# Loday-Quillen-Tsygan theorem on Quivers

Keyou Zeng  K. Zeng: Perimeter Institute for Theoretical Physics, Waterloo,
Canada<EMAIL_ADDRESS>
###### Abstract.

The well-known Loday-Quillen-Tsygan theorem calculates the Lie algebra
homology of the infinite general linear Lie algebra $\mathfrak{gl}(A)$ over an
unital associative algebra $A$. We generalize the Loday-Quillen-Tsygan theorem
to an infinite Lie algebra associated with a (framed) quiver, where we assign
to each vertex $v$ an infinite general linear Lie algebra
$\mathfrak{gl}(A_{v})$, to each edge $e$ an infinite matrix module and to each
framed vertex a (anti)-fundamental representation. Given this data, each loop
or path ending on framed vertices of the quiver defined a stratified
factorization algebra over $S^{1}$ or $[0,1]$ respectively. We show that the
corresponding Lie algebra homology can be expressed as summing the
factorization homology over all loops and framed paths of the quiver.

##  1\. Introduction

The celebrated Loday-Quillen-Tsygan (LQT) theorem [2], [3] states that for any
associative algebra $A$ in characteristic zero, the Lie algebra homology of
the infinite general linear Lie algebra over $A$ is canonically isomorphic, as
a Hopf algebra, to the exterior power of the cyclic homology of $A$:

$\lim_{\overset{\longrightarrow}{N}}H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathfrak{gl}_{N}(A))\cong\mathrm{Sym}(HC_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A)[1]).$

This theorem admit various generalization to many different setups. One can
replace the Lie algebra $\mathfrak{gl}_{N}(A)$ by the Lie algebra of
symplectic matrices $\mathfrak{sp}_{N}(A)$ or orthogonal matrices
$\mathfrak{o}_{N}(A)$. Then the corresponding Lie algebra homology is computed
by the dihedral homology [6]. There is also a generalization of the LQT
theorem where one considers the Lie algebra homology of $\mathfrak{gl}_{N}(A)$
with coefficient in the adjoint matrices representation [5].

In this paper, we consider an extension of the LQT theorem to an infinite Lie
algebra associated with a framed quiver. Recall that a quiver $Q$ is a
directed graph that consist of two sets $(Q_{0},Q_{1})$ and two functions

$s,t:Q_{1}\rightrightarrows Q_{0}.$

The set $Q_{0}$ is the set of vertices and $Q_{1}$ is the set of oriented
edges. An edge $e\in Q_{0}$ starts at the vertex $s(e)$ and ends on the vertex
$t(e)$.

A framed quiver $Q^{\text{fr}}$ consists of the data of a quiver
$Q=(Q_{0},Q_{1})$ together with two subsets $W^{+},W^{-}$ of $Q_{0}$ as the
framed vertices. Denote the inclusion map $j:W^{+}\sqcup W^{-}\to Q_{0}$. We
can think of the framed quiver $Q^{\text{fr}}$ as adding to $Q$ extra vertices
$W^{+},W^{-}$ and edges $w^{+}\to j(w^{+})$ for all $w^{+}\in W^{+}$, and
$j(w^{-})\to w^{-}$ for all $w^{-}\in W^{-}$.

For a framed quiver $Q^{\text{fr}}$, we assign to it the following set of data

$\left\\{\begin{tabular}[]{c}To each vertex $v\in Q_{0}$ we assign a unital
$k$ algebra $A_{v}$\\\ To each edge $e\in Q_{1}$ we assign a
$A_{s(e)}-A_{t(e)}$ bimodule $M_{e}$\\\ To each vertex $w^{+}\in W^{+}$ we
assign a right $A_{j(w^{+})}$ module $M_{w^{+}}$\\\ To each vertex $w^{-}\in
W^{-}$ we assign a left $A_{j(w^{-})}$ module
$M_{w^{-}}$\end{tabular}\right\\}.$

Given this data, we have the following Lie algebra

$\bigoplus_{v\in Q_{0}}\mathfrak{gl}_{N}(A_{v}),$

and its module

(1.1) $\bigoplus_{e\in Q_{1}}\mathrm{Mat}_{N}(M_{e})\bigoplus_{w^{-}\in
W^{-}}\mathbb{C}^{N}(M_{w^{-}})\bigoplus_{w^{+}\in
W^{+}}(\mathbb{C}^{N})^{*}(M_{w^{+}}).$

In this paper, we consider the Lie algebra homology of $\bigoplus_{v\in
Q_{0}}\mathfrak{gl}_{N}(A_{v})$ with coefficient in the symmetric power of the
module 1.1 in the $N\to\infty$ limit

(1.2)
$\lim_{\overset{\longrightarrow}{N}}H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\bigoplus_{v\in
Q_{0}}\mathfrak{gl}_{N}(A_{v}),\mathrm{Sym}(\bigoplus_{e\in
Q_{1}}\mathrm{Mat}_{N}(M_{e})\bigoplus_{w^{+}\in
W^{+}}\mathbb{C}^{N}(M_{w^{+}})\bigoplus_{w^{-}\in
W^{-}}(\mathbb{C}^{N})^{*}(M_{w^{-}}))).$

Equivalently, this is isomorphic to the Lie algebra homology of the following
super Lie algebra

$\left(\bigoplus_{v\in
Q_{0}}\mathfrak{gl}_{N}(A_{v})\right)\ltimes\left(\bigoplus_{e\in
Q_{1}}\mathrm{Mat}_{N}(M_{e})\bigoplus_{w^{+}\in
W^{+}}\mathbb{C}^{N}(M_{w^{+}})\bigoplus_{w^{-}\in
W^{-}}(\mathbb{C}^{N})^{*}(M_{w^{-}})\right)[-1].$

The main theorem we prove in this paper is the following

######  Theorem 1.1 (Theorem 4.2).

The Lie algebra homology 1.2 is isomorphic, as a Hopf algebra, to the
following

(1.3) $\begin{split}\mathrm{Sym}&\left(\bigoplus_{v\in
Q_{0}}HC_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A_{v})[1]\right.\\\
&\bigoplus_{\ell=(e_{1},\dots,e_{n})\in\mathrm{Cyc}(Q)}H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\left(M_{e_{1}}\otimes_{A_{t(e_{1})}}^{\mathbb{L}}M_{e_{2}}\otimes_{A_{t(e_{2})}}^{\mathbb{L}}\dots\otimes_{A_{t(e_{n-1})}}^{\mathbb{L}}M_{e_{n}}\otimes_{A_{t(e_{n})}}^{\mathbb{L}}\right)_{\mathbb{Z}_{\deg\ell}}\\\
&\left.\bigoplus_{\rho=(w^{+},e_{1},\dots,w^{-})\in\mathrm{fPath}(Q)}H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\left(M_{w^{+}}\otimes^{\mathbb{L}}_{A_{j(w^{+})}}M_{e_{1}}\otimes_{A_{t(e_{1})}}^{\mathbb{L}}\dots
M_{e_{n}}\otimes_{A_{j(w^{-})}}^{\mathbb{L}}M_{w^{-}}\right)\right),\end{split}$

where $\mathrm{Cyc}(Q)$ is the set of cycles in the quiver and
$\mathrm{fPath}$ is the set of paths ending on framed vertices of the quiver.
Here, the notation $\cdots\otimes_{A_{t(e_{n})}}$ with noting on the right
means tensor product with the first factor on the left.

There is a more concise way to express our above result in terms of stratified
factorization algebra [9]. In fact, a cycle
$\ell=(e_{1},\dots,e_{m})\in\mathrm{Cyc}(Q)$ gives us a set of associative
algebras $A_{s(e_{i})}$ and $A_{s(e_{i})}-A_{t(e_{i})}$ bimodules $M_{e_{i}}$.
This data $(A_{s(e_{i})},M_{e_{i}})$ defines a stratified factorization
algebra $\mathcal{A}_{\ell}$ on $S^{1}$ with $m$ marked points. Similarly, a
path $\rho\in\mathrm{fPath}(Q)$ defines a stratified factorization algebra
$\mathcal{A}_{\rho}$ on $[0,1]$. Then we can rewrite 1.3 (Theorem 4.3) as
follows

$\mathrm{Sym}\left(\bigoplus_{v\in
Q_{0}}\left(\int_{S^{1}}A_{v}\right)_{S^{1}}[1]\oplus\bigoplus_{\ell\in\mathrm{Cyc}(Q)}\left(\int_{S^{1}}\mathcal{A}_{\ell}\right)_{\mathbb{Z}_{\deg\ell}}\oplus\bigoplus_{\rho\in\mathrm{fPath}(Q^{\mathrm{fr}})}\left(\int_{[0,1]}\mathcal{A}_{\rho}\right)\right).$

Acknowledgment. I would like to thank Kevin Costello, Zhengping Gui, Si Li,
Lukas Mueller, Nuo Xu for illuminating discussions. Research at Perimeter
Institute is supported in part by the Government of Canada through the
Department of Innovation, Science and Economic Development Canada and by the
Province of Ontario through the Ministry of Colleges and Universities.

##  2\. Notation and convention

  * •

Through out we let $k$ be a field of characteristic zero. Without otherwise
mentioned, $\otimes$ refers to tensor product over $k$.

  * •

Let $G$ be a group and $V$ a left $k[G]$ module. We denote the space of
invariants as

$V^{G}:=\\{v\in V\mid g.v=v\text{ for all }g\in G\\},$

and the space of coinvariants as

$V_{G}:=k\otimes_{k[G]}V=V/\\{v-g.v\mid v\in V,\;g\in G\\}.$

  * •

A ($\mathbb{Z}$-) graded vector space is a vector space $V$ together with a
decomposition into a direct sum $V=\bigoplus\limits_{i\in\mathbb{Z}}V_{i}$.
For $k\in\mathbb{Z}$, we use the $k$-th shift notation $V[k]$ as follows

$(V[k])_{i}=V_{i-k}$

  * •

For an associative algebra $A$, we denote $A^{\text{op}}$ the opposite algebra
of $A$. We also denote $A^{e}=A\otimes A^{\text{op}}$.

  * •

Let $\mathrm{Mat}_{N}$ denote the space of $N\times N$ matrices over $k$. A
basis of $\mathrm{Mat}_{N}$ is given by $\\{E_{ij}\\}_{1\leq i,j\leq N}$,
where $E_{ij}$ is the matrix with $1$ in the $i$-th row and $j$-th column and
$0$ everywhere else. We also denote
$\mathrm{Mat}_{N}(A)=\mathrm{Mat}_{N}\otimes A$ and $E_{ij}^{a}:=E_{ij}\otimes
a\in\mathrm{Mat}_{N}(A)$.

  * •

Suppose $A$ is a $S$ bimodule. We use the notation
$A\otimes_{S}:=A\otimes_{S^{e}}S=A/[A,S]$. More generally, if $A_{1}$ is a
left $S$ module and $A_{n}$ is a right $S$ module, then we use the notation
$A_{1}\otimes\dots A_{n}\otimes_{S}:=A_{1}\otimes\dots A_{n}\otimes_{S^{e}}S$.
This gives $A_{1}\otimes\dots A_{n}$ factored by the relation
$(sa_{1},\dots,a_{n})=(a_{1},\dots,a_{n}s)$.

##  3\. Preliminaries

###  3.1. Fundamental Theorems of Invariant Theory

In this section, we briefly review some classical results on invariant theory.
We refer to [4] for more detail.

