# Symmetric homology is representation homology

Yuri Berest  Department of Mathematics, Cornell University, Ithaca, NY
14853-4201, USA<EMAIL_ADDRESS>and  Ajay C. Ramadoss  Department of
Mathematics, Indiana University, Bloomington, IN 47405, USA
<EMAIL_ADDRESS>
###### Abstract.

Symmetric homology is a natural generalization of cyclic homology, in which
symmetric groups play the role of cyclic groups. In the case of associative
algebras, the symmetric homology theory was introduced by Z. Fiedorowicz [19]
and was further developed in the work of S. Ault [1, 2]. In this paper, we
show that, for algebras defined over a field of characteristic $0$, the
symmetric homology theory is naturally equivalent to the (one-dimensional)
representation homology theory introduced by the authors (jointly with G.
Khachatryan) in [7]. Using known results on representation homology, we
compute symmetric homology explicitly for basic algebras, such as polynomial
algebras and universal enveloping algebras of (DG) Lie algebras. As an
application, we prove two conjectures of Ault and Fiedorowicz, including the
main conjecture of [3] on topological interpretation of symmetric homology of
polynomial algebras.

##  1\. Introduction and Main Results

Cyclic homology was introduced by A. Connes [15] (and independently by B.
Tsygan [31]) almost 40 years ago. In Connes’ approach, the key idea was to
extend the classical notion of a simplicial module to that of a cyclic module,
which he defined as a functor $\Delta C^{\rm op}\to\mathrm{Mod}_{k}$ on the
so-called cyclic category $\Delta C$. Connes observed that, for any
associative unital $k$-algebra $A$, the standard simplicial module $B^{\rm
Hoch}(A):\Delta^{\rm op}\to\mathrm{Mod}_{k}$, computing the Hochschild
homology of $A$, extends naturally to a cyclic module $B^{\rm cyc}(A):\Delta
C^{\rm op}\to\mathrm{Mod}_{k}$ called the cyclic bar construction of $A$. By
analogy with the well-known ‘Tor-formula’ for Hochschild homology:

(1.1) ${\rm{HH}}_{*}(A)\,=\,{\rm{Tor}}^{\Delta^{\rm op}}_{\ast}(k,\,B^{\rm
Hoch}A)$

he then defined the cyclic homology of $A$ by

(1.2) $\mathrm{HC}_{*}(A)\,:=\,{\rm{Tor}}^{\Delta C^{\rm
op}}_{\ast}(k,\,B^{\rm cyc}A)\,.$

Now, Connes’ cyclic category $\Delta C$ is an extension of the simplicial
category $\Delta$ obtained by taking a “crossed product” with the family
(groupoid) of cyclic groups $\\{C_{n}=\mathbb{Z}/(n+1)\mathbb{Z}\\}_{n\geq
0}\,$. This idea of a categorical crossed product was formalized by Loday and
Fiedorowicz [20] who introduced the notion of a crossed simplicial group
category $\Delta G$ and constructed new examples of such categories for
various families of groups $\\{G_{n}\\}_{n\geq 0}$. Apart from $\Delta$ and
$\Delta C$, the most interesting example111This and other examples of [20]
also appeared in the work of Feigin-Tsygan [18] and Krasauskas [24]. is the
symmetric category $\Delta S$ associated to the family of symmetric groups
$\\{\Sigma_{n+1}\\}_{n\geq 0}$. In the unpublished paper [19], Fiedorowicz
showed that, for any algebra $A$, there is a natural functor $B_{\rm
sym}(A):\Delta S\to\mathrm{Mod}_{k}$, called the symmetric bar
construction222Note that, unlike Hochschild and cyclic, the symmetric bar
construction is defined as a covariant functor on the category $\Delta S$,
which we indicate by placing ‘sym’ as a subscript in $B_{\rm sym}$. We recall
the precise definition of $\Delta S$ and the construction of $B_{\rm
sym}(A)\,$ in Section 2.3 below., and, by analogy with (1.1) and (1.2), he
then defined the symmetric homology of $A$ by the formula

(1.3) $\mathrm{HS}_{*}(A)\,:=\,{\rm{Tor}}^{\Delta S}_{\ast}(k,\,B_{\rm
sym}A)\,.$

It turns out that, despite its resemblance to (1.1) and (1.2), formula (1.3)
yields a much more complicated homology theory. While the Hochschild and
cyclic homology are explicitly known and can be easily calculated for many
different types of algebras (see [25]), the symmetric homology
$\mathrm{HS}_{*}(A)$ is not even known for the polynomial algebras
$A=k[x_{1},\ldots,x_{n}]$ in $n\geq 2$ variables! The main problem seems to be
that the $\mathrm{HS}_{*}$ theory is less accessible to algebraic methods than
that of ${\rm{HH}}_{*}$ and $\mathrm{HC}_{*}$, and — as was already recognized
in [19] — has to be approached through topology.

In 2007, Ault and Fiedorowicz [3] announced a number of general theorems on
the structure of $\mathrm{HS}_{*}(A)$ and an explicit conjecture on polynomial
algebras (see Conjecture 1 below). The proofs of most of the theorems of [3]
(along with results of the original preprint [19]) appeared in Ault’s Ph.D.
dissertation [1] and his subsequent paper [2]. However, the main conjecture of
[3] seems still to be open:

######  Conjecture 1 ([3], Conjecture 1).

For any $n\geq 1$, there is an isomorphism of graded algebras

(1.4) ${\rm
HS}_{\ast}(k[x_{1},\ldots,x_{n}])\,\cong\,\mathrm{H}_{\ast}\bigg{(}\prod_{i=1}^{n}\mathcal{C}_{\infty}({\mathbb{S}}^{0})\times\prod_{i=2}^{n}\,\Omega^{\infty}\Sigma^{\infty}({\mathbb{S}}^{i-1})^{\binom{n}{i}}\,,\,k\bigg{)}\
,$

where $\Omega^{\infty}\Sigma^{\infty}$ is the stable homotopy functor and
$\mathcal{C}_{\infty}$ is the monad associated to the little $\infty$-cubes
operad, both defined as functors on the category of based topological spaces
$\mathrm{Top}_{\ast}$ (see, e.g., [27]).

Another conjecture of [3] that apparently remains unproven333 Conjecture 2 is
stated in [3] as Theorem 8. However, as remarked in [2], this theorem remains
unproven; in fact, the author of [2] suspects that it is false in general (see
loc. cit., Remark 3.3). We will show, perhaps unsurprisingly, that Conjecture
2 does hold if $k$ is a field of characteristic $0$ (see Corollary 3.2).
provides a topological formula for the symmetric homology of an arbitrary
algebra:

######  Conjecture 2 ([3]).

For any $k$-algebra $A$, there is an isomorphism

(1.5) ${\rm HS}_{\ast}(A)\,\cong\,\mathrm{H}_{\ast}(B(D,T,A),\,k)\ ,$

where $B(D,T,A)$ stands for the two-sided bar construction associated to the
monad $D$ of an $E_{\infty}$-operad in the category of simplicial $k$-modules
corresponding to the classical Barratt-Eccles operad [4] and $T$ is the tensor
algebra functor on $k$-modules.

The goal of the present paper is to clarify the situation with symmetric
homology — in particular, to prove Conjecture 1 and Conjecture 2 stated above
— in the rational case: i.e., for algebras $A$ defined over a filed $k$ of
characteristic $0$. Our approach is based on Theorem 1.1 that provides a
natural identification of symmetric homology with (one-dimensional)
representation homology defined in [7] and studied in our subsequent papers
(see, e.g., [8, 5, 12, 11, 9]). Representation homology originates from
derived algebraic geometry; however, it can be defined algebraically, in a
straightforward way, using only classical homotopical algebra [29]. The
advantage of this algebraic approach is that it makes the relation to cyclic
homology quite natural. In the rest of the Introduction, we briefly review the
definition of representation homology and its relation to cyclic homology,
following [7], and then state our main theorems. Unlike in [7], however, we
will work with simplicial (rather than DG) algebras over an arbitrary
commutative ring.

Let $\mathrm{Alg}_{k}$ and $\mathrm{Comm}_{k}$ denote the categories of
associative and commutative algebras (with $1$), both defined over a fixed
commutative ring $k$. For an integer $n\geqslant 1$, consider the matrix
functor $\,M_{n}:\,\mathrm{Comm}_{k}\to\mathrm{Alg}_{k}\,$ that takes a
commutative $k$-algebra $C$ to the $k$-algebra $M_{n}(C)$ of $(n\times
n)$-matrices with entries in $C$. It is classically known and easy to prove
(see, e.g., [14] and [7, Prop. 2.1] for a general result) that $M_{n}$ has a
left adjoint

(1.6) $(\,\mbox{--}\,)_{n}:\ \mathrm{Alg}_{k}\to\mathrm{Comm}_{k}$

called the $n$-dimensional representation functor on $\mathrm{Alg}_{k}$. This
terminology is motivated by the fact that for any algebra $A$,
$\,(A)_{n}=\mathcal{O}[{\rm{Rep}}_{n}(A)]\,$ is the coordinate ring of the
affine moduli scheme ${\rm{Rep}}_{n}(A)$ parametrizing the $n$-dimensional
$k$-linear representations of $A$. Now, to define representation homology we
simply ‘derive’ the functor (1.6), following the standard procedure in
homotopical algebra: we prolong the adjunction
$\,(\,\mbox{--}\,)_{n}:\mathrm{Alg}_{k}\rightleftarrows\mathrm{Comm}_{k}:M_{n}\,$
to simplicial algebras, using degreewise extension:

