots,a_{i-1},u_{i},a_{i+1},\ldots,a_{k})\big{)},$

for $k\geq 1$, $y\in Y_{k}$ (which can be uniquely written as $y=y_{1}\vee
y_{2}$ for some $(i-1)$-tree $y_{1}\in Y_{i-1}$ and $(k-i)$-tree $y_{2}\in
Y_{k-i}$) and $(a_{1},u_{1}),\ldots,(a_{k},u_{k})\in A\oplus M.$

We denote the above $Diass_{\infty}$-algebra simply by
$A\oplus_{Diass_{\infty}}M$. Note that $A\oplus_{Diass_{\infty}}M$ generalizes
the diassociative algebra of Proposition 3.11 in the homotopy context. It is
important to mention that the converse of the above proposition is also true.
More precisely, let $A=\oplus_{i\in\mathbb{Z}}A_{i}$ and
$M=\oplus_{i\in\mathbb{Z}}M_{i}$ be two graded vector spaces equipped with two
collections $\\{\mu_{k}:A^{\otimes k}\rightarrow A\\}_{k=1}^{\infty}$ and
$\\{\eta_{k}:\mathcal{A}^{k-1,1}\rightarrow M\\}_{k=1}^{\infty}$ of degree $1$
graded linear maps. Then $(A,\\{\mu_{k}\\}_{k=1}^{\infty})$ is an
$A_{\infty}$-algebra and $(M,\\{\eta_{k}\\}_{k=1}^{\infty})$ is a
representation if and only if $(A\oplus M,\\{\pi_{k}\\}_{k=1}^{\infty})$ is a
$Diass_{\infty}$-algebra, where the maps $\pi_{k}$’s are given in (48).

In the following, we construct a graded Lie algebra whose Maurer-Cartan
elements correspond to $Diass_{\infty}$-algebra structures on a given graded
vector space. Let $D=\oplus_{i\in\mathbb{Z}}D_{i}$ be a graded vector space.
For each $n\in\mathbb{Z}$, we define the space
$CY^{n}(D,D):=\mathrm{Hom}_{n}({\bf k}[\overline{Y}]\otimes\overline{T}(D),D)$
whose elements are of the form $\pi=\sum_{k=1}^{\infty}\pi_{k}$, where
$\pi_{k}:{\bf k}[Y_{k}]\otimes D^{\otimes k}\rightarrow D$ is a degree $n$
linear map. For $\pi=\sum_{k=1}^{\infty}\pi_{k}\in CY^{m}(D,D)$ and
$\varpi=\sum_{l=1}^{\infty}\varpi_{l}\in CY^{l}(D,D)$, we define an element
$\\{\\![\pi,\varpi]\\!\\}\in CY^{m+n}(D,D)$ by

(49)
$\displaystyle\\{\\![\pi,\varpi]\\!\\}=\sum_{s=1}^{\infty}\sum_{k+l=s+1}(\pi_{k}\diamond\varpi_{l}-(-1)^{mn}\varpi_{l}\diamond\pi_{k}),~{}\text{
where }$
$\displaystyle(\pi_{k}\diamond\varpi_{l})(y;a_{1},\ldots,a_{s})=\sum_{i=1}^{s-l+1}(-1)^{|a_{1}|+\cdots+|a_{i-1}|}\pi_{k}\big{(}R_{0}^{k;i,l}(y);a_{1},\ldots,\varpi_{l}(R_{i}^{k;i,l}(y);a_{i},\ldots,a_{i+l-1}),\ldots,a_{s}\big{)},$

for $y\in Y_{s}=Y_{k+l-1}$ and homogeneous elements $a_{1},\ldots,a_{s}\in D$.
The bracket $\\{\\![~{},~{}[\\!\\}$ is the graded version of the Majumdar-
Mukherjee bracket given in (5). In particular, the bracket
$\\{\\![~{},~{}[\\!\\}$ makes the graded space
$\oplus_{n\in\mathbb{Z}}CY^{n}(D,D)=\oplus_{n\in\mathbb{Z}}\mathrm{Hom}_{n}({\bf
k}[\overline{Y}]\otimes\overline{T}(D),D)$ into a graded Lie algebra.