Let $V$ be a finite dimensional $k$-vector space and let $GL(V)$ be its group
of automorphisms. The left action of the symmetric group $S_{n}$ on
$V^{\otimes n}$ is given by place permutation

$\sigma(v_{1},\dots,v_{n})=(v_{\sigma^{-1}(1)},\dots,v_{\sigma^{-1}(n)}).$

The above map gives rise to a ring homomorphism

$\mu:k[S_{n}]\to\mathrm{End}(V^{\otimes n}).$

Let $GL(V)$ act on $V^{\otimes n}$ diagonally
$\alpha(v_{1},\dots,v_{n})=(\alpha v_{1},\dots,\alpha v_{n})$, and act on
$\mathrm{End}(V^{\otimes n})$ by conjugation $\alpha(f)=\alpha\circ
f\circ\alpha^{-1}$. We find that the image of $\mu$ lies in the invariant
algebra $\mathrm{End}(V^{\otimes n})^{GL(V)}$. Therefore, we have the ring
homomorphism:

$\mu:k[S_{n}]\to\mathrm{End}(V^{\otimes n})^{GL(V)}.$

We have the following theorem from classical invariant theory.

######  Theorem 3.1.

Let $V$ be a $k$-vector space with $\dim V=r\geq n$. Then
$\mu:k[S_{n}]\to\mathrm{End}(V^{\otimes n})^{GL(V)}$ is an isomorphism.

It will be useful to translate the fundamental theorems of invariant theory
into the coinvariant framework. The natural pairing on $V^{*}\otimes V$
introduces a trace map on $\mathrm{End}(V)$,
$\text{Tr}:\;\mathrm{End}(V)\overset{\approx}{\longrightarrow}V^{*}\otimes
V\overset{\langle-,-\rangle}{\longrightarrow}k$. This trace map induces the
nondegenerate pairing:

$\displaystyle\mathrm{End}(V)\otimes\mathrm{End}(V)\to k;$
$\displaystyle\alpha\otimes\beta\mapsto\text{Tr}(\alpha\beta).$

It follows immediately that $\mathrm{End}(V)^{\otimes
n}\approx(\mathrm{End}(V)^{\otimes n})^{*}$.

Note that for a $G$ module $W$, we have an isomorphism
$(W_{G})^{*}\cong(W^{*})^{G}$. Specifically, we consider the $GL(V)$ module
$\mathrm{End}(V^{\otimes n})$. Then we have

$\mathrm{End}(V^{\otimes n})^{GL(V)}\cong(\mathrm{End}(V)^{\otimes
n})^{GL(V)}\cong((\mathrm{End}(V)^{\otimes
n})^{*})^{GL(V)}\cong((\mathrm{End}(V)^{\otimes n})_{GL(V)})^{*}.$

This determines a map

$T:k[S_{n}]\to\mathrm{End}(V^{\otimes
n})^{GL(V)}\cong((\mathrm{End}(V)^{\otimes n})_{GL(V)})^{*}.$

######  Proposition 3.1.

Let
$\sigma=(\mu_{1},\dots,\mu_{j})(\nu_{1},\dots,\nu_{k})\dots(\rho_{1},\dots,\rho_{l})$
be the cycle decomposition of the permutation $\sigma$. Then the map $T$
defined above is explicitly given by

(3.1)
$T(\sigma)(\alpha_{1},\dots,\alpha_{n})=\operatorname{Tr}(\alpha_{\mu_{1}}\cdots\alpha_{\mu_{j}})\operatorname{Tr}(\alpha_{\nu_{1}}\cdots\alpha_{\nu_{k}})\cdots\operatorname{Tr}(\alpha_{\rho_{1}}\cdots\alpha_{\rho_{l}}).$

The $k$-module $k[S_{n}]$ is canonically isomorphic to its dual $k[S_{n}]^{*}$
since it comes with a preferred pairing $(\sigma,\tau)=\delta_{\sigma\tau}$.
Thus, we have a dual map

$T^{*}:(\mathrm{End}(V)^{\otimes n})_{GL(V)}\to k[S_{n}].$

It is clear that we have

$T^{*}(\alpha_{1},\dots,\alpha_{n})=\sum_{\sigma\in
S_{n}}T(\sigma)(\alpha_{1},\dots,\alpha_{n})\sigma,$

where $T(\sigma)(\alpha_{1},\dots,\alpha_{n})$ is given by the formula 3.1.
More explicitly, we have

######  Proposition 3.2.

Let $\omega$ and $\sigma$ be two permutations in $S_{n}$. The map $T^{*}$
satisfies

$T^{*}(E_{\omega(1)\sigma(1)},E_{\omega(2)\sigma(2)},\dots,E_{\omega(n)\sigma(n)})=\sigma\circ\omega^{-1}.$

As an associate algebra, $\mathrm{End}(V)$ inherits a structure of Lie
algebra, which we denote as $\mathfrak{gl}(V)$. The Lie algebra
$\mathfrak{gl}(V)$ is a module over itself, and so is the tensor product
$\mathfrak{gl}(V)^{\otimes n}$. We have a right action

$[(\alpha_{1},\dots,\alpha_{n}),\alpha]=\sum_{i}(\alpha_{1},\dots,\alpha_{i-1},[\alpha_{i},\alpha],\alpha_{i+1},\dots,\alpha_{n}).$

######  Lemma 3.1.

The module of coinvariants $(\mathfrak{gl}(V)^{\otimes n})_{\mathfrak{gl}(V)}$
is equal to the module $(\mathfrak{gl}(V)^{\otimes n})_{GL(V)}$.

Therefore, for the case $\dim V=r\geq n$, we have an isomorphism

$(\mathfrak{gl}(V)^{\otimes n})_{\mathfrak{gl}(V)}\cong k[S_{n}].$

This isomorphism plays an important role in the LQT theorem and its
generalization we present in this paper.

###  3.2. Hochschild and Cyclic homology

In this section, we review aspects of Hochschild and Cyclic homology.

A bimodule over $A$ is a $k$-module $M$ that is both a left and a right
$A$-module for which $(am)b=a(mb)$, for any $a,b\in A$, $m\in M$. We see that
$M$ being a $A$ bimodule is equivalent to $M$ being a $A^{e}$ left module,
where the action is defined by $(a\otimes b)m=amb$. The algebra $A$ itself is
a $A^{e}$-module given by the left and right multiplication: $(a\otimes
b)c=acb$.

######  Definition 3.1.

Let $M$ be an $A$ bimodule. The Hochschild homology of $A$ with coefficient in
$M$, denoted $H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A,M)$, is
defined as

(3.2)
$\mathrm{Tor}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}^{A^{e}}(A,M)=H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(M\otimes_{A^{e}}^{\mathbb{L}}A).$

For $M=A$, $H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A,A)$ is
also called the Hochschild homology of $A$. We denote it as
$H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A)$.

We use the standard bar construction to compute the Hochschild homology.

######  Definition 3.2.

Let $M$ be an $A$ bimodule. The Hochschild Bar complex of $A$ with coefficient
in $M$ is defined to be
$B_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A,M)=M\otimes(\oplus_{n\geq
0}(A[1])^{\otimes n})$. We define the following maps $B_{n}(A,M)\to
B_{n-1}(A,M)$:

$\begin{split}d_{1}(m\otimes a_{1}\otimes\dots\otimes a_{n})&=ma_{1}\otimes
a_{2}\otimes\dots\otimes a_{n},\\\ d_{i}(m\otimes a_{1}\otimes\dots\otimes
a_{n})&=m\otimes\dots\otimes a_{i}a_{i+1}\otimes\dots\otimes a_{n},\\\
d_{n}(m\otimes a_{1}\otimes\dots\otimes a_{n})&=a_{n}m\otimes
a_{1}\otimes\dots\otimes a_{n-1}.\end{split}$

Then the Hochschild differential $b:B_{n}(A,M)\to B_{n-1}(A,M)$ is given by
$b=\sum_{i=0}^{n}(-1)^{i}d_{i}$.

######  Proposition 3.3.

We have the following isomorphism

$H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A,M)\cong
H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(B_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A,M),b).$

It will be convenient for us to use a relative version of the Bar complex to
compute the Hochschild homology. Let $S$ be a subring of $A$. We define the
Bar complex over $S$ as follows:

$B^{S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A,M):=\bigoplus_{n\geq
0}M\otimes_{S}\underbrace{A[1]\otimes_{S}A[1]\otimes_{S}\dots
A[1]}_{n}\otimes_{S}.$

######  Remark 3.1.

Recall that the notation $M\otimes_{S}A\otimes_{S}\dots A\otimes_{S}$ means
$(M\otimes_{S}A\otimes_{S}\dots A)\otimes_{S^{e}}S$.

The Hochschild differential $b$ is compatible with this definition so there is
a well-defined complex
$(B^{S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A,M),b)$. Denote
its $n$-th homology by $HH^{S}_{n}(A,M)$. We are interested in the cases when
$HH^{S}_{n}(A,M)$ is isomorphic to the Hochschild homology.

######  Definition 3.3.

An unital $k$-algebra $S$ is called separable if there exists an element

$e=\sum u_{i}\otimes v_{i}\in S\otimes S^{op},$

such that $\sum u_{i}v_{i}=1$ and $(s\otimes 1)e=(1\otimes s)e$ for any $s\in
S$. Such an element $e$ is also called the idempotent.

We have the following slight generalization of [4] Theorem 1.2.13.

######  Theorem 3.2.

Let $S$ be a separable $k$ algebra. Then for any unital $S$-algebra $A$ there
is a canonical isomorphism

$HH_{n}(A,M)\cong HH^{S}_{n}(A,M).$

###### Proof.

There is an canonical projection $\phi:B_{n}(A,M)\to B^{S}_{n}(A,M)$.

Using the idempotent $e=\sum u_{i}\otimes v_{i}$ one can construct a map
$\psi:B^{S}_{n}(A,M)\to B_{n}(A,M)$:

(3.3) $\psi(m\otimes_{S}a_{1}\otimes_{S}\dots
a_{n}\otimes_{S})=\sum_{j_{0},j_{1},\dots,j_{n}}v_{j_{n}}mu_{j_{0}}\otimes
v_{j_{0}}a_{1}u_{j_{1}}\otimes\dots\otimes v_{j_{n-1}}a_{n}u_{j_{n}}.$

It is easy to check that $\phi\circ\psi=id$. On the other hand, we construct a
pre-simplicial homotopy between $\psi\circ\phi$ and $id$. Let

(3.4) $h_{i}(m\otimes a_{1}\otimes\dots
a_{n})=\sum_{j_{0},j_{1},\dots,j_{i}}mu_{j_{0}}\otimes
v_{j_{0}}a_{1}u_{j_{1}}\otimes\dots v_{j_{i-1}}a_{i}u_{j_{i}}\otimes
v_{j_{i}}\otimes a_{i+1}\otimes\dots a_{n}.$

We can check that $d_{0}h_{0}(m\otimes a_{1}\otimes\dots
a_{n})=d_{0}(\sum_{j}mu_{j}\otimes v_{j}\otimes a_{1}\dots a_{n})=m\otimes
a_{1}\otimes\dots a_{n}$. Therefore $d_{0}h_{0}=id$. We can also check that

$\displaystyle d_{n+1}h_{n}(m\otimes a_{1}\otimes\dots a_{n})$
$\displaystyle=d_{n+1}(\sum_{j_{0},j_{1},\dots,j_{n}}mu_{j_{0}}\otimes
v_{j_{0}}a_{1}u_{j_{1}}\otimes\dots\otimes v_{j_{n-1}}a_{n}u_{j_{n}}\otimes
v_{j_{n}})$ $\displaystyle=\psi\circ\phi(m\otimes a_{1}\otimes\dots a_{n}).$