(1.7) $(\,\mbox{--}\,)_{n}:\
\mathrm{sAlg}_{k}\,\rightleftarrows\,\mathrm{sComm}_{k}\ :M_{n}$

and then replace the adjoint functors (1.7) with their universal homotopy
approximations represented by derived functors. The key point here (observed
in [7] in the case of DG algebras) is that (1.7) is a Quillen adjunction with
respect to the standard projective model structures on simplicial
algebras444This follows from the fact that the underlying simplicial set of
$M_{n}(C)$ for any $C\in\mathrm{sComm}_{k}$ is simply the product of $n^{2}$
copies of $C$, and hence the matrix functor preserves both weak equivalences
and fibrations, i.e. it is a right Quillen functor on simplicial algebras
(see, e.g., [21]).. Hence the derived functors of (1.7) exist and are well
behaved: in particular, by Quillen’s Adjunction Theorem, they form an adjoint
pair at the level homotopy categories:

$\bm{L}(\,\mbox{--}\,)_{n}:\
{\mathtt{Ho}}(\mathrm{sAlg}_{k})\,\rightleftarrows\,{\mathtt{Ho}}(\mathrm{sComm}_{k})\,:M_{n}\
.$

For a (simplicial) $k$-algebra $A$, we can now define its $n$-dimensional
representation homology by

(1.8)
$\mathrm{HR}_{*}(A,k^{n})\,:=\,\bm{L}_{\ast}(A)_{n}\,=\,\pi_{\ast}[(QA)_{n}]\,,$

where $\,QA\,\stackrel{{\scriptstyle\sim}}{{\twoheadrightarrow}}\,A\,$ is a
cofibrant simplicial resolution of $A$. By definition,
$\mathrm{HR}_{\ast}(A,k^{n})$ is a graded commutative $k$-algebra, with
$\,\mathrm{HR}_{0}(A,k^{n})\cong(A)_{n}\,$, that does not depend on the choice
of the resolution $QA$. For $n=1$, the matrix functor is just the inclusion
$\mathrm{sComm}_{k}\,\hookrightarrow\,\mathrm{sAlg}_{k}$, and hence its left
adjoint — the $1$-dimensional derived representation functor
$\bm{L}(\,\mbox{--}\,)_{1}:\,{\mathtt{Ho}}(\mathrm{sAlg}_{k})\to{\mathtt{Ho}}(\mathrm{sComm}_{k})$
— coincides with the derived abelianization (‘commutativization’) of algebras.

Now, to relate representation homology to cyclic homology we recall that
$\,\mathrm{HC}_{0}(A)=A/[A,A]\,$ for any $A\in\mathrm{Alg}_{k}$, and observe
that there is an obvious trace map $\,{\rm{Tr}}_{n}(A):\
\mathrm{HC}_{0}(A)\,\to\,(A)_{n}\,$ induced by the character of the universal
$n$-dimensional representation of $A$. It turns out that this classical
character map extends naturally to higher cyclic homology taking values in
representation homology: i.e., for any $n\geq 1$, there is a canonical graded
$k$-linear map

(1.9) ${\rm{Tr}}_{n}(A)_{\ast}:\
\mathrm{HC}_{\ast}(A)\,\to\,\mathrm{HR}_{\ast}(A,k^{n})$

called the derived character map of $n$-dimensional representations of $A$.
The map (1.9) was originally constructed in [7] for associative DG algebras
over a field $k$ of characteristic zero. However, working simplicially, we can
construct (1.9) for $A$ defined over any field (or even an arbitrary
commutative ring $k$, provided $A$ is flat over $k$; see Section 3.1). For
$n=1$, to simplify the notation we denote (1.9) as

(1.10) ${\rm{Tr}}(A)_{\ast}:\
\mathrm{HC}_{\ast}(A)\,\to\,\mathrm{HR}_{\ast}(A,k)$

Next, recall that, for any algebra $A$, there is also a natural map

(1.11) $i^{*}:\ \mathrm{HC}_{\ast}(A)\,\to\,\mathrm{HS}_{*}(A)$

relating cyclic homology to symmetric homology via the inclusion of categories
$i:\Delta C^{\rm op}\,\hookrightarrow\,\Delta S$. (The existence of this map
has been noticed already in [19] (see, loc. cit., Remark 1.4), and it is
studied in [1, Section 12]; we will review its construction in Section 3.1
below.) Our first observation (see Proposition 3.1) is that the derived
character map (1.10) factors through (1.11), inducing a natural linear map
that relates symmetric homology to representation homology:

(1.12) ${\tt{SR}}(A)_{*}:\ \mathrm{HS}_{\ast}(A)\,\to\,\mathrm{HR}_{*}(A,k)$

Finally, to state our main theorem we recall a result of Ault (see [2, Theorem
4.1]) showing that the $\mathrm{HS}_{*}(A)$ for any $A$ admits natural
homology operations and a Pontryagin product making it a graded commutative
$k$-algebra.

######  Theorem 1.1.

For any associative unital algebra $A$ defined over a field $k$ of
characteristic $0$, the map (1.12) is an isomorphism of graded commutative
algebras:

(1.13) $\mathrm{HS}_{\ast}(A)\,\cong\,\mathrm{HR}_{*}(A,k)$

######  Remark 1.1.

One may wonder whether the isomorphism (1.13) holds in general, i.e. for
algebras defined over a field $k$ of positive characteristic or even an
arbitrary commutative ring. This is not the case: the simplest counterexample
is $A=k[x]$, the polynomial ring of one variable. Since $k[x]$ is free as a
$k$-algebra over any commutative ring $k$, we have
$\,\mathrm{HR}_{i}(k[x],k^{n})=0\,$ for all $i>0$ and all $n\geq 1$. On the
other hand, Ault’s (computer-assisted) calculations (see [1], Section 11.4
and, in particular, Conjecture 94) show that $\mathrm{HS}_{1}(k[x])\not=0$ if
$k={\mathbb{F}}_{2}$ or $k=\mathbb{Z}\,$.

######  Remark 1.2.

For $A=k[\Gamma]$, the group algebra of a (simplicial) group $\Gamma$, the map
(1.12) was constructed in our previous paper [9]. In this case, (1.12) has a
topological origin: it is induced (on homology) by a natural map of spaces:

(1.14) ${\tt{SR}}_{B\Gamma}:\
\Omega\,\Omega^{\infty}\Sigma^{\infty}\\!(B\Gamma)\,\to\,\Omega\,\mathrm{SP}^{\infty}\\!(B\Gamma)\,,$

where $\,\mathrm{SP}^{\infty}\\!(B\Gamma)\,$ denotes the classical Dold-Thom
space of the classifying space of $\Gamma\,$. The isomorphism (1.13) was
established in [9] for $A=k[\Gamma]$ under the additional assumption that
$B\Gamma$ has homotopy type of a simply connected CW complex of (locally)
finite type over $\mathbb{Q}$ (see [9, Corollary 5.1]). We strengthen this
last result of [9] by removing the above additional assumption (see Corollary
3.1).

Despite its modest appearance Theorem 1.1 is a useful result as it allows one
to translate known facts about representation homology to symmetric homology.
First of all, we can prove (see Section 3.3):

######  Theorem 1.2.

Let $V$ be a finite-dimensional vector space over a field $k$ of
characteristic $0$, and let $S_{k}(V)$ denote the symmetric algebra of $V$.
There is a natural isomorphism of graded commutative algebras

${\rm
HS}_{\ast}[S_{k}(V)]\,\cong\,\Lambda_{k}\big{(}\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}_{i=1}^{\dim
V}\wedge^{i}V[i-1]\big{)}\,,$

where $\Lambda_{k}$ stands for the graded symmetric algebra over $k$.

In fact, Theorem 1.2 is a trivial case of the following general result on
symmetric homology of universal enveloping algebras that we will prove using
results of [5] on representation homology of Lie algebras (see Theorem 3.2):

######  Theorem 1.3.

Let $U\mathfrak{a}$ be the universal enveloping algebra of a Lie algebra
$\mathfrak{a}$ defined over a field $k$ of characteristic $0$. There is a
natural isomorphism of graded commutative algebras

$\mathrm{HS}_{\ast}(U\mathfrak{a})\,\cong\,\Lambda_{k}[\,\overline{\mathrm{H}}_{\ast+1}(\mathfrak{a},\,k)\,]\,,$

where $\overline{\mathrm{H}}_{\ast}(\mathfrak{a},\,k)$ is the $($reduced$)$
Chevalley-Eilenberg Lie algebra homology of $\mathfrak{a}$ with trivial
coefficients.

As a consequence of Theorem 1.2, it is not hard to deduce (modulo well-known
topological results) that

######  Corollary 1.1 (see Corollary 3.3).

Conjecture 1 holds true over a field $k$ of characteristic $0$.

In a similar fashion, using Theorem 1.1 and results of [1], we will show that

######  Corollary 1.2 (see Corollary 3.2).

Conjecture 2 holds true over a field $k$ of characteristic $0$.

The paper is organized as follows. In Section 2, we review definitions and
prove some results on representation homology needed for the present paper. Of
independent interest here may be Theorem 2.1 that compares two different kinds
of representation homology: for groups and associated group algebras. The
proofs of our main results appear in Section 3. In Section 3.1, we construct
the derived character map (1.9) and show that, when $n=1$ and $k$ is a field,
it factors through symmetric homology (Proposition 3.1). In Section 3.2, we
prove Theorem 1.1 in a slightly more general form: for any simlicial
$k$-algebra (Theorem 3.1). The key fact behind this result is Proposition 3.2,
for which we decided to provide two independent proofs, both relying on
(different) topological arguments. In Section 3.2, we also verify Conjecture 2
(Corollary 3.2) and give another version of our main theorem: for associative
DG algebras (Proposition 3.3). Finally, in Section 3.3, we prove Theorem 1.3
(in a more general setting of chain DG Lie algebras) and deduce from it
Theorem 1.2 and Conjecture 1 (Corollary 3.3) stated above.