######  8.5 Proposition.

Let $D$ be a graded vector space. Then there is a one-to-one correspondence
between $Diass_{\infty}$-algebra structures on the graded vector space $D$ and
Maurer-Cartan elements of the graded Lie algebra
$(\oplus_{n\in\mathbb{Z}}CY^{n}(D,D),\\{\\![~{},~{}]\\!\\}).$

###### Proof.

Note that an element $\pi\in CY^{1}(D,D)=\mathrm{Hom}_{1}({\bf
k}[\overline{Y}]\otimes\overline{T}(D),D)$ is equivalent to having a
collection $\\{\pi_{k}:{\bf k}[Y_{k}]\otimes D^{\otimes k}\rightarrow
D\\}_{k=1}^{\infty}$ of degree $1$ graded linear maps. Then it follows from
(49) that $\\{\\![\pi,\pi]\\!\\}=0$ if and only if the collection
$\\{\pi_{k}\\}_{k=1}^{\infty}$ defines a $Diass_{\infty}$-algebra structure on
$D$. ∎

Given an $A_{\infty}$-algebra and a representation of it, we will now
introduce the notion of a homotopy relative averaging operator. Let
$(A,\\{\mu_{k}\\}_{k=1}^{\infty})$ be an $A_{\infty}$-algebra and
$(M,\\{\eta_{k}\\}_{k=1}^{\infty})$ be a representation of it. Consider the
graded Lie algebra

$\displaystyle\mathfrak{g}=\big{(}\oplus_{n\in\mathbb{Z}}CY^{n}(A\oplus
M,A\oplus M)=\oplus_{n\in\mathbb{Z}}\mathrm{Hom}_{n}({\bf
k}[\overline{Y}]\otimes\overline{T}(A\oplus M),A\oplus
M),\\{\\![~{},~{}]\\!\\}\big{)}$

associated to the graded vector space $A\oplus M.$ Then it is easy to see that
the graded subspace
$\mathfrak{a}=\oplus_{n\in\mathbb{Z}}CY^{n}(M,A)=\oplus_{n\in\mathbb{Z}}\mathrm{Hom}_{n}({\bf
k}[\overline{Y}]\otimes\overline{T}(M),A)$ is an abelian Lie subalgebra of
$\mathfrak{g}$. Let $p:\mathfrak{g}\rightarrow\mathfrak{g}$ be the projection
map onto the subspace $\mathfrak{a}$. On the other hand, since
$A\oplus_{Diass_{\infty}}M=(A\oplus M,\\{\pi_{k}\\}_{k=1}^{\infty})$ is a
$Diass_{\infty}$-algebra, it defines a Maurer-Cartan element
$\pi=\sum_{k=1}^{\infty}\pi_{k}\in CY^{1}(A\oplus M,A\oplus M)$ of the graded
Lie algebra $\mathfrak{g}$ (i.e. $\\{\\![\pi,\pi]\\!\\}=0$). Further, the
element $\pi\in\mathrm{ker}(p)_{1}$. Hence we obtain a $V$-data
$(\mathfrak{g},\mathfrak{a},p,\pi)$. Therefore, by Theorem 5.4 (i), the graded
vector space $\mathfrak{a}$ inherits a $L_{\infty}$-algebra structure with the
operations $\\{l_{k}:\mathfrak{a}^{\otimes
k}\rightarrow\mathfrak{a}\\}_{k=1}^{\infty}$ given by

$\displaystyle
l_{k}(\gamma_{1},\ldots,\gamma_{k})=p\\{\\![\cdots\\{\\![\\{\\![\pi,\gamma_{1}]\\!\\},\gamma_{2}]\\!\\},\ldots,a_{k}]\\!\\},$

for homogeneous $\gamma_{1},\ldots,\gamma_{k}\in\mathfrak{a}$. This
$L_{\infty}$-algebra can be seen as the homotopy analogue of the graded Lie
algebra given in Theorem 4.1. Our next definition is motivated by the Maurer-
Cartan characterization of a relative averaging operator given in Theorem 4.1.