Therefore $d_{n+1}h_{n}=\psi\circ\phi$. It is also immediate to check the
remaining relation

(3.5) $d_{i}h_{j}=\begin{cases}h_{j}d_{i},&\text{ for }i<j,\\\
d_{i}h_{i-1},&\text{ for }i=j\neq 0,\\\ h_{j}d_{i-1},&\text{ for
}i>j+1\,.\end{cases}$

Let $h=\sum_{i=0}^{n}(-1)^{i}h_{i}$. Then $hd+dh=id-\psi\circ\phi$. ∎

For $L$ a left $A$ module and $M$ a right $A$ module, $L\otimes M$ has a
canonical $A$ bimodule structure. Note that

$(L\otimes M)\otimes_{A^{e}}^{\mathbb{L}}A\cong M\otimes_{A}^{\mathbb{L}}L.$

We have the following

######  Corollary 3.1.

Let $L$ be a left $A$ module and $M$ be a right $A$ module. Let $S$ be a
separable subalgebra of $A$. We define the complex

$B^{S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(M,A,L)=\bigoplus_{n\geq
0}N\otimes_{S}(A[1])^{\otimes_{S}n}\otimes_{S}L.$

Then there is a canonical isomorphism

$H_{n}(B^{S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(M,A,L))=\mathrm{Tor}_{n}^{A}(M,L).$

Now we introduce the cyclic homology. The cyclic group $\mathbb{Z}_{n+1}$ acts
on $B_{n}(A,A)$ via permuting the tensor factor. Explicitly, let $t$ be the
generator of $\mathbb{Z}_{n+1}$. Then its action on $B_{n}(A,A)$ is given by

$t(a_{0},\dots,a_{n})=(-1)^{n}(a_{n},a_{0},\dots,a_{n-1}).$

It is called the Connes’ cyclic operator. Let $N=1+t=t^{2}+\dots+t^{n}$ be the
corresponding norm operator.

The cyclic bicomplex
$CC_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}},\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A)$
is defined as follows

${A^{\otimes 3}}$${A^{\otimes 3}}$${A^{\otimes 3}}$ ${A^{\otimes
2}}$${A^{\otimes 2}}$${A^{\otimes 2}}$ ${A}$${A}$${A}$
$\scriptstyle{b}$$\scriptstyle{-b^{\prime}}$$\scriptstyle{1-t}$$\scriptstyle{b}$$\scriptstyle{N}$$\scriptstyle{1-t}$$\scriptstyle{b}$$\scriptstyle{-b^{\prime}}$$\scriptstyle{1-t}$$\scriptstyle{b}$$\scriptstyle{N}$$\scriptstyle{1-t}$$\scriptstyle{1-t}$$\scriptstyle{N}$$\scriptstyle{1-t}$

The differential $b$ is the Hochschild differential and the differential
$b^{\prime}$ is defined as $b^{\prime}=\sum_{i=0}^{n-1}(-1)^{i}d_{i}$. One can
check from the definition that all the squares commute and the above is indeed
a bicomplex.

######  Definition 3.4.

The cyclic homology $HC_{n}(A)$ of the associative algebra $A$ is defined as
the homology of the total complex of
$CC_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}},\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A)$

$HC_{n}(A)=H_{n}(\mathrm{Tot}\,CC_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}},\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A)).$

Consider the space of coinvariants of $B_{n}(A,A)$ under the action of
$\mathbb{Z}_{n+1}$. We denote this space by
$C^{\lambda}_{n}(A)=(A^{\otimes(n+1)})_{\mathbb{Z}_{n+1}}$. We have a well-
defined complex

$C^{\lambda}_{\bullet}(A):\;\;\dots\overset{b}{\longrightarrow}C^{\lambda}_{n}(A)\overset{b}{\longrightarrow}C^{\lambda}_{n-1}(A)\overset{b}{\longrightarrow}\dots\overset{b}{\longrightarrow}C^{\lambda}_{0}(A).$

called the Connes complex. Its $n$-th homology group is denoted
$H^{\lambda}_{n}(A)$. We have the following result

######  Theorem 3.3.

Let $k$ be a field of characteristic $0$, then we have an isomorphism

$H^{\lambda}_{n}(A)\cong HC_{n}(A).$

Similar to Theorem 3.2, we can also consider the cyclic bicomplex and Connes’
complex over a ground ring $S$.

######  Lemma 3.2.

Let $S$ be a separable subalgebra of $A$. We define the cyclic bicomplex
$CC^{S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}},\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A)$
over $S$ as follows

${A\otimes_{S}A\otimes_{S}A\otimes_{S}}$${A\otimes_{S}A\otimes_{S}A\otimes_{S}}$${A\otimes_{S}A\otimes_{S}A\otimes_{S}}$
${A\otimes_{S}A\otimes_{S}}$${A\otimes_{S}A\otimes_{S}}$${A\otimes_{S}A\otimes_{S}}$
${A\otimes_{S}}$${A\otimes_{S}}$${A\otimes_{S}}$
$\scriptstyle{b}$$\scriptstyle{-b^{\prime}}$$\scriptstyle{1-t}$$\scriptstyle{b}$$\scriptstyle{N}$$\scriptstyle{1-t}$$\scriptstyle{b}$$\scriptstyle{-b^{\prime}}$$\scriptstyle{1-t}$$\scriptstyle{b}$$\scriptstyle{N}$$\scriptstyle{1-t}$$\scriptstyle{1-t}$$\scriptstyle{N}$$\scriptstyle{1-t}$

Then we have an isomorphism

(3.6)
$H_{n}(\mathrm{Tot}\,CC^{S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}},\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A))\cong
HC_{n}(A).$

###### Proof.

The projection map $\phi:B_{n}(A,A)\to B_{n}^{S}(A,A)$ and the map
$\psi:B_{n}^{S}(A,A)\to B_{n}(A,A)$ defined by 3.3 are compatible with the
cyclic operator.

$\displaystyle\psi\circ t(a_{0}\otimes_{S}a_{1}\otimes_{S}\dots
a_{n}\otimes_{S})$
$\displaystyle=(-1)^{n}\sum_{j_{0},j_{1},\dots,j_{n}}v_{j_{n}}a_{n}u_{j_{0}}\otimes
v_{j_{0}}a_{0}u_{j_{1}}\otimes\dots\otimes v_{j_{n-1}}a_{n-1}u_{j_{n}}$
$\displaystyle=t\circ\psi(a_{0}\otimes_{S}a_{1}\otimes_{S}\dots
a_{n}\otimes_{S}).$

As a result, they define morphisms between the bicomplex
$CC_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}},\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A)$
and
$CC^{S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}},\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A)$.
Using Theorem 3.2 we see that they are quasi-isomorphisms when restrict to the
$b$-columns. All the $b^{\prime}$-columns are acyclic. Therefore $\phi$ and
$\psi$ are also quasi-isomorphism when restricted to the $b^{\prime}$-columns.
So the total complexes are quasi-isomorphic. ∎

######  Corollary 3.2.

Let $S$ be a separable subalgebra of $A$. We consider the Connes’ complex over
$S$:

(3.7) $C^{\lambda,S}_{n}(A)=(\underbrace{A\otimes_{S}A\otimes_{S}\dots
A\otimes_{S}}_{n+1})_{\mathbb{Z}_{n+1}},$

with the differential $b$. Then there is a canonical isomorphism

(3.8)
$H_{n}(C^{\lambda,S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A))=HC_{n}(A).$

###### Proof.

Using the isomorphism 3.6, we only need to prove
$H_{n}(C^{\lambda,S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A))\cong
H_{n}(\mathrm{Tot}\,CC^{S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}},\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A))$.
This proof is the same as the proof of Theorem 3.3, which we recall now. We
construct homotopy from $\mathrm{id}$ to $0$ for each row complex. Let

$h^{\prime}:=\frac{1}{n+1}\mathrm{id},\quad h=-\frac{1}{n+1}\sum_{i=1}it^{i}$

be maps from $B_{n}^{S}(A)$ to itself. One can verifies that

$h^{\prime}N+(1-t)h=\mathrm{id},\quad Nh^{\prime}+h(1-t)=\mathrm{id}.$

Therefore, each row of the bicomplex
$CC^{S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}},\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A)$
has homology concentrated in degree $0$ and with
$H_{0}(CC^{S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}},n}(A),d_{v})=C^{\lambda,S}_{n}(A)$.
∎

###  3.3. Loday-Quillen-Tsygan theorem

Let $\mathfrak{gl}_{N}(A)=A\otimes\mathfrak{gl}_{N}$ be the Lie algebra of
matrices valued in $A$. For each $N$, we have a natural inclusion of Lie
algebra $\mathfrak{gl}_{N}(A)\to\mathfrak{gl}_{N+1}(A)$ given by
$\alpha\to\begin{pmatrix}\alpha&0\\\ 0&0\end{pmatrix}$. With respect to these
inclusions we define direct limit

$\mathfrak{gl}(A)=\lim_{\overset{\longrightarrow}{N}}\mathfrak{gl}_{N}(A),$

and consider the Lie algebra homology

$H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathfrak{gl}(A))=\lim_{\overset{\longrightarrow}{N}}H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathfrak{gl}_{N}(A)).$

In this section, we sketch the proof of the Loday-Quillen-Tsygan theorem. We
refer to [4] for the full detail.

######  Theorem 3.4.

There is a canonical isomorphism of graded Hopf algebras

$H_{\bullet}(\mathfrak{gl}(A))\cong\mathrm{Sym}(HC_{\bullet}(A)[1]).$

###### Proof.

(sketch) Step 1. We can decompose the Chevalley-Eilenberg complex by the
adjoint action of $\mathfrak{gl}_{N}$, and only consider the trivial
representation in the homology. This is due to the following proposition.

######  Proposition 3.4.

Let $\mathfrak{g}$ be a Lie algebra and $V$ a $\mathfrak{g}$ module. Let
$\mathfrak{h}$ be a reductive sub-Lie algebra of $\mathfrak{g}$. Then the
surjective map
$(V\otimes\mathrm{Sym}(\mathfrak{g}[1]))\to(V\otimes\mathrm{Sym}(\mathfrak{g}[1]))_{\mathfrak{h}}$
induces an isomorphism on homology

$H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathfrak{g},V)\cong
H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}((V\otimes\mathrm{Sym}(\mathfrak{g}[1]))_{\mathfrak{h}},d_{CE})\,.$

In our case, we get a quasi-isomorphism of complex

$(\mathrm{Sym}(\mathfrak{gl}_{N}(A)[1]),d_{CE})\to(\mathrm{Sym}(\mathfrak{gl}_{N}(A)[1])_{\mathfrak{gl}_{N}},d_{CE}).$

Step 2. Note that $\mathrm{Sym}^{n}(\mathfrak{gl}_{N}(A)[1])$ can be regarded
as the coinvariant of $\mathfrak{gl}_{N}(A)^{\otimes n}$ under the signed
action of $S_{n}$. The action of $S_{n}$ commutes with the adjoint action of
$\mathfrak{gl}_{N}$. We have

$(\mathrm{Sym}^{n}(\mathfrak{gl}_{N}(A)[1]))_{\mathfrak{gl}_{N}}\cong((\mathfrak{gl}_{N}^{\otimes
n}\otimes(A[1])^{\otimes
n})_{\mathfrak{gl}_{N}})_{S_{n}}\cong((\mathfrak{gl}_{N}^{\otimes
n})_{\mathfrak{gl}_{N}}\otimes(A[1])^{\otimes n})_{S_{n}}.$

Classical invariant theory tells us that for $N\geq n$, there is an
isomorphism

$T^{*}:(\mathfrak{gl}_{N}^{\otimes n})_{\mathfrak{gl}_{N}}\to k[S_{n}].$

As a consequence, we have an isomorphism

$\Theta:(\mathrm{Sym}^{n}(\mathfrak{gl}(A)[1]))_{\mathfrak{gl}}\cong(k[S_{n}]\otimes(A[1])^{\otimes
n})_{S_{n}}.$

According to Proposition 3.2, this isomorphism $\Theta$ is given as follows:

$\Theta(E^{a_{1}}_{1\sigma(1)}\wedge E^{a_{2}}_{2\sigma(2)}\dots\wedge
E^{a_{n}}_{n\sigma(n)})=(\sigma\otimes(a_{1},a_{2},\dots
a_{n}))\in(k[S_{n}]\otimes(A[1])^{\otimes n})_{S_{n}}.$

Under this isomorphism, $S_{n}$ acts on $k[S_{n}]$ by conjugation and acts on
$A^{\otimes n}$ by signed permutation.