### Notation and conventions

Throughout the paper, $k$ denotes a commutative ground ring. Beginning with
Section 3.2, we will specialize $k$ to be a field of characteristic zero.
$\mathrm{Mod}_{k}$ denotes the category of $k$-modules; an unadorned tensor
product $\,\otimes\,$ stands for the tensor product $\,\otimes_{k}\,$ over
$k$. An algebra means an associative $k$-algebra with $1$; the category of
such algebras is denoted $\mathrm{Alg}_{k}$. A commutative algebra means a
commutative associative $k$-algebra with $1$; the category of such algebras is
denoted $\mathrm{Comm}_{k}$. The corresponding categories of simplicial and
differential-graded algebras are denoted $\mathrm{sAlg}_{k}$,
$\mathrm{sComm}_{k}$, $\mathrm{dgAlg}_{k}$, and $\mathrm{dgComm}_{k}$,
respectively. If $k$ is a field and $V$ is a graded vector space over $k$, we
write $\Lambda_{k}(V)$ for its graded symmetric algebra over $k\,$: thus
$\,\Lambda_{k}(V):=S_{k}(V_{\rm ev})\otimes E_{k}(V_{\rm odd})$, where
$S_{k}(V_{\rm ev})$ and $E_{k}(V_{\rm odd})$ are the symmetric and exterior
algebras of the even and odd subspaces of $V$, respectively.

##  2\. Preliminaries

###  2.1. Functor tensor products

We begin by recalling the classical construction of functor tensor products
(see [25, Appendix C.10]). For a small category $\mathcal{C}$, we let ${\rm
Mod}_{k}(\mathcal{C})$ and ${\rm Mod}_{k}(\mathcal{C}^{\rm op})$ denote the
categories of all covariant and contravariant functors from $\mathcal{C}$ to
${\rm Mod}_{k}$, respectively. It is well known that these are abelian
categories with sufficiently many projective and injective objects. Moreover,
there is a natural bi-additive functor

$\mbox{--}\otimes_{\mathcal{C}}\mbox{--}\,:\,{\rm Mod}_{k}(\mathcal{C}^{\rm
op})\times{\rm Mod}_{k}(\mathcal{C})\to{\rm Mod}_{k}$

called the functor tensor product. Explicitly, for
$\mathcal{M}:\mathcal{C}\to\mathrm{Mod}_{k}$ and $\mathcal{N}:\mathcal{C}^{\rm
op}\to\mathrm{Mod}_{k}$, it is defined by

(2.1)
$\mathcal{N}\otimes_{\mathcal{C}}\mathcal{M}\,:=\,\big{[}\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}_{c\in\mathcal{C}}\mathcal{N}(c)\otimes\mathcal{M}(c)\big{]}/R$

where $R$ is the $k$-submodule generated by elements of the form
$\mathcal{N}(\varphi)(x)\otimes y-x\otimes\mathcal{M}(\varphi)(y)$ for all
$x\in\mathcal{N}(c^{\prime})$, $y\in\mathcal{M}(c)$ and $\varphi\in{\rm
Hom}_{\mathcal{C}}(c,c^{\prime})$. We extend the bifunctor (2.1) to the
categories of chain complexes of $\mathcal{C}$-modules (which we denote by
${\rm Ch}({\rm Mod}_{k}(\mathcal{C}^{\rm op}))$ and ${\rm Ch}({\rm
Mod}_{k}(\mathcal{C}))$) and define

(2.2) ${\rm Tor}^{\mathcal{C}}_{\ast}(\mathcal{N},\mathcal{M})\,:=\,{\rm
H}_{\ast}(\mathcal{N}\otimes^{\bm{L}}_{\mathcal{C}}\mathcal{M})$

for $\mathcal{N}\in{\rm Ch}({\rm Mod}_{k}(\mathcal{C}^{\rm op}))$ and
$\mathcal{M}\in{\rm Ch}({\rm Mod}_{k}(\mathcal{C}))$. Note that
$\mathcal{N}\otimes^{\bm{L}}_{\mathcal{C}}\mathcal{M}$ is an object in the
(unbounded) derived category $\mathscr{D}(k):=\mathscr{D}({\rm Mod}_{k})$ of
$k$-modules, and (2.2) is just the usual hyper-Tor functor on chain complexes.
Next, we observe that there is a natural functor transforming the
$\mathcal{C}^{\rm op}$-diagrams in $\mathrm{sSet}$ (simplicial presheaves on
$\mathcal{C}$) to chain complexes over ${\rm Mod}_{k}(\mathcal{C}^{\rm op})$:

(2.3) $\mathrm{sSet}^{\mathcal{C}^{\rm
op}}\,\xrightarrow{\,k[\,\mbox{--}\,]\,}\,{\rm sMod}_{k}({\mathcal{C}}^{\rm
op})\xrightarrow{N}\,{\rm Ch}({\rm Mod}_{k}(\mathcal{C}^{\rm op}))\,,$

where $k[\,\mbox{--}\,]$ is the ($\mathcal{C}$-objectwise extension of the)
free module functor on simplicial sets and $N$ is the classical Dold-Kan
normalization functor. Abusing notation, we will write (2.3) simply as
$\,k[\,\mbox{--}\,]\,$.

Finally, we recall one well-known result on derived functor tensor products
that we will use repeatedly in this paper. Let $p:\mathcal{C}\to\mathcal{D}$
be a map of small categories. The restriction functor
$p^{*}:\mathrm{Mod}_{k}(\mathcal{D})\to\mathrm{Mod}_{k}(\mathcal{C})$
associated to $p$ is exact and has a left adjoint
$\,p_{!}:\mathrm{Mod}_{k}(C)\to\mathrm{Mod}_{k}(\mathcal{D})\,$, which is a
right exact functor on chain complexes. Replacing $p_{!}$ with its (abelian)
left derived functor $\bm{L}p_{!}$, we get a natural adjunction between
derived categories:

$\bm{L}p_{!}\,:\,\mathscr{D}({\rm
Mod}_{k}(\mathcal{C}))\rightleftarrows\mathscr{D}({\rm
Mod}_{k}(\mathcal{D}))\,:\,p^{\ast}\ .$

######  Lemma 2.1.

For $\mathcal{N}\in{\rm Mod}_{k}(\mathcal{D}^{\rm op})$, $\mathcal{M}\in{\rm
Mod}_{k}(\mathcal{C})$ there is a natural isomorphism in $\mathscr{D}(k)$:

$p^{\ast}\mathcal{N}\otimes^{\bm{L}}_{\mathcal{C}}\mathcal{M}\,\cong\,\mathcal{N}\otimes^{\bm{L}}_{\mathcal{D}}\bm{L}p_{!}\mathcal{M}\
.$

The proof of Lemma 2.1 and some of its consequences can be found, for example,
in [17, Prop. 1.1].

###  2.2. Representation homology

We review the definition of representation homology of (simplicial) groups and
monoids, following the approach of [12]. Let $\mathfrak{G}$ denote the small
category whose objects $\langle n\rangle$ are the finitely generated free
groups $\mathbb{F}_{n}={\mathbb{F}}\langle x_{1},x_{2},\ldots,x_{n}\rangle$,
one for each $n\geq 0$ (with convention that $\langle 0\rangle$ is the trivial
group), and the morphisms are arbitrary group homomorphisms. Every simplicial
group $\Gamma\in\mathrm{sGr}$ (i.e. a simplicial object in $\mathrm{Gr}$)
defines a functor $\underline{\Gamma}:\mathfrak{G}^{\rm
op}\to\mathrm{sSet}\,$,$\,\langle n\rangle\mapsto\Gamma^{n}$, where
$\Gamma^{n}$ denotes the product of $n$ copies of (the underlying simplicial
set of) $\Gamma$ in $\mathrm{sSet}$. Composing this functor with
$k[\mbox{--}]$ gives the simplicial $\mathfrak{G}^{\rm op}$-module

(2.4) $\underline{k[\Gamma]}:\mathfrak{G}^{\rm
op}\to\mathrm{sMod}_{k}\,,\qquad\langle n\rangle\mapsto k[\Gamma^{n}]\cong
k[\Gamma]^{\otimes n}\ .$

More generally, for any simplicial (or DG) cocommutative Hopf algebra
${\mathcal{A}}$, we can define

(2.5) $\underline{{\mathcal{A}}}:\mathfrak{G}^{\rm op}\to{\rm Ch}({\rm
Mod}_{k})\,,\qquad\langle n\rangle\mapsto{\mathcal{A}}^{\otimes n}\ .$

where $\mathrm{sMod}_{k}$ is identified with ${\rm Ch}(\mathrm{Mod}_{k})$ via
the Dold-Kan normalization functor. Dually (see, e.g., [30, Prop. 14.1.6]),
every commutative Hopf algebra algebra ${\mathcal{H}}$ defines a covariant
functor (left $\mathfrak{G}$-module) by the rule

(2.6) $\underline{{\mathcal{H}}}:\mathfrak{G}\to{\rm Mod}_{k}\ ,\quad\langle
n\rangle\mapsto{\mathcal{H}}^{\otimes n}\ .$

For example, if $G$ is an affine algebraic group (e.g., $G={\rm GL}_{n}(k)$)
with coordinate ring ${\mathcal{H}}=\mathcal{O}(G)$, then (2.6) can be written
in the form $\langle n\rangle\mapsto\mathcal{O}[{\rm Rep}_{G}(\langle
n\rangle)]$ which makes the functoriality clear.