######  8.6 Definition.

A homotopy relative averaging operator on $(M,\\{\eta_{k}\\}_{k=1}^{\infty})$
over the $A_{\infty}$-algebra $(A,\\{\mu_{k}\\}_{k=1}^{\infty})$ is a Maurer-
Cartan element of the $L_{\infty}$-algebra
$(\mathfrak{a},\\{l_{k}\\}_{k=1}^{\infty}).$

It follows from the above definition that a homotopy relative averaging
operator is an element $P=\sum_{k=1}^{\infty}P_{k}\in\mathrm{Hom}_{0}({\bf
k}[\overline{Y}]\otimes\overline{T}(M),A)$ that satisfies

(50) $\displaystyle\sum_{k=1}^{\infty}\frac{1}{k!}l_{k}(P,\ldots,P)=0.$

In other words, $P$ must satisfy
$\sum_{k=1}^{\infty}\frac{1}{k!}p\\{\\![\cdots\\{\\![\\{\\![\pi,P]\\!\\},P]\\!\\},\ldots,P]\\!\\}=0$,
which is equivalent to the condition that $p(e^{\\{\\![-,P]\\!\\}}\pi)=0$.
Note that a homotopy relative averaging operator can be equivalently described
by a collection $P=\\{P_{k}:{\bf k}[Y_{k}]\otimes M^{\otimes k}\rightarrow
A\\}_{k=1}^{\infty}$ of degree $0$ linear maps satisfying
$p(e^{\\{\\![-,P]\\!\\}}\pi)=0$.

######  8.7 Definition.

A homotopy relative averaging algebra is a triple $(A,M,P)$ consisting of an
$A_{\infty}$-algebra $A=(A,\\{\mu_{k}\\}_{k=1}^{\infty})$, a representation
$M=(M,\\{\eta_{k}\\}_{k=1}^{\infty})$ and a homotopy relative averaging
operator $P=\\{P_{k}\\}_{k=1}^{\infty}$. We often denote a homotopy relative
averaging algebra as above by $M\xrightarrow{\\{P_{k}\\}_{k=1}^{\infty}}A$.

######  8.8 Proposition.

Let $M\xrightarrow{\\{P_{k}\\}_{k=1}^{\infty}}A$ be a homotopy relative
averaging algebra. Then $(M,\\{\pi_{k}^{P}\\}_{k=1}^{\infty})$ is a
$Diass_{\infty}$-algebra, where

$\displaystyle\pi_{k}^{P}(y;u_{1},\ldots,u_{k})=(e^{\\{\\![-,P]\\!\\}}\pi)(y;u_{1},\ldots,u_{k}),\text{
for }k\geq 1,y\in Y_{k}\text{ and }u_{1},\ldots,u_{k}\in M.$

###### Proof.

Note that

$\displaystyle\\{\\![e^{\\{\\![-,P]\\!\\}}\pi,e^{\\{\\![-,P]\\!\\}}\pi]\\!\\}=e^{\\{\\![-,P]\\!\\}}\\{\\![\pi,\pi]\\!\\}=0~{}~{}~{}(\text{as
}\\{\\![\pi,\pi]\\!\\}=0).$

This shows that $e^{\\{\\![-,P]\\!\\}}\pi$ is a Maurer-Cartan element of the
graded Lie algebra $\mathfrak{g}$. Hence the collection of maps
$\\{\pi_{k}\\}_{k=1}^{\infty}$ defines a $Diass_{\infty}$-algebra structures
on $M$, where $\pi_{k}=(e^{\\{\\![-,P]\\!\\}}\pi)|_{{\bf k}[Y_{k}]\otimes
M^{\otimes k}}$, for $k\geq 1$. This completes the proof. ∎

A homotopy relative averaging operator $\\{P_{k}\\}_{k=1}^{\infty}$ is said to
be strict if $P_{k}=0$ for $k\neq 1$. It follows from (50) that a strict
homotopy relative averaging operator is a degree $0$ linear map
$P:M\rightarrow A$ that satisfies