Step 3. The diagonal map
$\Delta:\mathfrak{gl}(A)\to\mathfrak{gl}(A)\times\mathfrak{gl}(A)$ induces a
map on the Chevalley-Eilenberg complex
$\Delta:(\mathrm{Sym}^{n}(\mathfrak{gl}(A)[1]))_{\mathfrak{gl}}\to(\mathrm{Sym}^{n}(\mathfrak{gl}(A)[1]))_{\mathfrak{gl}}\otimes(\mathrm{Sym}^{n}(\mathfrak{gl}(A)[1]))_{\mathfrak{gl}}$.
This map defines a coassociative and cocommutative coproduct on the complex
$(\mathrm{Sym}(\mathfrak{gl}(A)[1]))_{\mathfrak{gl}}$.

There is also a map
$\oplus:\mathfrak{gl}(A)\times\mathfrak{gl}(A)\to\mathfrak{gl}(A)$ which can
be schematically described as

(3.9) $\begin{pmatrix}\star&\star&\star&\cdots\\\ \star&\star&\star&\cdots\\\
\star&\star&\star&\cdots\\\
\vdots&\vdots&\vdots&\vdots\end{pmatrix}\oplus\begin{pmatrix}\times&\times&\times&\cdots\\\
\times&\times&\times&\cdots\\\ \times&\times&\times&\cdots\\\
\vdots&\vdots&\vdots&\vdots\end{pmatrix}=\begin{pmatrix}\star&0&\star&0&\star&0&\cdots\\\
0&\times&0&\times&0&\times&\cdots\\\ \star&0&\star&0&\star&0&\cdots\\\
0&\times&0&\times&0&\times&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\end{pmatrix}.$

It induces a map of complex
$\mathrm{Sym}(\mathfrak{gl}(A)[1])\otimes\mathrm{Sym}(\mathfrak{gl}(A)[1])\to\mathrm{Sym}(\mathfrak{gl}(A)[1])$.
Although this map is neither associative nor commutative, it induces a graded
associative and commutative product on the $\mathfrak{gl}$ coinvariant complex

$\oplus_{*}:(\mathrm{Sym}(\mathfrak{gl}(A)[1]))_{\mathfrak{gl}}\otimes(\mathrm{Sym}(\mathfrak{gl}(A)[1]))_{\mathfrak{gl}}\to(\mathrm{Sym}(\mathfrak{gl}(A)[1]))_{\mathfrak{gl}}.$

Therefore, $((\mathrm{Sym}(\mathfrak{gl}(A)[1]))_{\mathfrak{gl}},d_{CE})$
equipped with $\Delta$ and $\oplus_{*}$ is a commutative and cocommutative
differential graded Hopf algebra. We have the following theorem

######  Theorem 3.5.

Let $\mathcal{H}$ be a cocommutative differential graded Hopf algebra over a
field of characteristic $0$. We have that the homology and primitive functors
commute,

$H(\mathrm{Prim}\mathcal{H})=\mathrm{Prim}H(\mathcal{H}).$

######  Theorem 3.6.

Let $\mathcal{H}$ be a connected cocommutative graded Hopf (resp. differential
graded Hopf) algebra over a field of characteristic $0$. Then the canonical
map

$U(\mathrm{Prim}\mathcal{H})\to\mathcal{H}$

is an isomorphism of graded Hopf (resp. differential graded Hopf) algebras.

As a corollary, $H_{\bullet}(\mathfrak{gl}(A))$ is a commutative and
cocommutative graded Hopf algebra whose primitive part is the homology of
$(\text{Prim}(\mathrm{Sym}(\mathfrak{gl}(A)[1]))_{\mathfrak{gl}},d_{CE})$.
Moreover, since $H_{\bullet}(\mathfrak{gl}(A))$ is commutative, it is
isomorphic to the graded symmetric algebra of
$H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathrm{Prim}(\mathrm{Sym}(\mathfrak{gl}(A)[1])_{\mathfrak{gl}}))$.

Step 4. We need to determine the primative part of $L_{\bullet}:=\oplus_{n\geq
1}(k[S_{n}]\otimes(A[1])^{\otimes n})_{S_{n}}$. The coproduct on
$\mathrm{Sym}(\mathfrak{gl}(A)[1])_{\mathfrak{gl}}$ induces a coproduct on
$L_{\bullet}$ which we also denote as $\Delta$.

(3.10)
$\Delta(\sigma\otimes(\alpha_{1},\dots\alpha_{n}))=\sum_{I,J}(\sigma_{I}\otimes(\alpha_{i_{1}},\alpha_{i_{2}}\dots))\otimes(\sigma_{J}\otimes(\alpha_{j_{1}},\alpha_{j_{2}}\dots)),$

where the sum go through all ordered partitions $(I,J)$ of $S_{n}$ such that
$\sigma(I)=I,\sigma(J)=J$.

Let $U_{n}\subset S_{n}$ be the conjugacy class of the cycle $(1,2,\dots,n)$.
We see that $P_{\bullet}:=\oplus_{n\geq 1}(k[U_{n}]\otimes(A[1])^{\otimes
n})_{S_{n}}$ is a subspace of $\mathrm{Prim}L_{\bullet}$. Moreover, any
permutation is a product of cycles. We thus have a surjection
$\mathrm{Sym}P_{\bullet}\to L_{\bullet}$. This tell us that
$P_{\bullet}=\text{Prim}L_{\bullet}$. Therefore, we have an isomorphism

$\text{Prim}(\mathrm{Sym}(\mathfrak{gl}(A)[1]))_{\mathfrak{gl}}\cong\oplus_{n\geq
1}(k[U_{n}]\otimes(A[1])^{\otimes n})_{S_{n}}.$

As a $S_{n}$ module, $k[U_{n}]$ is isomorphic to the induced representation of
the trivial representation of $\mathbb{Z}_{n}$ to $S_{n}$,
$\mathrm{Ind}_{\mathbb{Z}_{n}}^{S_{n}}k$. We have

(3.11) $(\mathrm{Ind}_{\mathbb{Z}_{n}}^{S_{n}}k\otimes(A[1])^{\otimes
n})_{S_{n}}\cong((A[1])^{\otimes n})_{\mathbb{Z}_{n}}=C_{n-1}^{\lambda}(A).$

We see that
$\text{Prim}(\mathrm{Sym}(\mathfrak{gl}(A)[1]))_{\mathfrak{gl}}\cong
C_{\bullet-1}^{\lambda}(A)$. One can also verify that the Chevalley-Eilenberg
differential $d_{CE}$ induces the Hochschild differential $b$. ∎

##  4\. Homology of Lie algebra of matrices associated to quiver

###  4.1. Adding fundamental representation

In this section, we consider a generalization of the LQT theorem by
considering Lie algebra homology with coefficient in the symmetric powers of
the fundamental and the dual of fundamental representation of
$\mathfrak{gl}_{N}(A)$. While we believe that the results in this section have
not appeared in the literature, similar statement should already be known to
some experts (c.f. [7]).

Let $L$ be a left $A$ module and $M$ a right $A$ module. Let $\mathbb{C}^{N}$
be the fundamental representation of $\mathfrak{gl}_{N}$. Then
$\mathbb{C}^{N}(L):=\mathbb{C}^{N}\otimes L$ is a $\mathfrak{gl}_{N}(A)$
module given by the following

(4.1) $(A\otimes a)(v\otimes x)=(Av\otimes ax).$

Similarly, for the dual representation $(\mathbb{C}^{N})^{*}$,
$(\mathbb{C}^{N})^{*}(M):=(\mathbb{C}^{N})^{*}\otimes M$ is a
$\mathfrak{gl}_{N}(A)$ module given by

(4.2) $(A\otimes a)(v^{*}\otimes x)=(-A^{t}v^{*}\otimes xa).$

We consider the following super Lie algebra

$\mathcal{L}_{N}:=\mathfrak{gl}_{N}(A)\ltimes(\mathbb{C}^{N}(L)\oplus(\mathbb{C}^{N})^{*}(M))[-1].$

The natural inclusion $\mathcal{L}_{N}\to\mathcal{L}_{N+1}$ defines direct
limit
$\mathcal{L}:=\lim\limits_{\overset{\longrightarrow}{N}}\mathcal{L}_{N}$. We
have

$\mathcal{L}=\mathfrak{gl}(A)\ltimes(\mathcal{V}(L)\oplus\mathcal{V}^{*}(M))[-1],$

where $\mathcal{V}=\lim\limits_{\overset{\longrightarrow}{N}}\mathbb{C}^{N}$.
In this section, we consider the Lie algebra homology

$H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathcal{L})=\lim_{\overset{\longrightarrow}{N}}H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathcal{L}_{N}).$

By the definition, there is a natural isomorphism between the Lie algebra
homology of $\mathcal{L}$ and the Lie algebra homology of $\mathfrak{gl}(A)$
valued in $\mathrm{Sym}(\mathcal{V}(L)\oplus\mathcal{V}^{*}(M))$

$H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathcal{L})=H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathfrak{gl}(A),\mathrm{Sym}(\mathcal{V}(L)\oplus\mathcal{V}^{*}(M))).$

######  Theorem 4.1.

There is a canonical isomorphism of graded Hopf algebras

$H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathfrak{gl}(A),\mathrm{Sym}(\mathcal{V}(L)\oplus\mathcal{V}^{*}(M)))\cong\mathrm{Sym}(HC_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A)[1]\oplus\mathrm{Tor}^{A}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(M,L)).$

###### Proof.

Step 1. Applying Proposition 3.4. We only need to compute the
$\mathfrak{gl}_{N}(k)$ coinvariant of the Chevalley-Eilenberg complex
$\mathrm{Sym}(\mathcal{L}_{N}[1])_{\mathfrak{gl}_{N}}$.