######  Definition 2.1.

The representation homology of a group $\Gamma\in\mathrm{sGr}$ with
coefficients in ${\mathcal{H}}$ is defined by

${\rm HR}_{\ast}(\Gamma,{\mathcal{H}})\,:=\,{\rm
Tor}^{\mathfrak{G}}_{\ast}(\underline{k[\Gamma]},\,\underline{{\mathcal{H}}})\
.$

In the special case when $G$ is an affine algebraic group over $k$ and
${\mathcal{H}}=\mathcal{O}(G)$, we write ${\rm HR}_{\ast}(\Gamma,G)$ instead
of ${\rm HR}_{\ast}(\Gamma,\mathcal{O}(G))\,$ to simplify the notation.

Next, recall that there are classical adjoint functors (originally constructed
in [23]) that relate the (model) category of simplicial groups to that of
(reduced) simplicial sets:

(2.7)
$\mathbb{G}\,:\,\mathrm{sSet}_{0}\rightleftarrows\mathrm{sGr}\,:\,\bar{W}$

The left adjoint $\mathbb{G}$ is called the Kan loop group functor, and the
right adjoint $\bar{W}$ is the classifying complex functor on simplicial
groups. The properties of these functors are well known and described, for
example, in [22, Chap. V] (see also [12, Sect. 2.2]). We mention only that the
pair (2.7) is a Quillen equivalence, both $\mathbb{G}$ and $\bar{W}$ being
homotopy functors (see [22, V.6.4]). By [9, Lemma 3.2],
$\mathrm{HR}_{\ast}(\Gamma,{\mathcal{H}})$ depends only on the homotopy type
of the classifying space $\bar{W}\Gamma$.

Representation homology can be defined naturally — in a way similar to
Definition 2.1 — for other types of algebraic objects (more precisely,
simplicial algebras over arbitrary algebraic theories, see [10]). Here, for
purposes of comparison, we will give such a definition for the theory of
monoids.

Let $\mathfrak{M}\subset{\rm Mon}$ denote the category whose objects are
finitely generated free monoids555Abusing notation, we will use the same
symbols to denote the objects of $\mathfrak{M}$ and $\mathfrak{G}$. $\langle
n\rangle$, one for each $n\geq 0$. Group completion yields a natural functor
(map of algebraic theories)$\,p:\mathfrak{M}\to\mathfrak{G}\,$ that takes the
free monoid on $n$ generators to the free group on $n$ generators (for
$n\geqslant 0$). Now, for any fixed integer $n\geqslant 1$, the affine monoid
$k$-scheme $M_{n}$ (of $n\times n$-matrices) defines the covariant functor,
i.e. the left $\mathfrak{M}$-module

$\underline{\mathcal{O}}(M_{n})\,:\,\mathfrak{M}\to{\rm
Mod}_{k}\,,\qquad\langle m\rangle\mapsto\mathcal{O}[M_{n}(k)]^{\otimes m}\ .$

where $\mathcal{O}[M_{n}(k)]$ denotes the coordinate ring of $M_{n}$. On the
other hand, any simplicial monoid $N$ yields a right $\mathfrak{M}$-module
$k[N]$ with $\langle n\rangle\mapsto k[N]^{\otimes n}$, where $k[N]$ is the
monoid algebra of $N$. Thus we may define

$\mathrm{HR}_{\ast}^{\rm Mon}(N,M_{n}(k))\,:=\,{\rm
Tor}^{\mathfrak{M}}_{\ast}(k[N],\underline{\mathcal{O}}(M_{n}))\ .$

The next proposition is a special case of a general comparison result proved
in [10].

######  Proposition 2.1.

Let $k$ be a field. For any simplicial monoid $N$, there is a natural
isomorphism

(2.8) $\mathrm{HR}_{\ast}^{\rm
Mon}(N,M_{n}(k))\,\cong\,\mathrm{HR}_{\ast}(k[N],k^{n})\,,$

where the right-hand side is the $n$-dimensional representation homology of
the simplicial algebra $A=k[N]$ as defined in the Introduction.

###### Proof.

Let $R\stackrel{{\scriptstyle\sim}}{{\to}}N$ be a semi-free simplicial
resolution. Since $k$ is a field, $k[R]\to k[N]$ is a weak equivalence of
simplicial (right) $\mathfrak{M}$-modules by the Künneth theorem. As in the
proof of [12, Theorem 4.2], the right $\mathfrak{M}$-module
$k[R]_{m}=k[R_{m}]$ of $m$-simplices is flat, since the module $k[S]$
corresponding to any finitely generated free monoid $S$ is projective, and
$k[{R}_{m}]$ is a colimit of such modules. As in loc. cit., it follows that

(2.9) $\mathrm{HR}_{\ast}^{\rm
Mon}(N,M_{n}(k))\,\cong\,\pi_{\ast}[k[R]\otimes_{\mathfrak{M}}\underline{\mathcal{O}}(M_{n})]\
.$

Using the same argument as in [12, Prop. 4.1], it is easy to show that the
tensor product in the right-hand side of (2.9) is naturally isomorphic to
$\,(k[R])_{n}$, where $(\,\mbox{--}\,)_{n}$ denotes the (degreewise simplicial
extension of) representation functor (1.6). Since the algebra $k[R]$ is a
semi-free simplicial resolution of $k[N]$, we have
$\mathrm{HR}_{\ast}(k[N],k^{n})\cong\pi_{\ast}[(k[R])_{n}]$ by definition
(1.8). This proves the desired proposition. ∎

Next, we compare the representation homology of (simplicial) monoids to that
of (simplicial) groups.

######  Proposition 2.2.

Let $k$ be a field of characteristic $0$. Then, for any simplicial group
$\Gamma\in\mathrm{sGr}$, there is a natural isomorphism

(2.10)
$\mathrm{HR}_{\ast}(\Gamma,{\rm{GL}}_{n}(k))\,\cong\,\mathrm{HR}_{\ast}^{\rm
Mon}(\Gamma,M_{n}(k))\ .$

###### Proof.

We have

$\mathrm{HR}_{\ast}^{\rm
Mon}(\Gamma,M_{n}(k)):={\rm{Tor}}^{\mathfrak{M}}_{\ast}(k[\Gamma],\underline{\mathcal{O}}(M_{n}))={\rm{Tor}}^{\mathfrak{M}}_{\ast}(p^{\ast}\underline{k[\Gamma]},\underline{\mathcal{O}}(M_{n}))\cong{{\rm{Tor}}}^{\mathfrak{G}}_{\ast}(\underline{k[\Gamma]},\,p_{!}\underline{\mathcal{O}}(M_{n}))\,,$

where the last isomorphism is due to Lemma 2.1. Proposition follows now from
Lemma 2.2, which we prove below. ∎

######  Lemma 2.2.

Let $k$ be a field of characteristic $0$. There is an isomorphism of left
$\mathfrak{G}$-modules in the derived category
${\mathscr{D}}(\mathrm{Mod}_{k}(\mathfrak{G}))$:

(2.11)
$\bm{L}p_{!}\underline{\mathcal{O}}(M_{n})\cong\underline{\mathcal{O}}({\rm{GL}}_{n})\
.$

###### Proof.

Note that, for each $m\geq 0$, there are natural isomorphisms in
${\mathscr{D}}(k)$

$\bm{L}p_{!}\underline{\mathcal{O}}(M_{n})(\langle
m\rangle)\,\cong\,\underline{k\langle
m\rangle}\otimes^{\bm{L}}_{\mathfrak{G}}\bm{L}p_{!}\underline{\mathcal{O}}(M_{n})\,\cong\,p^{\ast}\underline{k\langle
m\rangle}\otimes^{\bm{L}}_{\mathfrak{M}}\underline{\mathcal{O}}(M_{n})\ .$

The first isomorphism above follows from the fact that $k\langle m\rangle$ is
projective as a right $\mathfrak{G}$-module; the second one is a consequence
of Lemma 2.1. Now, by Proposition 2.1, we conclude

$\mathrm{H}_{\ast}[p^{\ast}\underline{k\langle
m\rangle}\otimes^{\bm{L}}_{\mathfrak{M}}\underline{\mathcal{O}}(M_{n})]\,=:\,\mathrm{HR}_{\ast}^{\rm
Mon}(p^{\ast}\langle m\rangle,M_{n})\,\cong\,\mathrm{HR}_{\ast}(k\langle
x_{1}^{\pm 1},\ldots,x_{m}^{\pm 1}\rangle,k^{n})$

Since the algebra $\,k\langle x_{1}^{\pm 1},\ldots,x_{m}^{\pm 1}\rangle\,$ is
quasi-free (formally smooth in the category $\mathrm{Alg}_{k}$), when $k$ is a
field of characteristic zero (see, e.g., [16, Prop. 5.3(2)]) and has a semi-
free resolution with finitely many generators in each homological degree, we
can use [6, Theorem 21] to compute

$\mathrm{HR}_{i}(k\langle x_{1}^{\pm 1},\ldots,x_{m}^{\pm
1}\rangle,k^{n})\,=\,\begin{cases}\,\mathcal{O}[{\rm{GL}}_{n}(k)^{m}]&\mbox{if}\
i=0\\\\[4.30554pt] \,0&\mbox{if}\ i>0\end{cases}$

It follows that $\,\bm{L}p_{!}\underline{\mathcal{O}}(M_{n})(\langle
m\rangle)\cong\mathcal{O}[{\rm{GL}}_{n}(k)^{m}]\cong\underline{\mathcal{O}}({\rm{GL}}_{n})(\langle
m\rangle)$ for all $m\geqslant 0$. This implies (2.11), completing the proof
of the lemma. ∎

As a consequence of Proposition 2.1 and Proposition 2.2, we get the following
result that identifies group representation homology with the representation
homology of its group algebra.