$\displaystyle\mu_{k}\big{(}P(u_{1}),\ldots,P(u_{k})\big{)}=P\big{(}\eta_{k}(P(u_{1}),\ldots,u_{i},\ldots,P(u_{k}))\big{)},\text{
for }k\geq 1\text{ and }1\leq i\leq k.$

A strict homotopy relative averaging algebra is a triple that consists of an
$A_{\infty}$-algebra, a representation and a strict homotopy relative
averaging operator. In this case, Theorem 8.8 reads as follows.

######  8.9 Theorem.

Let $M\xrightarrow{P}A$ be a strict homotopy relative averaging algebra. Then
$(M,\\{\pi_{k}^{P}\\}_{k=1}^{\infty})$ is a $Diass_{\infty}$-algebra, where

$\displaystyle\pi_{k}^{P}(y;u_{1},\ldots,u_{k}):=\eta_{k}(P(u_{1}),\ldots,u_{i},\ldots,P(u_{k})),$

for $k\geq 1$, $y\in Y_{k}$ (which can be uniquely written as $y=y_{1}\vee
y_{2}$ for some $(i-1)$-tree $y_{1}\in Y_{i-1}$ and $(k-i)$-tree $y_{2}\in
Y_{k-i}$) and $u_{1},\ldots,u_{k}\in M$.

In the following, we show that any $Diass_{\infty}$-algebra is always induced
from a strict strict homotopy relative averaging algebra. Let
$(D,\\{\pi_{k}\\}_{k=1}^{\infty})$ be a given $Diass_{\infty}$-algebra.
Consider the graded vector space $D/I$ which is obtained from $D$ quotient by
the homogeneous ideal $I$ generated by the set

$\displaystyle\\{\pi_{k}(y;a_{1},\ldots,a_{k})-\pi_{k}(y^{\prime};a_{1},\ldots,a_{k})~{}|~{}k\geq
1,~{}y,y^{\prime}\in Y_{k}\text{ and }a_{1},\ldots,a_{k}\in D\\}.$

It is easy to see that the graded vector space $D/I$ carries an
$A_{\infty}$-algebra structure with the operations $\\{\mu_{k}:(D/I)^{\otimes
k}\rightarrow D/I\\}_{k=1}^{\infty}$ given by

$\displaystyle\mu_{k}([a_{1}],\ldots,[a_{k}])=[\pi_{k}(y;a_{1},\ldots,a_{k})],\text{
for }k\geq 1\text{ and }[a_{1}],\ldots,[a_{k}]\in D/I.$

We denote this $A_{\infty}$-algebra structure simply by
$D_{\mathrm{Ass}_{\infty}}$. It is also easy to check that the
$A_{\infty}$-algebra $D_{\mathrm{Ass}_{\infty}}$ has a representation on the
graded vector space $D$ with the action maps

$\displaystyle\eta_{k}([a_{1}],\ldots,[a_{i-1}],a_{i},[a_{i+1}],\ldots,[a_{k}])=\pi_{k}(y;a_{1},\ldots,a_{k}),$

for $k\geq 1$, $[a_{1}],\ldots,[a_{i-1}],[a_{i+1}],\ldots,[a_{k}]\in
D_{\mathrm{Ass}_{\infty}}$ and $a_{i}\in D$. Here $y\in Y_{k}$ is any $k$-tree
which is the grafting of some $(i-1)$-tree and $(k-i)$-tree. Moreover,
$D\xrightarrow{q}D_{\mathrm{Ass}_{\infty}}$ is a strict homotopy relative
averaging algebra, where $q$ is the quotient map. Further, the induced
$Diass_{\infty}$-algebra structure on $D$ coincides with the given one.

Data availability statement. Data sharing does not apply to this article as no
datasets were generated or analysed during the current study.

Acknowledgements. The author would like to thank Indian Institute of
Technology (IIT) Kharagpur for providing the beautiful academic environment
where the research has been carried out.

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