Step 2. Application of invariant theory. We observe that

$(\mathrm{Sym}^{k}(\mathfrak{gl}_{N}(A)[1])\otimes\mathrm{Sym}^{l}(\mathbb{C}^{N}(L))\otimes\mathrm{Sym}^{i}((\mathbb{C}^{N})^{*}(M)))_{\mathfrak{gl}_{N}}.$

is only non zero when $i=l$. In this case, we have

$\displaystyle\left(\mathrm{Sym}^{k}(\mathfrak{gl}_{N}(A)[1])\otimes\mathrm{Sym}^{l}(\mathbb{C}^{N}(L))\otimes\mathrm{Sym}^{l}((\mathbb{C}^{N})^{*}(M))\right)_{\mathfrak{gl}_{N}}$
$\displaystyle=$ $\displaystyle\left(\left((\mathfrak{gl}_{N}(A)[1]))^{\otimes
k}\otimes(\mathbb{C}^{N}(L))^{\otimes
l}\otimes((\mathbb{C}^{N})^{*}(M))^{\otimes l}\right)_{S_{k}\times S_{l}\times
S_{l}}\right)_{\mathfrak{gl}_{N}}$ $\displaystyle=$
$\displaystyle\left((\mathfrak{gl}_{N}^{\otimes
k}\otimes(\mathbb{C}^{N})^{\otimes l}\otimes((\mathbb{C}^{N})^{*})^{\otimes
l})_{\mathfrak{gl}}\otimes(A[1]^{\otimes k}\otimes L^{\otimes l}\otimes
M^{\otimes l})\right)_{S_{k}\times S_{l}\times S_{l}}$ $\displaystyle=$
$\displaystyle\left((\mathfrak{gl}_{N}^{\otimes
k}\otimes\mathfrak{gl}_{N}^{\otimes
l})_{\mathfrak{gl}_{N}}\otimes(A[1]^{\otimes k}\otimes L^{\otimes l}\otimes
M^{\otimes l})\right)_{S_{k}\times S_{l}\times S_{l}}.$

Using classical invariant theory, we have an isomorphism

$(\mathfrak{gl}_{N}^{\otimes k}\otimes\mathfrak{gl}_{N}^{\otimes
l})_{\mathfrak{gl}_{N}}\cong\mathbb{C}[S_{k+l}].$

We consider the composite of the above isomorphisms, and denote it as
$\widetilde{\Theta}$ at the $N\to\infty$ limit.

$\displaystyle\widetilde{\Theta}:\quad$
$\displaystyle\left(\mathrm{Sym}^{k}(\mathfrak{gl}(A)[1])\otimes\mathrm{Sym}^{l}(\mathcal{V}(L))\otimes\mathrm{Sym}^{l}(\mathcal{V}^{*}(M))\right)_{\mathfrak{gl}}$
$\displaystyle\to\left(\mathbb{C}[S_{k+l}]\otimes(A[1]^{\otimes k}\otimes
L^{\otimes l}\otimes M^{\otimes l})\right)_{S_{k}\times S_{l}\times S_{l}}.$

Using Proposition 3.2, we have

(4.3) $\displaystyle\widetilde{\Theta}$
$\displaystyle(E^{a_{1}}_{1\sigma(1)}\otimes E^{a_{2}}_{2\sigma(2)}\dots
E^{a_{k}}_{k\sigma(k)}\otimes e^{x_{1}}_{k+1}\dots e^{x_{l}}_{k+l}\otimes
f^{y_{1}}_{\sigma(k+1)}\dots f^{y_{l}}_{\sigma(k+l)})$
$\displaystyle=\sigma\otimes((a_{1},\dots,a_{k})\otimes(x_{1},\dots,x_{l})\otimes(y_{1},\dots,y_{l})),$

where $\\{e_{i}\\}$ (resp. $\\{f_{i}\\}$) is the canonical basis of
$\mathcal{V}$ (resp. $\mathcal{V}^{*}$). We also use the notation
$e^{x}_{i}:=e_{i}\otimes x\in\mathcal{V}\otimes L$ and
$f^{y}_{i}:=f_{i}\otimes y\in\mathcal{V}^{*}\otimes M$.

Using the above isomorphism, we can also identify the action of $S_{k}\times
S_{l}\times S_{l}$ on $\mathbb{C}[S_{k+l}]$. Let $\rho_{k}:S_{k}\to S_{k+l}$
the inclusion that sends $\omega\in S_{k}$ to the element in $S_{k+l}$ that
permutes $\\{1,2,\dots,k\\}\subset\\{1,2,\dots,k+l\\}$. Then $\omega\in S_{k}$
acts on $k[S_{k+l}]$ by the conjugate action

$\sigma\in
S_{k+l}\mapsto\rho_{k}(\omega)\circ\sigma\circ\rho_{k}(\omega^{-1}).$

Let $\rho_{l}:S_{l}\to S_{k+l}$ be the inclusion that sends $\tau\in S_{l}$ to
the element in $S_{k+l}$ that permutes
$\\{k+1,k+2,\dots,k+l\\}\subset\\{1,2,\dots,k+l\\}$. Then
$(\tau_{1},\tau_{2})\in S_{l}\times S_{l}$ acts on $k[S_{k+l}]$ as follows

$\sigma\in
S_{k+l}\mapsto\rho_{k}(\tau_{2})\circ\sigma\circ\rho_{k}(\tau_{1}^{-1}).$

We have the following proposition characterizing the orbits of $k[S_{k+l}]$
under the $S_{l}$ action

######  Proposition 4.1.

We define the set $S_{k,l}\subset S_{k+l}$ as the set of elements whose cycle
decomposition has at most one number in $\\{k+1,k+2,\dots,k+l\\}$ in each
cycle.

For any $\sigma\in S_{k+l}$, there exist an $\tau\in S_{l}$ such that
$\sigma\circ\rho_{k}(\tau^{-1})\in S_{k,l}$. Moreover, such a
$\sigma\circ\rho_{k}(\tau^{-1})\in S_{k,l}$ is the permutation that has the
most number of cycles in the orbit of $\sigma\in S_{k+l}$ under the action of
$S_{l}$.

###### Proof.

We consider the cycle decomposition of $\sigma$. Let $(l_{1},\cdots
l_{2},\cdots l_{m},\cdots)$ be a cycle of $\sigma$, where $l_{1},l_{2},\dots
l_{m}\in\\{k+1,k+2,\dots,k+l\\}$ and $\cdots$ refer to elements in
$\\{1,\dots,k\\}$. We define $\rho_{k}(\tau^{-1})$ to act as $l_{1}\mapsto
l_{m},l_{2}\mapsto l_{1},\dots,l_{m}\mapsto l_{m-1}$. For product of cycles,
$\rho_{k}(\tau^{-1})$ is defined similarly for each cycle. Then we can check
that $\sigma\circ\rho_{k}(\tau^{-1})\in S_{k,l}$ has cycle decomposition
$(l_{1},\cdots)(l_{2},\cdots)\dots(l_{m},\cdots)$, where $\cdots$ are elements
in $\\{1,\dots,k\\}$. We see that $\sigma\circ\rho_{k}(\tau^{-1})\in S_{k,l}$.
∎

Step 3. Hopf algebra structure.

The diagonal map
$\Delta:\mathfrak{gl}(A)\to\mathfrak{gl}(A)\times\mathfrak{gl}(A)$ and
$\Delta:\mathcal{V}(L)\oplus\mathcal{V}^{*}(M)\to(\mathcal{V}(L)\oplus\mathcal{V}^{*}(M))\times(\mathcal{V}(L)\oplus\mathcal{V}^{*}(M))$
induce a map of complexes
$\mathrm{Sym}(\mathcal{L}[1])\to\mathrm{Sym}(\mathcal{L}[1]\times\mathcal{L}[1])_{\mathfrak{gl}}$.
Composite this map with the map
$\mathrm{Sym}(\mathcal{L}[1]\times\mathcal{L}[1])_{\mathfrak{gl}}\to\mathrm{Sym}(\mathcal{L}[1])\otimes\mathrm{Sym}(\mathcal{L}[1])$
gives us the comultiplication map. One can check that this comultiplication
map is coassociative and cocommutative.

In order to define the multiplication map, we consider the direct sum of
matrices as in 3.9 and the following maps

$\tilde{\oplus}:\mathcal{V}(M)\times\mathcal{V}(M)\to\mathcal{V}(M).$

given by

(4.4) $\begin{pmatrix}\star\\\ \star\\\ \star\\\
\vdots\end{pmatrix}\tilde{\oplus}\begin{pmatrix}\times\\\ \times\\\ \times\\\
\vdots\end{pmatrix}\to\begin{pmatrix}\star\\\ \times\\\ \star\\\ \times\\\
\vdots\end{pmatrix}.$

The map
$\tilde{\oplus}:\mathcal{V}^{*}(N)\times\mathcal{V}^{*}(N)\to\mathcal{V}^{*}(N)$
is defined similarly. These maps induce a product map of the complexes
$\mathrm{Sym}(\mathcal{L}[1])\otimes\mathrm{Sym}(\mathcal{L}[1])\to\mathrm{Sym}(\mathcal{L}[1])$,
and a map on the $\mathfrak{gl}$ coinvariant complex
$\tilde{\oplus}_{*}:(\mathrm{Sym}(\mathcal{L}[1]))_{\mathfrak{gl}}\otimes(\mathrm{Sym}(\mathcal{L}[1]))_{\mathfrak{gl}}\to(\mathrm{Sym}(\mathcal{L}[1]))_{\mathfrak{gl}}$.
We want to show that this product is associative and commutative. From the
proof of the original LQT theorem, we know that this map restricts to an
associative and commutative product on
$(\mathrm{Sym}(\mathfrak{gl}(A)[1]))_{\mathfrak{gl}}$. We only need to prove
that it also restricts to an associative and commutative product on
$\mathrm{Sym}(\mathcal{V}(L)\oplus\mathcal{V}^{*}(M)))_{\mathfrak{gl}}$. This
follows from the fact that for any $u,v,w\in\mathcal{V}(M)$,
$((u\tilde{\oplus}v)\tilde{\oplus}w)$ and
$(u\tilde{\oplus}(v\tilde{\oplus}w))$ only differs by a permutation.
Similarly, $u\tilde{\oplus}v$ and $v\tilde{\oplus}u$ only differs by a
permutation. We see that $\tilde{\oplus}$ induces an associative and
commutative product after we pass to the $\mathfrak{gl}$ coinvariant.

As a result, the Lie algebra homology
$H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathrm{Sym}(\mathcal{L}[1]),d_{CE})$
is isomorphic, as a Hopf algebra, to the symmetric algebra of the homology of
the primitive part
$\mathrm{Prim}(\mathrm{Sym}(\mathcal{L}[1]))_{\mathfrak{gl}}$.

Step 4. Computation of the primitive part. The coalgebra structure on
$\mathbb{C}[S_{k+l}]\otimes(A[1]^{\otimes k}\otimes M^{\otimes l}\otimes
N^{\otimes l})$ takes a similar form as 3.10. Let

(4.5) $P_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}=\bigoplus_{k\geq
1}(\mathbb{C}[U_{k}]\otimes(A[1]^{\otimes
k}))_{S_{k}}\oplus(\mathbb{C}[U_{k+1}]\otimes(A[1]^{\otimes k}\otimes M\otimes
N))_{S_{k}}.$

We see that
$P_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\subset\mathrm{Prim}\left(\mathbb{C}[S_{k+l}]\otimes(A[1]^{\otimes
k}\otimes M^{\otimes l}\otimes N^{\otimes l})\right)_{S_{k}\times S_{l}\times
S_{l}}$. To see that the map
$\mathrm{Sym}P_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\to\left(\mathbb{C}[S_{k+l}]\otimes(A[1]^{\otimes
k}\otimes M^{\otimes l}\otimes N^{\otimes l})\right)_{S_{k}\times S_{l}\times
S_{l}}$ is surjective, we use Proposition 4.1 and observe that elements of
$S_{k,l}$ are written as product of cycles that contain at most one number in
$\\{k+1,\dots,k+l\\}\subset\\{1,\dots,k+l\\}$. Thus we have
$P_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}=\mathrm{Prim}\,\mathrm{Sym}(\mathcal{L}[1])_{\mathfrak{gl}}$.