######  Theorem 2.1.

Let $k$ be a field of characteristic $0$. For any $\Gamma\in\mathrm{sGr}$,
there is a natural isomorphism

$\mathrm{HR}_{\ast}(\Gamma,{\rm{GL}}_{n}(k))\,\cong\,\mathrm{HR}_{\ast}(k[\Gamma],\,k^{n})\
.$

###### Proof.

Indeed,
$\,\mathrm{HR}_{\ast}(\Gamma,{\rm{GL}}_{n}(k))\,\cong\,\mathrm{HR}_{\ast}^{\rm
Mon}(\Gamma,M_{n}(k))\,\cong\,\mathrm{HR}_{\ast}(k[\Gamma],k^{n})\,$, where
the first isomorphism is by Proposition 2.2, and the second by Proposition
2.1. ∎

######  Remark 2.1.

Although not stated explicitly, the result of Theorem 2.1 can be deduced from
the proof of [12, Theorem 3.1] for $G={\rm{GL}}_{n}(k)$ (see loc. cit., Sect.
3.3). Our arguments above give a different, more direct proof of this result.

###  2.3. Symmetric bar construction

We recall (see [25, 6.1.4]) that the symmetric category $\Delta S$ is defined
to be the extension of $\Delta$ having the same objects as $\Delta$ (and
$\Delta C$), with morphisms satisfying the two properties:

  1. (1)

For each $n\geqslant 0$, ${\rm Aut}_{\Delta S}([n])\cong\Sigma_{n+1}^{\rm
op}\,$, where $\Sigma_{n+1}$ is the $(n+1)$-th symmetric group.

  2. (2)

Any morphism $f:[n]\to[m]$ in $\Delta S$ can be factored uniquely as the
composite $f=g\circ\sigma$ with $g\in{\rm Hom}_{\Delta}([n],[m])$ and
$\sigma\in{\rm Aut}_{\Delta S}([n])\cong\Sigma_{n+1}^{\rm op}$.

It is convenient use the following notation for the morphisms in $\Delta S$
(see [1, Sect. 1.1]). Any
$\,f:\,[n]\xrightarrow{\sigma}[n]\xrightarrow{g}[m]\,$ in $\Delta S$ can be
written uniquely as the ‘tensor product’ of $m+1$ noncommutative monomials
$\,X_{0},\,X_{1},\,\ldots,X_{m}\,$ in $n+1$ formal variables
$\\{x_{0},x_{1},\ldots,x_{n}\\}$:

(2.12) $f=X_{0}\otimes X_{1}\otimes\ldots\otimes X_{m}$

in such a way that each $X_{i}$ is the product $\,x_{i_{1}}x_{i_{2}}\ldots
x_{i_{r}}\,$ of $r=|f^{-1}(i)|$ variables whose indices $i_{k}$ occur in the
fibre $f^{-1}(i)$ and that are ordered in the same way as numbers in
$\\{\sigma(0),\ldots,\sigma(n)\\}$, i.e.,
$\sigma(i_{1})<\sigma(i_{2})<\ldots<\sigma(i_{r})$.

Using the above notation, the symmetric bar construction of an algebra
$A\in\mathrm{Alg}_{k}$ is defined as the functor $\,B_{\rm sym}A:\Delta
S\to\mathrm{Mod}_{k}\,$ that is given on objects by

(2.13) $B_{\rm sym}A([n])=A^{\otimes(n+1)}\ ,\quad\forall\,n\geq 0\,,$

and on morphisms by the following ‘substitution formula’:

(2.14) $B_{\rm sym}A(f)(a_{0}\otimes\ldots\otimes a_{n})\,:=\,(a_{i_{1}}\ldots
a_{i_{r}})\otimes\ldots\otimes(a_{k_{1}}\ldots a_{k_{s}})\ .$

where $f\in{\rm{Hom}}_{\Delta S}([n],[m])$ is represented by
$\,f=(x_{i_{1}}\ldots x_{i_{r}})\otimes\cdots\otimes(x_{k_{1}}\ldots
x_{k_{s}})\,$ (cf. [25, 6.1.12]).

The symmetric bar construction extends naturally to differential graded and
simplicial $k$-algebras: for $A\in\mathrm{dgAlg}_{k}$, we have $B_{\rm
sym}A\in{\rm dgMod}_{k}(\Delta S)\cong{\rm Ch}({\rm Mod}_{k}(\Delta S))$ and
for $A\in\mathrm{sAlg}_{k}$, $\,B_{\rm sym}A\in{\rm sMod}_{k}(\Delta
S)\cong{\rm Ch}({\rm Mod}_{k}(\Delta S))$, where in the last case the
identification is given via the Dold-Kan normalization functor $N$. In this
way, using the same formula (1.3), we can extend the definition of symmetric
homology $\mathrm{HS}_{*}(A)$ to both DG and simplicial algebras.

For cocommutative Hopf algebras, we can give the following interpretation of
the symmetric bar construction. By [9, Lemma 4.2], there is a natural functor

(2.15) $\Psi_{\rm sym}:\ \Delta S\to\mathfrak{G}^{\rm op}$

defined on objects by $\Psi_{\rm sym}([n])=\langle n+1\rangle$ and on
morphisms by the following formula: if $f\in{\rm{Hom}}_{\Delta S}([n],[m])$ is
represented by $f=(x_{i_{1}}\ldots
x_{i_{r}})\otimes\cdots\otimes(x_{k_{1}}\ldots x_{k_{s}})\ ,$, then

(2.16) $\Psi_{\rm sym}(f):\langle m+1\rangle\to\langle n+1\rangle\,,\qquad
X_{0}\mapsto x_{i_{1}}\ldots x_{i_{r}}\,,\ \ldots\ ,\,X_{m}\mapsto
x_{k_{1}}\ldots x_{k_{s}}\ ,$

where $\langle m+1\rangle=\mathbb{F}\langle X_{0},\ldots,X_{m}\rangle$ and
$\langle n+1\rangle=\mathbb{F}\langle x_{0},\ldots,x_{n}\rangle$. Now, every
(simplicial/DG) cocommutative Hopf algebra ${\mathcal{A}}$ defines DG right
$\mathfrak{G}$-module $\underline{{\mathcal{A}}}$ by formula (2.5). As in the
case of group algebras (see [9, Lemma 3.3]), it is straightforward to verify
that

(2.17) $B_{\rm sym}{\mathcal{A}}\,=\,\Psi_{\rm
sym}^{\ast}(\underline{{\mathcal{A}}})\ ,$

where $\Psi_{\rm sym}^{\ast}$ denotes the restriction functor via (2.15).

##  3\. Proofs of Main Results

###  3.1. Derived character maps

We now construct derived character maps relating symmetric homology to
representation homology. As in the Introduction, we will work with simplicial
algebras.

Let $R\stackrel{{\scriptstyle\sim}}{{\to}}A$ be a semi-free simplicial
resolution of $A\in\mathrm{sAlg}_{k}$. Applying the unit of the adjunction
(1.7) to $R$, we obtain a map of simplicial $k$-algebras $\,\pi_{n}:R\to
M_{n}[(R)_{n}]\,$. Composing $\pi_{n}$ with the trace
$\,M_{n}[(R)_{n}]\to(R)_{n}\,$ gives a map of simplicial $k$-modules
$\,R\to(R)_{n}\,$. The latter factors through the quotient $\,R/[R,R]\,$,
inducing the simplicial $k$-linear map

(3.1) ${\rm{Tr}}_{n}(R):\ R/[R,R]\to(R)_{n}\ .$

######  Lemma 3.1.

There is a natural isomorphism of simplicial $k$-modules

$k\otimes_{\Delta C^{\rm op}}B^{\rm cyc}R\,\cong\,R/[R,R]\ .$

###### Proof.

It suffices to check that $\,k\otimes_{\Delta C^{\rm op}}B^{\rm cyc}R_{m}\cong
R_{m}/[R_{m},R_{m}]\,$ for each $m$, but this follows immediately from formula
(1.2) and the fact that $\mathrm{HC}_{0}(B)\cong B/[B,B]$ for any
$B\in\mathrm{Alg}_{k}$. ∎

By Lemma 3.1, we obtain the following composition of morphisms in
$\mathscr{D}(k)$, where ${\rm C}$ stands for the associated chain complex:

(3.2) $k\otimes^{\bm{L}}_{\Delta C^{\rm op}}{\rm C}(B^{\rm cyc}R)\to
k\otimes_{\Delta C^{\rm op}}{\rm C}(B^{\rm cyc}R)\cong{\rm
C}(R/[R,R])\,\xrightarrow{{\rm{Tr}}_{n}(R)}\,{\rm C}[(R)_{n}]\,,$

where the last map is defined by (3.1). Note that, if $k$ is a field, $B^{\rm
cyc}R$ is weakly equivalent to $B^{\rm cyc}A$ by Künneth’s Theorem. In this
case, (3.2) induces on homology a natural graded $k$-linear map

(3.3)
${\rm{Tr}}_{n}(A)_{\ast}:\mathrm{HC}_{\ast}(A)\to\mathrm{HR}_{\ast}(A,k^{n})\
,$

which is the derived character map (1.10) referred to in the Introduction. On
the other hand, there is an inclusion functor $i:\Delta C^{\rm
op}\hookrightarrow\Delta S$ defined in [25, 6.1.11], for which it is easy to
check

(3.4) $B^{\rm cyc}A=i^{\ast}B_{\rm sym}A$

for any algebra $A\in\mathrm{sAlg}_{k}$. Hence, the functor $i$ induces a
natural map

(3.5) $i^{\ast}:\ \mathrm{HC}_{\ast}(A)={\rm Tor}^{\Delta C^{\rm
op}}_{\ast}(k,B^{\rm cyc}A)\,\to\,{\rm Tor}^{\Delta S}_{\ast}(k,B_{\rm
sym}A)={\rm HS}_{\ast}(A)\ .$

relating cyclic homology to symmetric homology. This is the map (1.11) in the
Introduction.

Next, we recall that, for any algebra $A\in\mathrm{Alg}_{k}$, Ault [2, Theorem
4.1] constructed a graded commutative $k$-algebra structure on
$\mathrm{HS}_{\ast}(A)$. The following lemma extends this structure to
simplicial algebras, i.e., to any $A\in\mathrm{sAlg}_{k}$.

######  Lemma 3.2.

Ault’s structure of a graded commutative algebra on ${\rm HS}_{\ast}(A)$
extends naturally to simplicial $k$-algebras $($defined over an arbitrary
commutative ring $k)$.

###### Proof.

Recall (see [2, Sect. 3]) that Ault constructed a functor
$C(\mbox{--})_{\ast}:\mathrm{Alg}_{k}\to\mathrm{sE}_{\infty}\mathrm{Alg}_{k}$
such that $\pi_{\ast}[CA_{\ast}]\cong{\rm HS}_{\ast}(A)$ for any
$A\in\mathrm{Alg}_{k}$. Here, $\mathrm{sE}_{\infty}\mathrm{Alg}_{k}$ denotes
the category of $E_{\infty}$-algebra objects in the category of simplicial
$k$-modules (which we shall refer to as simplicial $E_{\infty}$-algebras).
Here, the term ‘$E_{\infty}$-algebra’ refers to an algebra over the simplicial
$E_{\infty}$-operad $\mathcal{D}_{\bf Mod}$ of [2, Remark 2.8]. Now let
$A\in\mathrm{sAlg}_{k}$ be an arbitrary simplicial $k$-algebra. Then, $CA$ is
a bisimplicial $E_{\infty}$-algebra whose diagonal $k$-module is an
$E_{\infty}$-algebra. Let $\Delta S_{+}$ be the category obtained by adjoining
an initial object $[-1]$ to $\Delta S$. Now recall that, by definition,

$CA=k[N(\mbox{--}\downarrow\Delta S_{+})]\otimes_{\Delta S_{+}}B_{\rm
sym,+}A\,,$

where $N(\mbox{--})$ stands for the simplicial nerve functor and $B_{\rm
sym,+}A$ is the extension of $B_{\rm sym}A$ to a $\Delta S_{+}$-module as in
[1, Def. 30]. Since the simplicial right $\Delta S_{+}$-module
$k[N(\mbox{--}\downarrow\Delta S_{+})]$ is a levelwise projective resolution
of $k$, the homology of (the Dold-Kan normalization of) the diagonal of $CA$
is precisely

${\rm Tor}^{\Delta S_{+}}_{\ast}(k,B_{\rm sym,+}A)\,\cong\,{\rm Tor}^{\Delta
S}_{\ast}(k,B_{\rm sym}A)=:\mathrm{HS}_{\ast}(A)\ .$

The above isomorphism follows from [1, Theorem 31]. This proves the desired
lemma. ∎

Now, we come to the main result of this section.

######  Proposition 3.1.

Assume that $k$ is a field. The one-dimensional derived character map (3.3)
factors through (3.5):

$\begin{diagram}$

inducing a homomorphism of graded commutative $k$-algebras with respect to the
Ault structure on $\mathrm{HS}_{*}(A)$:

(3.6) ${\tt{SR}}(A)_{*}:\ \mathrm{HS}_{\ast}(A)\,\to\,\mathrm{HR}_{*}(A,k)\ $

###### Proof.

Given an algebra $A$, we can construct the map (3.6) as follows. Fix a semi-
free simplicial resolution $R\stackrel{{\scriptstyle\sim}}{{\to}}A$ in
$\mathrm{sAlg}_{k}$ and consider the natural map

(3.7) $k\otimes^{\bm{L}}_{\Delta S}{\rm C}(B_{\rm sym}R)\to k\otimes_{\Delta
S}{\rm C}(B_{\rm sym}R)$

where ${\rm C}$ stands for the associated chain complex. By the Künneth
Theorem, $B_{\rm sym}R\xrightarrow{\sim}B_{\rm sym}A$ is a weak equivalence.
Hence $\mathrm{H}_{\ast}[k\otimes^{\bm{L}}_{\Delta S}{\rm C}(B_{\rm
sym}R)]\cong\mathrm{HS}_{\ast}(A)$. By [1, Theorem 86], there is a natural
isomorphism of simplicial $k$-vector spaces $k\otimes_{\Delta S}B_{\rm
sym}R\cong R_{\rm ab}=:(R)_{1}$. Hence $\mathrm{H}_{\ast}[k\otimes_{\Delta
S}{\rm C}(B_{\rm sym}R)]\cong\mathrm{HR}_{\ast}(A,1)$.Thus, (3.7) induces on
homology a graded $k$-linear map
${\tt{SR}}_{\ast}(A):\mathrm{HS}_{\ast}(A)\to\mathrm{HR}_{\ast}(A,1)$.

Next, note that the map ${\rm{Tr}}_{1}(R):R/[R,R]\to(R)_{1}$ inducing
${\rm{Tr}}(A)_{\ast}$ is just the canonical quotient to the abelianization,
and the map $i^{\ast}:k\otimes_{\Delta C^{\rm op}}B^{\rm cyc}R\to
k\otimes_{\Delta S}B_{\rm sym}R$ coincides with ${\rm{Tr}}_{1}(R)$. Hence the
following diagram commutes in ${\mathscr{D}}(k)$:

$\begin{diagram}$

which gives the natural isomorphism ${\tt{SR}}_{*}(A)\circ
i^{*}\cong{\rm{Tr}}(A)_{*}$ on homology.

It remains to prove that the map (3.6) constructed above is an algebra
homomorphism. To this end, consider the bisimplicial $E_{\infty}$-algebra
$C{R}_{\ast}$, where
$C(\mbox{--})_{\ast}:\mathrm{Alg}_{k}\to\mathrm{sE}_{\infty}\mathrm{Alg}_{k}$
is Ault’s functor (see the proof of Lemma 3.2). The total (i.e. diagonal)
simplicial vector space of $CR_{\ast}$ is a (simplicial) $E_{\infty}$-algebra.
By (the proof of) [2, Lemma 3.1],

$(C{R})_{i}\ \cong\bigoplus_{[n]\stackrel{{\scriptstyle\rm
id}}{{\to}}[n]\stackrel{{\scriptstyle
f_{1}}}{{\to}}[n_{1}]\stackrel{{\scriptstyle
f_{2}}}{{\to}}\ldots\stackrel{{\scriptstyle f_{i}}}{{\to}}[n_{i}]}{R}^{\otimes
i}\,,$

where the direct sum runs over the chains of morphisms
$[n]\stackrel{{\scriptstyle\rm id}}{{\to}}[n]\stackrel{{\scriptstyle
f_{1}}}{{\to}}[n_{1}]\stackrel{{\scriptstyle
f_{2}}}{{\to}}\ldots\stackrel{{\scriptstyle f_{i}}}{{\to}}[n_{i}]$ of length
$i+1$ in the category $\Delta S_{+}$ obtained by adjoining an initial object
$[-1]$ to $\Delta S$. Since

$(C{R})_{i}\ \cong\bigoplus_{[n]\stackrel{{\scriptstyle\rm
id}}{{\to}}[n]\stackrel{{\scriptstyle
f_{1}}}{{\to}}[n_{1}]\stackrel{{\scriptstyle
f_{2}}}{{\to}}\ldots\stackrel{{\scriptstyle f_{i}}}{{\to}}[n_{i}]}{R}^{\otimes
i}\,,$

and

$(C{A})_{i}\ \cong\bigoplus_{[n]\stackrel{{\scriptstyle\rm
id}}{{\to}}[n]\stackrel{{\scriptstyle
f_{1}}}{{\to}}[n_{1}]\stackrel{{\scriptstyle
f_{2}}}{{\to}}\ldots\stackrel{{\scriptstyle f_{i}}}{{\to}}[n_{i}]}{A}^{\otimes
i}\,,$

it follows from the Künneth theorem that $(CR)_{i}\to(CA)_{i}$ is a weak
equivalence of simplicial $E_{\infty}$-algebras for all $i$. Hence, ${\tt
diag}(C{R})_{\ast}\xrightarrow{\sim}{\tt diag}(CA)_{\ast}$ is a quasi-
isomorphism of $E_{\infty}$-algebras. On the other hand, the natural map
$C(R_{i})\to\mathrm{HS}_{0}(R_{i})\cong(R_{i})_{\rm ab}$ is an
$E_{\infty}$-algebra homomorphism by [2, Proposition 4.4], where $R_{i}$ is
the algebra of $i$-simplices on $R$. Hence the map ${\tt diag}(CR)\to R_{\rm
ab}$ is a map of simplicial $E_{\infty}$-algebras. Since the map
$C(R_{i})\to\mathrm{HS}_{0}(R_{i})\cong(R_{i})_{\rm ab}$ represents the map
$k\otimes^{\bm{L}}_{\Delta S}B_{\rm sym}R_{i}\to k\otimes_{\Delta S}B_{\rm
sym}R_{i}$, the map induced on homology (of the associated chain complexes) by
the map ${\tt diag}(CR)\to R_{\rm ab}$ is the same as that induced on
homologies by (3.7), which is ${\tt{SR}}_{\ast}(A)$. Since the map ${\tt
diag}(CR)\to R_{\rm ab}$ is a homomorphism of simplicial
$E_{\infty}$-algebras, ${\tt{SR}}_{\ast}(A)$ is a morphism of graded
commutative algebras. This finishes the proof of the proposition. ∎

######  Remark 3.1.

The result of Proposition 3.1 holds true in greater generality: for any
algebra $A$ that is flat over a commutative ring $k$. This last condition is
automatic when $k$ is a field.