Now we analyze $P_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}$. The
first summand $(\mathbb{C}[U_{k}]\otimes(A[1]^{\otimes k}))_{S_{k}}$ in 4.5 is
isomorphic to $C^{\lambda}_{k-1}(A)$. For the second summand in 4.5, note that
$U_{k+1}$ is isomorphic to $S_{k}$ as an $S_{k}$ set. Therefore we have

$\mathrm{Prim}\,(\mathrm{Sym}(\mathcal{L}[1]))_{\mathfrak{gl}}=C^{\lambda}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A)[1]\oplus
B_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A,M\otimes N),$

In order to finish the proof it remains to identify the Hochschild
differential on
$B_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A,M\otimes N)$ with the
Chevalley-Eilenberg differential.

By the isomorphism 4.3, the image of the element

$s=E^{a_{1}}_{12}\otimes E^{a_{2}}_{23}\dots E^{a_{n}}_{n\,n+1}\otimes
e^{x}_{n+1}\otimes f^{y}_{1}$

under $\widetilde{\Theta}$ represent the class of $(a_{1}\otimes a_{2}\dots
a_{n}\otimes x\otimes y)\in B_{n}(A,M\otimes N)$.

The Chevalley-Eilenberg differential is computed as follows:

$\displaystyle d_{CE}(s)=$ $\displaystyle E^{a_{2}}_{23}\otimes
E^{a_{3}}_{34}\dots E^{a_{n}}_{n\,n+1}\otimes e^{x}_{n+1}\otimes
f^{ya_{1}}_{2}$
$\displaystyle+\sum_{i=1}^{n-1}(-1)^{i}E^{a_{1}}_{12}\dots\otimes
E^{a_{i}a_{i+1}}_{i,i+2}\otimes\dots E^{a_{n}}_{n\,n+1}\otimes
e^{x}_{n+1}\otimes f^{y}_{1}$ $\displaystyle+(-1)^{n}E^{a_{1}}_{12}\otimes
E^{a_{2}}_{23}\dots E^{a_{n-1}}_{n-1\,n}\otimes e^{a_{n}x}_{n}\otimes
f^{y}_{1}.$

Its image under $\widetilde{\Theta}$ is exactly the Hochschild differential

$b(a_{1}\otimes\dots a_{n}\otimes x\otimes y).$

Finally, note that

$HH_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A,M\otimes
N)\cong\mathrm{Tor}^{A}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(N,M).$

∎

###  4.2. Unframed quiver

Recall that a quiver $Q$ is a directed graph that consist of two sets $Q_{0}$
(the set of vertices of $Q$), $Q_{1}$ (the set of edges of $Q$) and two
functions

$s,t:Q_{1}\rightrightarrows Q_{0},$

called the source and target functions.

Given a quiver $Q$, we consider the following data of algebras and modules
associated with $Q$.

(4.6) $\left\\{\begin{tabular}[]{c}To each vertex $v\in Q_{0}$ we assign a
unital $k$ algebra $A_{v}$\\\ To each edge $e\in Q_{1}$ we assign a
$A_{s(e)}-A_{t(e)}$ bimodule $M_{e}$\end{tabular}\right\\}.$

This allows us to define the direct sum Lie algebra $\bigoplus_{v\in
Q_{0}}\mathfrak{gl}_{N}(A_{v})$ and the $\bigoplus_{v\in
Q_{0}}\mathfrak{gl}_{N}(A_{v})$ module $\bigoplus_{e\in
Q_{1}}\mathrm{Mat}_{N}(M_{e})$. Denote
$\mathcal{M}:=\lim\limits_{\overset{\longrightarrow}{N}}\mathrm{Mat}_{N}$. We
consider the Lie algebra homology:

(4.7) $H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\bigoplus_{v\in
Q_{0}}\mathfrak{gl}(A_{v}),\mathrm{Sym}(\bigoplus_{e\in
Q_{1}}\mathcal{M}(M_{e})))=\lim_{\overset{\longrightarrow}{N}}H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\bigoplus_{v\in
Q_{0}}\mathfrak{gl}_{N}(A_{v}),\mathrm{Sym}(\bigoplus_{e\in
Q_{1}}\mathrm{Mat}_{N}(M_{e}))).$

This can be identified with the Lie algebra homology of the following
semidirect product Lie algebra

$\mathfrak{g}_{Q,N}:=\bigoplus_{v\in
Q_{0}}\mathfrak{gl}_{N}(A_{v})\ltimes\bigoplus_{e\in
Q_{1}}\mathrm{Mat}_{N}(M_{e})[-1].$

We also define a (super) algebra as an abelian extension of $\bigoplus_{v\in
Q_{0}}A_{v}$ by $\bigoplus_{e\in Q_{1}}M_{e}[-1]$.

$\mathcal{A}_{Q}:=\bigoplus_{v\in Q_{0}}A_{v}\bigoplus_{e\in Q_{1}}M_{e}[-1].$

By the definition of $\mathcal{A}_{Q}$, we have the following identification
of Lie algebras

$\mathfrak{g}_{Q,N}=\mathfrak{gl}_{N}(\mathcal{A}_{Q}).$

Then we can further identify the homology 4.7 with the Lie algebra homology of
$\mathfrak{gl}(\mathcal{A}_{Q})$. As a corollary of the Loday-Quillen-Tsygan
theorem, we have

######  Corollary 4.1.

There is a canonical isomorphism of graded Hopf algebras

$H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\bigoplus_{v\in
Q_{0}}\mathfrak{gl}(A_{v}),\mathrm{Sym}(\bigoplus_{e\in
Q_{1}}\mathcal{M}(M_{e})))\cong\mathrm{Sym}(HC_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathcal{A}_{Q})[1]).$

The aim of this section is to compute the cyclic homology
$HC_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathcal{A}_{Q})$.

Recall that a path of length $n$ in a quiver $Q$ is a sequence
$(e_{1},e_{2},\dots,e_{n})$ of edges such that $s(e_{i+1})=t(e_{i})$ for
$i=1,\dots n-1$. A pointed cycle is a path that satisfies the additional
relation $s(e_{1})=t(e_{n})$. Denote $\mathrm{Cyc}^{\circ}_{n}(Q)$ the set of
pointed cycles of length $n$ in $Q$. The cyclic group $\mathbb{Z}_{n}$ acts on
$\mathrm{Cyc}^{\circ}_{n}$ via the action
$(e_{1},e_{2},\dots,e_{n})\to(e_{n},e_{1},\dots,e_{n-1})$. We define a cycle
as an equivalent class of pointed cycles module the cyclic permutation. We
denote $\mathrm{Cyc}_{n}(Q):=\mathrm{Cyc}^{\circ}_{n}(Q)/\mathbb{Z}_{n}$, and
$\mathrm{Cyc}(Q)=\bigcup_{n\geq 1}\mathrm{Cyc}_{n}(Q)$.

We define the degree $\deg\ell$ of a cycle $\ell$ to be the largest integer
$d$ such that there exist a cycle $\ell_{0}$ and

$\ell=\ell_{0}^{d}\,,$

where the multiplication is taken in the path algebra of the quiver. For each
cycle $\ell$ represented by $(e_{1},e_{2},\dots,e_{n})$ we define the
following space

(4.8)
$\mathcal{F}_{\ell}=H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\left(M_{e_{1}}\otimes_{A_{t(e_{1})}}^{\mathbb{L}}M_{e_{2}}\otimes_{A_{t(e_{2})}}^{\mathbb{L}}\dots\otimes_{A_{t(e_{n-1})}}^{\mathbb{L}}M_{e_{n}}\otimes_{A_{t(e_{n})}}^{\mathbb{L}}\right)_{\mathbb{Z}_{\deg\ell}}.$

For any other representative
$(e_{1}^{\prime},e_{2}^{\prime},\dots,e_{n}^{\prime})$ of $\ell$, we have a
cyclic action $g\in\mathbb{Z}_{n}$ such that
$(e_{1}^{\prime},e_{2}^{\prime},\dots,e_{n}^{\prime})=g(e_{1},e_{2},\dots,e_{n})$.
Then we have a canonical isomorphism

$\left(M_{e_{1}^{\prime}}\otimes_{A_{t(e_{1}^{\prime})}}^{\mathbb{L}}\dots\otimes_{A_{t(e_{n-1}^{\prime})}}^{\mathbb{L}}M_{e_{n}^{\prime}}\otimes_{A_{t(e_{n}^{\prime})}}^{\mathbb{L}}\right)_{\mathbb{Z}_{\deg\ell}}\cong\left(M_{e_{1}}\otimes_{A_{t(e_{1})}}^{\mathbb{L}}\dots\otimes_{A_{t(e_{n-1})}}^{\mathbb{L}}M_{e_{n}}\otimes_{A_{t(e_{n})}}^{\mathbb{L}}\right)_{\mathbb{Z}_{\deg\ell}},$

induced by the cyclic action $g\in\mathbb{Z}_{n}$. Therefore, the definition
of $\mathcal{F}_{\ell}$ does not depend on the choice of representation of the
cycle $\ell$.

In the special case of $\ell=(e)$, we have an algebra $A_{s(e)}$ (note that
$s(e)=t(e)$), and a $A_{s(e)}$ bimodule $M_{e}$. Then the above definition
gives us the Hochschild homology with coefficient in the bimodule $M_{e}$,
$\mathcal{F}_{\ell}=HH_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A_{s(e)},M_{e})$.

In this section, we prove the following result

######  Proposition 4.2.

For any quiver $Q$, let $\mathcal{A}_{Q}$ be defined as above. There is an
isomorphism

(4.9)
$HC_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathcal{A}_{Q})\cong\bigoplus_{v\in
Q_{0}}HC_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A_{v})\bigoplus_{\ell\in\mathrm{Cyc}(Q)}\mathcal{F}_{\ell}[-1].$

###### Proof.

Denote $\mathbb{1}_{v}:=\mathbb{1}_{A_{v}}$ the unit of the algebra $A_{v}$.
We have $\mathbb{1}_{v}\cdot\mathbb{1}_{v^{\prime}}=0$ if $v\neq v^{\prime}$
and $\mathbb{1}_{v}^{2}=\mathbb{1}_{v}$. Let $S=\bigoplus_{v\in
Q_{0}}k\mathbb{1}_{v}\subset\mathcal{A}_{Q}$. $S$ is a separable algebra with
idempotent $e=\sum_{v\in Q_{0}}\mathbb{1}_{v}\otimes\mathbb{1}_{v}$. To
compute the cyclic homology, we use Corollary 3.2 and consider the Connes’
complex
$C^{\lambda,S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathcal{A}_{Q})$
over $S$. We have

$HC_{n}(\mathcal{A}_{Q})\cong
H_{n}(C_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}^{\lambda,S}(\mathcal{A}_{Q})).$

Let $Q^{+}$ be the quiver constructed from $Q$ by adding to each vertex $v\in
Q_{0}$ an edge $v^{*}$ that has $v$ as both its source and target. In other
words,

(4.10) $Q^{+}=(Q_{0},Q_{1}\sqcup Q_{0}).$

For each vertex $v\in Q_{0}$, denote $v^{*}$ the corresponding edge in
$Q^{+}$. We have $t(v^{*})=s(v^{*})=v$.