######  Remark 3.2.

In general, for $n>1$, the derived character map
${\rm{Tr}}_{n}(A)_{\ast}:\mathrm{HC}_{*}(A)\to\mathrm{HR}_{*}(A,k^{n})$ does
not seem to factor through $\mathrm{HS}_{*}(A)$.

###  3.2. Symmetric homology vs representation homology

From now on, we will assume that $k$ is a field of characteristic $0$. Theorem
1.1 stated in the Introduction is a special case of the following more general
result.

######  Theorem 3.1.

For any simplicial algebra $A\in\mathrm{sAlg}_{k}$, the map
${\tt{SR}}_{\ast}(A)$ of Proposition 3.1 is an isomorphism of graded
commutative $k$-algebras.

The proof of Theorem 3.1 is based on the following key proposition.

######  Proposition 3.2.

Let $V$ be a $k$-vector space, and let $TV=T_{k}(V)$ denote the tensor algebra
of $V$. Then

${\rm Tor}^{\Delta S}_{i}(k,B_{\rm
sym}TV)\,\cong\,\begin{cases}S_{k}(V)&i=0\\\ 0&i\neq 0\end{cases}$

We will give two different proofs of Proposition 3.2, both relying on
topological results. Our first proof is parallel to that of [11, Proposition
2.1]: it is based on [11, Theorem 3.2], its Corollary [11, Corollary 3.2] as
well as [12, Proposition 5.3] and [9, Corollary 5.1], the last two involving
topological arguments.

###### First proof of Proposition 3.2.

By [1, Theorem 86], we know that $\,\mathrm{HS}_{0}(A)\cong A_{\rm ab}\,$ for
any $A\in\mathrm{Alg}_{k}$. Hence $\,k\otimes_{\Delta S}B_{\rm sym}TV\cong
S_{k}(V)\,$. We need only to prove the vanishing of higher homology:

${\rm Tor}^{\Delta S}_{i}(k,B_{\rm sym}TV)=0\quad\text{for all}\quad i>0\,$

To this end we assign $V$ homological degree $2$, and let $\mathcal{L}V$
denote the free graded Lie algebra generated by $V$. Then $TV\cong
U(\mathcal{L}V)$, where $U(\mbox{--})$ stands for universal enveloping
algebra. This gives $TV$ the structure of a (graded) cocommutative Hopf
algebra. Let $\underline{TV}$ denote the corresponding graded right
$\mathfrak{G}$-module, whose component in homological degree $2q$ is
$\underline{TV}_{q}$, the component of $\underline{TV}$ of weight $q$. More
explicitly, $\underline{TV}_{q}(\langle m\rangle)$ is the span (over $k$) of
elements of the form $w_{d_{1}}\otimes\ldots\otimes w_{d_{m}}\in(TV)^{\otimes
m}$ such that $d_{1}+\ldots+d_{m}=q$ and $w_{d_{i}}\in V^{\otimes
d_{i}}\subset TV$. By (2.17), $B_{\rm sym}TV$ acquires the structure of a
graded $\Delta S$ module whose component in homological degree $2q$ is

$(B_{\rm sym}TV)_{q}\,:=\,\Psi_{\rm sym}^{\ast}\underline{TV}_{q}\ .$

Thus,

$\mathrm{H}_{n}[k\otimes^{\bm{L}}_{\Delta S}B_{\rm
sym}TV]\,\cong\,\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}_{2q+i=n}{\rm
Tor}^{\Delta S}_{i}(k,(B_{\rm sym}TV)_{q})\ .$

The desired proposition will follow once we show that

(3.8) $\mathrm{H}_{\ast}[k\otimes^{\bm{L}}_{\Delta S}B_{\rm
sym}TV]\,\cong\,\Lambda_{k}(V)\ .$

By [11, Theorem A.1], there are Quillen equivalences refining the Dold-Kan
equivalence $N:\mathtt{Com}_{k}\rightleftarrows\mathtt{sVect}_{k}:N^{-1}$

$N^{\ast}\,:\,\mathrm{dgLie}_{k}^{+}\rightleftarrows\mathtt{sLie}_{k}\,:\,N\,,\qquad
N^{\ast}\,:\,\mathrm{dgAlg}_{k}^{+}\rightleftarrows\mathrm{sAlg}_{k}\,:\,N$

where $\mathtt{s}\mathcal{C}$ denotes the category of simplicial objects in a
category $\mathcal{C}$. Equip $N^{\ast}(TV)\cong T(N^{-1}V)$ (see formula A.4
in the Appendix of loc. cit.) with simplicial cocommutative Hopf algebra
structure given by its identification with the universal enveloping algebra
$U(\mathcal{L}(N^{-1}V))$ of the free Lie algebra generated by $N^{-1}V$. We
denote the corresponding simplicial right $\mathfrak{G}$-module by
$\underline{N^{\ast}TV}$. Let $B_{\rm sym}N^{\ast}(TV)=\Psi_{\rm
sym}^{\ast}(\underline{N^{\ast}TV})$. Since $V$ has homological degree $2$,
$N^{\ast}\mathcal{L}V\cong\mathcal{L}(N^{-1}V)$ (see formula A.4 in the
appendix of [11]) is a semi-free simplicial model of the space $X$ given by
the wedge of ${\rm dim}_{k}V$ copies of the three sphere $\mathbb{S}^{3}$. By
[11, Theorem 3.2] the right $\mathfrak{G}$-modules $k[\mathbb{G}({X})]$ and
$\underline{N^{\ast}TV}$ are (naturally) weakly equivalent, where
$\mathbb{G}({X})$ is the Kan loop group of $X$. Hence $B_{\rm sym}N^{\ast}TV$
and $B_{\rm sym}k[\mathbb{G}(X)]$ are (naturally) weakly equivalent as
simplicial $\Delta S$-modules. As in loc. cit., Corollary 3.2, it follows that
there is a natural isomorphism on homologies

$\mathrm{H}_{\ast}[k\otimes^{\bm{L}}_{\Delta S}N(B_{\rm
sym}N^{\ast}TV)]\,\cong\,\mathrm{H}_{\ast}[k\otimes^{\bm{L}}_{\Delta
S}N(k[\mathbb{G}(X)])]\ .$

The right hand side is isomorphic to the symmetric homology ${\rm
HS}_{\ast}(X)$ of a(ny) simplicial group model of $X$ by definition (see [9,
Section 4.2]). Hence there are isomorphisms

(3.9) $\mathrm{H}_{\ast}[k\otimes^{\bm{L}}_{\Delta S}N(B_{\rm
sym}N^{\ast}TV)]\,\cong\,{\rm
HS}_{\ast}(X)\,\cong\,\mathrm{HR}_{\ast}(X,\mathbb{G}_{m}(k))\,\cong\,\Lambda_{k}(V)\
.$

The second isomorphism above is by [9, Corollary 5.1] (since $X$ is indeed
simply connected), while the last isomorphism above follows from our earlier
computation of the representation homology of wedges of spheres [12,
Proposition 5.3]. Finally, as observed in (the proof of) [11, Proposition
2.1], there is a quasi-isomorphism of right $\mathfrak{G}$-modules
$\underline{\varepsilon}:\underline{TV}\to N(\underline{N^{\ast}TV})$ induced
by the unit of the adjunction between the functors $N$ and $N^{\ast}$. It
follows that $\Psi^{\ast}_{\operatorname{sym}}(\underline{\varepsilon}):B_{\rm
sym}TV\to N(B_{\rm sym}N^{\ast}TV)$ is a quasi-isomorphism of $\Delta
S$-modules. By (3.9), $\mathrm{H}_{\ast}[k\otimes^{\bm{L}}_{\Delta S}B_{\rm
sym}TV]\,\cong\,{\Lambda}_{k}V$. This verifies (3.8), completing the proof of
the desired proposition. ∎

Our second proof of Proposition 3.2 is shorter than the first, but it relies
heavily on results of Fiedorowicz and Ault (in particular, on rather technical
computations of Lemma 33 and Lemma 36 of [1]).