We observe that the tensor product over $S$ gives either $0$ or the tensor
product over $k$. Denote $M_{v*}:=A_{v}[1]$. We have

(4.11) $M_{e}\otimes_{S}M_{e^{\prime}}=\begin{cases}0&\text{ if }t(e)\neq
s(e^{\prime})\text{ in }Q^{+},\\\ M_{e}\otimes M_{e^{\prime}}&\text{ if
}t(e)=s(e^{\prime})\text{ in }Q^{+}.\end{cases}$

Therefore, we can rewrite the $n$-fold tensor product

$(\mathcal{A}_{Q}[1])^{\otimes_{S}n}\otimes_{S}:=\underbrace{\mathcal{A}_{Q}[1]\otimes_{S}\mathcal{A}_{Q}[1]\otimes_{S}\dots\mathcal{A}_{Q}[1]\otimes_{S}}_{n}$

as follows

$(\mathcal{A}_{Q}[1])^{\otimes_{S}n}\otimes_{S}=\bigoplus_{\tilde{\ell}=(\tilde{e}_{1},\dots,\tilde{e}_{n})\in\mathrm{Cyc}^{\circ}_{n}(Q^{+})}M_{\tilde{e}_{1}}\otimes
M_{\tilde{e}_{2}}\otimes\dots M_{\tilde{e}_{n}}.$

Under the above decomposition, the group $\mathbb{Z}_{n}$ acts on the summand
by isomorphism

(4.12) $t:M_{\tilde{e}_{1}}\otimes M_{\tilde{e}_{2}}\otimes\dots
M_{\tilde{e}_{n}}\overset{\simeq}{\to}M_{\tilde{e}_{n}}\otimes
M_{\tilde{e}_{1}}\otimes\dots M_{\tilde{e}_{n-1}},$

which is compatible with the cyclic action on the pointed cycle,
$t:(\tilde{e}_{1},\tilde{e}_{2},\dots,\tilde{e}_{n})\mapsto(\tilde{e}_{n},\tilde{e}_{1},\dots,\tilde{e}_{n-1})$.
Then by passing to the coinvariant space, we can express the above direct sum
as summing over cycles in $\mathrm{Cyc}(Q^{+})$.

Note that when $\deg\tilde{\ell}=d\geq 2$, $\tilde{\ell}=\ell_{0}^{d}$ for
some $\ell_{0}$. In this case a representative of $\tilde{\ell}$ is fixed by a
nontrivial subgroup $\mathbb{Z}_{d}\subset\mathbb{Z}_{n}$, generated by
$t^{n/d}:\tilde{\ell}\mapsto\tilde{\ell}$. Therefore, we find that the Connes’
complex is isomorphic to the following

(4.13)
$C^{\lambda,S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}-1}(\mathcal{A}_{Q})=\bigoplus_{\tilde{\ell}=(\tilde{e}_{1},\dots,\tilde{e}_{n})\in\mathrm{Cyc}(Q^{+})}(M_{\tilde{e}_{1}}\otimes
M_{\tilde{e}_{2}}\otimes\dots
M_{\tilde{e}_{n}})_{\mathbb{Z}_{\deg\tilde{\ell}}}.$

where we can choose any representative $(\tilde{e}_{1},\dots,\tilde{e}_{n})$
of the class $\tilde{\ell}\in\mathrm{Cyc}(Q^{+})$ using the isomorphism 4.12.

By the construction of $Q^{+}$, any $\tilde{\ell}\in\mathrm{Cyc}(Q^{+})$ can
be represented either by

$(v^{*})^{i}\quad\text{ for some }v\in Q_{0},$

or by

(4.14) $(s(e_{1})^{*})^{i_{1}}e_{1}(s(e_{2})^{*})^{i_{2}}e_{2}\dots
e_{m}\,\;\;\ \text{for some
}\ell=(e_{1},e_{2},\dots,e_{m})\in\mathrm{Cyc}_{m}(Q).$

In the first case, we obtain the complex
$C^{\lambda}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}-1}(A_{v})$
that computes the cyclic homology of $A_{v}$. We focus on the second case in
the following.

We would like to use 4.14 to rewrite the summation 4.13 as a summation over
$\ell\in\mathrm{Cyc}_{m}(Q)$ and $i_{1},\dots,i_{m}\in\mathbb{Z}_{+}$.
However, the map

$\varphi:\bigcup_{m\geq
1}\mathrm{Cyc}_{m}(Q)\times\mathbb{Z}_{+}^{m}\to\mathrm{Cyc}(Q^{+})$

defined by 4.14 is not injective. To correct this, we use a larger group for
the coinvariant. We take any $\tilde{\ell}\in\mathrm{Cyc}(Q^{+})$ and the
corresponding $\ell\in\mathrm{Cyc}(Q)$ such that
$\varphi(\ell,i_{1},\dots,i_{m})=\tilde{\ell}$. We observe that the group
$\mathbb{Z}_{\deg\ell}$ acts transitively on the set of preimages
$\varphi^{-1}(\tilde{\ell})$. Moreover, the subgroup that fix
$(\ell,i_{1},\dots,i_{m})$ is exact $\mathbb{Z}_{\deg\tilde{\ell}}$.
Therefore, for $\ell\in\mathrm{Cyc}(Q)$, we define the following
$\text{length}(\ell)$-fold complex

$C(Q,\ell)_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}},\dots,\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}=\left(\bigoplus_{i_{1},\dots,i_{m}\geq
0}(A_{s(e_{1})}[1])^{\otimes i_{1}}\otimes
M_{e_{1}}\otimes(A_{s(e_{2})}[1])^{\otimes i_{2}}\otimes\dots
M_{e_{m}}\right)_{\mathbb{Z}_{\deg\ell}}.$

Combining with the first case when $\tilde{\ell}=(v^{*})^{i}$, we have the
following isomorphism

$\bigoplus_{\tilde{\ell}=(\tilde{e}_{1},\dots,\tilde{e}_{n})\in\mathrm{Cyc}(Q^{+})}(M_{\tilde{e}_{1}}\otimes
M_{\tilde{e}_{2}}\otimes\dots
M_{\tilde{e}_{n}})_{\mathbb{Z}_{\deg\tilde{\ell}}}\cong\bigoplus_{v\in
Q_{0}}C^{\lambda}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathcal{A}_{v})[1]\bigoplus_{m\geq
1}\bigoplus_{\ell\in\mathrm{Cyc}_{m}(Q)}\mathrm{Tot}\,C(Q,\ell).$

Then we find that the Connes’ cyclic complex is isomorphic to

$C^{\lambda,S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathcal{A}_{Q})=\bigoplus_{v\in
Q_{0}}C^{\lambda}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathcal{A}_{v})\bigoplus_{\ell\in\mathrm{Cyc}(Q)}\mathrm{Tot}\,C(Q,\ell)[-1].$

Finally, we analyze the differential of the complex $C(Q,\ell)$. The
differential is induced from the differential $b$ on the Connes’ complex
$C_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}^{\lambda,S}(\mathcal{A}_{Q})$.
For any $x_{j}\in M_{e_{j}}$ and $a^{j}_{1},\dots,a^{j}_{i_{j}}\in
A_{s(e_{j})},a^{j+1}_{1},\dots,a^{j+1}_{i_{j+1}}\in A_{s(e_{j+1})}$ , we have

(4.15) $\displaystyle b(\dots\otimes a^{j}_{1}\otimes
a^{j}_{2}\otimes\dots\otimes a^{j}_{i_{j}}\otimes x_{j}\otimes
a^{j+1}_{1}\otimes\dots\otimes a^{j+1}_{i_{j+1}}\otimes\dots)$
$\displaystyle=\dots\dots\otimes a^{j}_{1}a^{j}_{2}\otimes\dots\otimes
a^{j}_{i_{j}}\otimes x_{j}\otimes a^{j+1}_{1}\otimes\dots\otimes
a^{j+1}_{i_{j+1}}\otimes\dots$ $\displaystyle+\dots$
$\displaystyle+\dots\otimes a^{j}_{1}\otimes a^{j}_{2}\otimes\dots\otimes
a^{j}_{i_{j}}x_{j}\otimes a^{j+1}_{1}\otimes\dots\otimes
a^{j+1}_{i_{j+1}}\otimes\dots$ $\displaystyle+\dots\otimes a^{j}_{1}\otimes
a^{j}_{2}\otimes\dots\otimes a^{j}_{i_{j}}\otimes
x_{j}a^{j+1}_{1}\otimes\dots\otimes a^{j+1}_{i_{j+1}}\otimes\dots$
$\displaystyle+\dots$

We find that the differential on $C(Q,\ell)$ correspond to the differential of
the Bar complex computing the derived tensor product
$\left(M_{e_{1}}\otimes_{A_{t(e_{1})}}^{\mathbb{L}}M_{e_{2}}\otimes_{A_{t(e_{2})}}^{\mathbb{L}}\dots\otimes_{A_{t(e_{n-1})}}^{\mathbb{L}}M_{e_{n}}\otimes_{A_{t(e_{n})}}^{\mathbb{L}}\right)_{\mathbb{Z}_{\deg\ell}}$.

∎

######  Example 4.1.

In the simplest case, we consider the quiver with one vertex $v$ and one edge
$e$ starting and ending on $v$. For this quiver, we assign an associative
algebra $A$ to the vertex and a $A$-$A$ bimodule $M$ to the edge.

$A$$M$

In this case, we find that the Lie algebra homology
$H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathfrak{gl}(A),\mathrm{Sym}(\mathcal{M}(M)))$
is given by the following

$\mathrm{Sym}(HC_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A)[1]\oplus
H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A,M)\oplus
H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(M\otimes^{\mathbb{L}}_{A}M\otimes^{\mathbb{L}}_{A})_{\mathbb{Z}_{2}}\oplus\cdots).$

###  4.3. Framed quiver

A framed quiver $Q^{\text{fr}}$ consists of the data of a quiver
$Q=(Q_{0},Q_{1})$ together with two subsets $W^{+},W^{-}$ of $Q_{0}$. Denote
the inclusion map $j:W^{+}\sqcup W^{-}\to Q_{0}$. We can think of the framed
quiver $Q^{\text{fr}}$ as adding to $Q$ extra vertices $W^{+},W^{-}$ and edges
$w^{+}\to j(w^{+})$ for all $w^{+}\in W^{+}$, and $j(w^{-})\to w^{-}$ for all
$w^{-}\in W^{-}$.

For a framed quiver $Q^{\text{fr}}=(Q,W^{+},W^{-})$, we consider the same set
of data 4.6 for $Q$, together with the following

(4.16) $\left\\{\begin{tabular}[]{c}To each vertex $w^{+}\in W^{+}$ we assign
a right $A_{j(w^{+})}$ module $M_{w^{+}}$\\\ To each vertex $w^{-}\in W^{-}$
we assign a left $A_{j(w^{-})}$ module $M_{w^{-}}$\end{tabular}\right\\}.$

In the last section, we defined the algebra $\mathcal{A}_{Q}$ associated with
the quiver $Q$. Given the above data associated with the framing, we denote

$M_{+}=\bigoplus_{w^{+}\in W^{+}}M_{w^{+}},\quad M_{-}=\bigoplus_{w^{-}\in
W^{-}}M_{w^{-}}.$

Then $M_{+}$ is a right $\mathcal{A}_{Q}$ module and $M_{-}$ is a left
$\mathcal{A}_{Q}$ module. Using the construction of 4.1 and 4.2, we have a
$\mathfrak{gl}_{N}(\mathcal{A}_{Q})$ module
$\mathbb{C}^{N}(M_{-})\bigoplus(\mathbb{C}^{N})^{*}(M_{+}).$ In this section,
we compute the following Lie algebra homology

$H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathfrak{gl}(\mathcal{A}_{Q}),\mathrm{Sym}(\mathcal{V}(M_{-})\oplus\mathcal{V}^{*}(M_{+})))=\lim_{\overset{\longrightarrow}{N}}H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathfrak{gl}_{N}(\mathcal{A}_{Q}),\mathrm{Sym}(\mathbb{C}^{N}(M_{-})\oplus(\mathbb{C}^{N})^{*}(M_{+}))).$

As a corollary of Theorem 4.1, we have the following

######  Corollary 4.2.

We have a canonical isomorphism of graded Hopf algebras

$H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathfrak{gl}(\mathcal{A}_{Q}),\mathrm{Sym}(\mathcal{V}(M_{-})\oplus\mathcal{V}^{*}(M_{+})))\cong\mathrm{Sym}(HC_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathcal{A}_{Q})[1]\oplus
H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(M_{+}\otimes_{\mathcal{A}_{Q}}^{\mathbb{L}}M_{-})).$

The cyclic homology
$HC_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\mathcal{A}_{Q})$ is
computed in the last section. In this section, we compute
$M_{+}\otimes_{\mathcal{A}_{Q}}^{\mathbb{L}}M_{-}$.

Denote $\mathrm{fPath}_{w^{+},w^{-}}(Q^{\mathrm{fr}})$ the set of paths in
$Q^{\text{fr}}$ that start with $w^{+}$ and end with $w^{-}$. Let
$\mathrm{fPath}(Q^{\mathrm{fr}})=\cup_{w^{+}\in W^{+},w^{-}\in
W^{-}}\mathrm{fPath}_{w^{+},w^{-}}$. To each
$\rho=(w^{+},e_{1},\dots,e_{n},w^{-})\in\mathrm{fPath}(Q^{\mathrm{fr}})$, we
assign the space

(4.17)
$\mathcal{F}_{\rho}=H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}\left(M_{w^{+}}\otimes^{\mathbb{L}}_{A_{j(w^{+})}}M_{e_{1}}\otimes_{A_{t(e_{1})}}^{\mathbb{L}}\dots
M_{e_{n}}\otimes_{A_{j(w^{-})}}^{\mathbb{L}}M_{w^{-}}\right).$

Then we have the following result

######  Proposition 4.3.

(4.18)
$H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(M_{+}\otimes^{\mathbb{L}}_{\mathcal{A}_{Q}}\otimes
M_{-})=\bigoplus_{\rho\in\mathrm{fPath}}\mathcal{F}_{\rho}.$

###### Proof.

To compute the derived tensor product
$M_{+}\otimes^{\mathbb{L}}_{\mathcal{A}_{Q}}\otimes M_{-}$, we use Corollary
3.1 and consider the Bar complex over $S$

$B^{S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(M_{+},\mathcal{A}_{Q},M_{-})=\bigoplus_{n\geq
0}M_{+}\otimes_{S}(\mathcal{A}_{Q}[1])^{\otimes_{S}n}\otimes_{S}M_{-}.$

Let $Q^{\mathrm{fr},+}=(Q^{+},W^{+},W^{-})$, where $Q^{+}$ is constructed from
$Q$ by 4.10. Namely, $Q^{\mathrm{fr},+}$ is constructed from the quiver
$Q^{\mathrm{fr}}$ by adding to each non-framing vertex $v\in Q_{0}$ an edge
$v^{*}$ that has $v$ as both its source and target. The tensor product
$\otimes_{S}$ has the same property as 4.11. Therefore, we have the following
isomorphism

(4.19)
$B^{S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(M_{+},\mathcal{A}_{Q},M_{-})=\bigoplus_{\rho^{+}=(w^{+},e_{1},\dots,e_{n},w^{-})\in\mathrm{fPath}(Q^{\mathrm{fr},+})}M_{w^{+}}\otimes
M_{e_{1}}\otimes\dots M_{w^{-}},$

where we denote $M_{v^{*}}=A_{v}[1]$ as before.

By definition, each framed path $\rho^{+}\in\mathrm{fPath}(Q^{\mathrm{fr},+})$
can be expressed as follows

$w^{+}(s(e_{1})^{*})^{i_{1}}e_{1}(s(e_{2})^{*})^{i_{2}}e_{2}\dots
e_{m}(t(e_{m})^{*})^{i_{m+1}}w^{-}\,\;\;\ \text{for some
}\ell=(e_{1},e_{2},\dots,e_{m})\in\mathrm{Cyc}_{m}(Q).$

As a result, the Bar complex can be written as follows

$B^{S}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(M_{+},\mathcal{A}_{Q},M_{-})=\bigoplus_{\rho\in\mathrm{fPath}(Q^{\mathrm{fr}})}C(Q^{\mathrm{fr}},\rho),$

where $C(Q^{\mathrm{fr}},\rho)$ is the complex

$C(Q^{\mathrm{fr}},\rho)_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}},\dots,\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}=\bigoplus_{i_{1},\dots,i_{m}\geq
0}M_{w^{+}}\otimes(A_{s(e_{1})}[1])^{\otimes i_{1}}\otimes
M_{e_{1}}\otimes\dots\otimes(A_{t(e_{m})}[1])^{\otimes i_{m+1}}\otimes
M_{w^{-}}.$

We find that the complex
$C(Q^{\mathrm{fr}},\rho)_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}},\dots,\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}$
is exactly the Bar complex computing the derived tensor product
$M_{w^{+}}\otimes^{\mathbb{L}}_{A_{j(w^{+})}}M_{e_{1}}\otimes_{A_{t(e_{1})}}^{\mathbb{L}}\dots
M_{e_{n}}\otimes_{A_{j(w^{-})}}^{\mathbb{L}}M_{w^{-}}$. Therefore, we have the
following

(4.20)
$\mathrm{Tor}^{\mathcal{A}_{Q}}_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(M_{+},M_{-})=\bigoplus_{\rho\in\mathrm{fPath}(Q^{\mathrm{fr},+})}\mathcal{F}_{\rho}.$

∎

Together with the Proposition 4.2 that compute the cyclic homology of
$\mathcal{A}_{Q}$, we conclude that

######  Theorem 4.2.

We have an isomorphism of graded Hopf algebras

$\displaystyle
H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\bigoplus_{v\in
Q_{0}}\mathfrak{gl}(A_{v}),$ $\displaystyle\mathrm{Sym}(\bigoplus_{e\in
E}\mathcal{M}(M_{e})\bigoplus_{w^{-}\in
W^{-}}\mathcal{V}(M_{w^{-}})\bigoplus_{w^{+}\in
W^{+}}\mathcal{V}^{*}(M_{w^{+}})))$
$\displaystyle\cong\mathrm{Sym}\left(\bigoplus_{v\in
Q_{0}}HC_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A_{v})[1]\bigoplus_{\ell\in\mathrm{Cyc}(Q)}\mathcal{F}_{\ell}\bigoplus_{\rho\in\mathrm{fPath}}\mathcal{F}_{\rho}\right).$

###  4.4. Factorization homology

In this section, we show that our previous result can be succinctly
reformulated in terms of (stratified) factorization homology.

As a first step, we see that the data of 4.6 assigns a vertex $v\in Q_{0}$ an
associative algebra $A_{v}$, which can be further identified with a one
dimensional factorization algebra. We have the isomorphism between the
Hochschild homology of $A_{v}$ and the factorization homology of $A_{v}$ over
$S^{1}$[8]:

(4.21) $\int_{S^{1}}A_{v}\cong
HH_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A).$

The natural $S^{1}$ action that rotates the circle induces an action of
$S^{1}$ on the factorization homology $\int_{S^{1}}A_{v}$. Moreover, this
action can be identified with the Connes’ cyclic operator on the Hochschild
bar complex [10]. Taking homotopy orbits (coinvariant) of the $S^{1}$-action
on the Hochschild homology gives us the cyclic homology. We have

$\left(\int_{S^{1}}A_{v}\right)_{S^{1}}\cong
HC_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(A_{v}).$

Next, we consider a cycle $\ell=(e_{1},\dots,e_{m})\in\mathrm{Cyc}(Q)$ of the
quiver. This gives us a set of associative algebras
$(A_{t(e_{i})})_{i=1,\dots,n}$ together with the set of
$A_{s(e_{i})}-A_{t(e_{i})}$ bimodules $M_{e_{i}}$. In fact, this data
$(A_{t(e_{i})},M_{e_{i}})$ defines a stratified factorization algebra
$\mathcal{A}_{\ell}$ on $S^{1}$ with $m$ marked points.

$M_{e_{1}}$$A_{t(e_{1})}$$M_{e_{2}}$$A_{t(e_{2})}$$\dots$$M_{e_{m}}$$A_{t(e_{m})}$

Using the excision theorem for stratified factorization algebra [9], we have
the following result.

######  Proposition 4.4.

There is an equivalence

$\int_{S^{1}}\mathcal{A}_{\ell}=M_{e_{1}}\otimes_{A_{t(e_{1})}}^{\mathbb{L}}M_{e_{2}}\otimes_{A_{t(e_{2})}}^{\mathbb{L}}\dots\otimes_{A_{t(e_{n-1})}}^{\mathbb{L}}M_{e_{n}}\otimes_{A_{t(e_{n})}}^{\mathbb{L}}.$

In the presence of the marked points, only the discrete subgroup
$\mathbb{Z}_{\deg\ell}\subset S^{1}$ acts on the factorization homology
$\int_{S^{1}}\mathcal{A}_{\ell}$. Taking the coinvariant of this action gives
us 4.8.

Finally, we consider a path
$\rho=(w^{+},e_{1},\dots,e_{n},w^{-})\in\mathrm{fPath}$ that ends on framed
vertices. This gives us a set of associative algebras
$(A_{s(e_{1})},A_{s(e_{2})},\dots,A_{s(e_{n})},A_{t(e_{n})})$, a set of
$A_{s(e_{i})}-A_{t(e_{i})}$ bimodules $M_{e_{i}}$, together with a right
$A_{s(e_{1})}$ module $M_{w^{+}}$ and a left $A_{t(e_{n})}$ module
$M_{w^{-}}$. This data defines a stratified factorization algebra over the
interval $[0,1]$ with $n$ additional marked points.

$M_{w^{+}}$$A_{s(e_{1})}$$M_{e_{1}}$$\cdots$$M_{e_{n}}$$A_{t(e_{n})}$$M_{w^{-}}$

Again, we use the excision theorem for stratified factorization algebra [9]
and find the following result.

######  Proposition 4.5.

There is an equivalence

$\int_{[0,1]}\mathcal{A}_{\rho}=M_{w^{+}}\otimes^{\mathbb{L}}_{A_{j(w^{+})}}M_{e_{1}}\otimes_{A_{t(e_{1})}}^{\mathbb{L}}\dots
M_{e_{n}}\otimes_{A_{j(w^{-})}}^{\mathbb{L}}M_{w^{-}}.$

Combining our previous analysis, we can now reformulate the main Theorem 4.2
of this paper using factorization homology

######  Theorem 4.3.

We have an isomorphism

$\displaystyle
H_{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\bullet$}}}}}(\bigoplus_{v\in
Q_{0}}\mathfrak{gl}(A_{v}),\mathrm{Sym}(\bigoplus_{e\in
E}\mathcal{M}(M_{e})\bigoplus_{w^{-}\in
W^{-}}\mathcal{V}(M_{w^{-}})\bigoplus_{w^{+}\in
W^{+}}\mathcal{V}^{*}(M_{w^{+}})))$
$\displaystyle\cong\mathrm{Sym}\left(\bigoplus_{v\in
Q_{0}}\left(\int_{S^{1}}A_{v}\right)_{S^{1}}[1]\oplus\bigoplus_{\ell\in\mathrm{Cyc}(Q)}\left(\int_{S^{1}}\mathcal{A}_{\ell}\right)_{\mathbb{Z}_{\deg\ell}}\oplus\bigoplus_{\rho\in\mathrm{fPath}(Q)}\left(\int_{[0,1]}\mathcal{A}_{\rho}\right)\right).$

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