###### Second proof of Proposition 3.2.

Choose a basis in $V$, say a set $X\subset V$, so that $TV\cong T(k[X])\cong
k[J(X_{+})]$, where $J(X_{+})$ denotes the free monoid generated by the set
$X_{+}=X\amalg\\{\ast\\}$, with ‘$\ast$’ identified with the identity element.
We regard $J(X_{+})$ as the classical James construction of the discrete space
$X$. Then, by [1, Lemma 33], we have

${\rm HS}_{\ast}(TV)\,\cong\,{\rm
HS}_{\ast}(k[J(X_{+})])\,\cong\,\mathrm{H}_{\ast}[\mathcal{N}X;k]\ ,$

where

$\mathcal{N}X\,:=\,\coprod_{n\geq-1}N([n]\downarrow\Delta
S_{+})\times_{\Sigma_{n+1}^{\rm op}}X^{n+1}\ .$

Here, $\Delta S_{+}$ is the category obtained by adding to $\Delta S$ the
initial object $[-1]$, and $N(\mbox{--})$ in the formula above stands for the
nerve. By (the proof of) [1, Lemma 36], there is a weak equivalence
$\mathcal{N}X\simeq\mathcal{D}X$, where $\mathcal{D}X$ is the free
$\mathcal{D}$-algebra generated by $X$ for a specific simplicial
$E_{\infty}$-operad $\mathcal{D}$. Hence

${\rm
HS}_{\ast}(TV)\,\cong\,\mathrm{H}_{\ast}[\mathcal{N}X;k]\,\cong\,\mathrm{H}_{\ast}[\mathcal{D}X;k]\
.$

Now, recall that $\mathcal{D}X=\coprod_{n\geqslant 0}E_{\ast}\Sigma_{n}^{\rm
op}\times_{\Sigma_{n}}X^{n}$. Hence,

$\mathrm{H}_{\ast}[\mathcal{D}X;k]\,\cong\,\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}_{n=0}^{\infty}k[E_{\ast}\Sigma_{n}^{\rm{op}}]\otimes_{k[\Sigma_{n}]}k[X]^{\otimes
n}\,\cong\,\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}_{n=0}^{\infty}k[E_{\ast}\Sigma_{n}^{\rm{op}}]\otimes_{k[\Sigma_{n}]}V^{\otimes
n}\ .$

The last isomorphism because $X$ is a discrete basis of $V$. Since
$E_{\ast}\Sigma_{n}^{\rm op}$ is contractible for all $n$,
$k[E_{\ast}\Sigma_{n}^{\rm op}]$ is weakly equivalent to $k$. Further,
$k[E_{\ast}\Sigma_{n}^{\rm op}]$ is a levelwise projective simplicial right
$\Sigma_{n}$-module. Hence,

${\rm
HS}_{\ast}(TV)\,\cong\,\mathrm{H}_{\ast}[\mathcal{D}X;k]\,\cong\,\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}_{n=0}^{\infty}\mathrm{H}_{\ast}[k\otimes^{\bm{L}}_{k[\Sigma_{n}]}V^{\otimes
n}]\,\cong\,\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}_{n=0}^{\infty}\mathrm{H}_{\ast}(\Sigma_{n},V^{\otimes
n})\,\cong\,S_{k}(V)\ .$

where the last isomorphism holds because $k$ is a field of characteristic $0$.
∎

Now, we can proceed with

###### Proof of Theorem 3.1.

Let $R\stackrel{{\scriptstyle\sim}}{{\to}}A$ be a semi-free simplicial
resolution. Then, the (left) $\Delta S$-module of $n$-simplices in $B_{\rm
sym}{R}$ is of the form $B_{\rm sym}TV$ for some vector space $V$. It follows
from Proposition 3.2 that the following natural chain map is a quasi-
isomorphism:

(3.10) $k\otimes^{\bm{L}}_{\Delta S}{\rm C}(B_{\rm sym}R)\to k\otimes_{\Delta
S}{\rm C}(B_{\rm sym}R)\cong{\rm C}(R_{\rm ab})\,,$

where last isomorphism is a consequence of [1, Theorem 86]. Since (3.10) is
precisely the map that induces ${\tt{SR}}_{\ast}(A)$ on homology, the
${\tt{SR}}_{\ast}(A)$ is an isomorphism of graded $k$-vector spaces. By
Proposition 3.1, we then conclude that this isomorphism of vector spaces is
actually an isomorphism of $k$-algebras. ∎

As a first application of Theorem 3.1, we state the following result that
strengthens [9, Corollary 5.1] (see Remark 1.2 in the Introduction). To avoid
confusion we recall that, unlike representation homology of group algebras,
the representation homology of groups is defined in terms of Tor-functors (see
Definition 2.1). However, when ${\mathcal{H}}=\mathcal{O}[{\rm{GL}}_{n}(k)]$
and $k$ is a field of characteristic $0$, Theorem 2.1 proved in Section 2.2
shows that the two definitions agree. In combination with Theorem 3.1, this
gives

######  Corollary 3.1.

For any simplicial group $\Gamma\in\mathrm{sGr}$, there is a natural algebra
isomorphism

$\mathrm{HS}_{*}(k[\Gamma])\,\cong\,\mathrm{HR}_{*}(\Gamma,{\mathbb{G}}_{m}(k))\,,$

where ${\mathbb{G}}_{m}(k)={\rm{GL}}_{1}(k)$ is the multiplicative group of
$k$.

Next, we apply Theorem 3.1 to verify Conjecture 2 stated in the Introduction.
For a detailed explanation of the notation used in this conjecture, we refer
to [3, Theorem 8] and [2, Remark 2.8]. Here, we only recall that $D$ denotes
the monad associated to an $E_{\infty}$-operad in the category of simplicial
$k$-modules that, for $V\in\mathrm{sMod}_{k}$, can be explicitly defined by

(3.11)
$DV=\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}_{n=0}^{\infty}k[E_{\ast}\Sigma_{n}^{\rm{op}}]\otimes_{k[\Sigma_{n}]}V^{\otimes
n}$

Given a $k$-algebra $A$, $\,B(D,T,A)\,$ stands then for the two-sided
topological bar construction, i.e. the geometric realization of a simplicial
bar constriction $B_{\ast}(D,T,A)$, associated to the monad $D$ and the tensor
algebra functor $T:\mathrm{Mod}_{k}\to\mathrm{Mod}_{k}$.

######  Corollary 3.2.

For $A\in\mathrm{Alg}_{k}$, there is an isomorphism of graded commutative
$k$-algebras

${\rm HS}_{\ast}(A)\cong\mathrm{H}_{\ast}[B(D,T,A)]\ .$

###### Proof.

When $k$ is a field of characteristic $0$, we have shown (see the second proof
of Proposition 3.2) that there is a natural homology isomorphism $\,DV\to
S_{k}(V)\,$ for any $k$-vector space $V$. This induces a homology isomorphism
of the bar constructions: $\,B(D,T,A)\to B(S,T,A)\,$, where $S$ is the monad
associated with the commutative operad. Hence

$\mathrm{H}_{\ast}[B(D,T,A)]\,\cong\,\mathrm{H}_{\ast}[B(S,T,A)]\,\cong\,\pi_{\ast}[B_{\ast}(S,T,A)]\,\cong\,\pi_{\ast}[B_{\ast}(T,T,A)_{\rm
ab}]\,\cong\,\mathrm{HR}_{\ast}(A,k)\,,$

where the last isomorphism follows from the well-known fact that
$B_{\ast}(T,T,A)$ is a semi-free simplicial resolution of $A$. Theorem 3.1 now
implies the desired result. ∎

For further applications, it will be convenient for us to restate the result
of Theorem 3.1 for differential graded algebras. To this end we will use
monoidal versions of the classical Dold-Kan correspondence in the form proved
in [11, Theorem A.1]. Specifically, if $k$ is a field of characterisitc zero,
there are natural Quillen equivalences:

(3.12)
$N^{\ast}\,:\,\mathrm{dgAlg}_{k}^{+}\rightleftarrows\mathrm{sAlg}_{k}\,:\,N\,,\qquad
N^{\ast}\,:\,\mathrm{dgComm}_{k}^{+}\rightleftarrows\mathrm{sComm}_{k}\,:\,N\,,$

where $\mathrm{dgAlg}_{k}^{+}$ (respectively, $\mathrm{dgComm}_{k}^{+}$)
denote the model categories of non-negatively graded (chain) DG $k$-algebra
(respectively, commutative DG algebras) over $k$. In the above adjunctions,
the right adjoint functors are given by the Dold-Kan normalization functor $N$
restricted to the corresponding subcategories of algebras in
$\mathrm{sMod}_{k}$. Since $N$ reflects weak equivalences and the unit ${\rm
Id}\to NN^{\ast}$ is a natural weak equivalence, the left adjoint functors
$N^{\ast}$ preserve weak equivalences.

Now, recall that, in [7, Section 2], we defined the representation homology
$\mathrm{HR}_{\ast}(A,k^{n})$ for any $A\in\mathrm{dgAlg}_{k}^{+}$. Apart from
the model categories $\mathrm{sAlg}_{k}$ and $\mathrm{sComm}_{k}$ being
replaced by $\mathrm{dgAlg}_{k}^{+}$ and $\mathrm{dgComm}_{k}^{+}$
respectively, the definition of [7] is identical to the definition of
representation homology of simplicial algebras that we presented in the
Introduction. To deduce the following proposition from Theorem 3.1 we need
only to check that the equivalence
$N^{\ast}:\mathrm{dgAlg}_{k}^{+}\to\mathrm{sAlg}_{k}$ preserves both symmetric
and representation homology.

######  Proposition 3.3.

For any $A\in\mathrm{dgAlg}_{k}^{+}$, there is an isomorphism of graded
$k$-vector spaces $\mathrm{HS}_{\ast}(A)\cong\mathrm{HS}_{\ast}(N^{\ast}A)$.
As a consequence, there is an isomorphism of graded commutative $k$-algebras

${\rm HS}_{\ast}(A)\cong\mathrm{HR}_{\ast}(A,k)\,,$

where $\mathrm{HS}_{\ast}(A)$ inherits its algebra structure from
$\mathrm{HS}_{\ast}(N^{\ast}A)$.

###### Proof.

There are natural quasi-isomorphisms of complexes of $\Delta S$-modules

$B_{\rm sym}A\stackrel{{\scriptstyle\sim}}{{\to}}B_{\rm
sym}(NN^{\ast}A)\stackrel{{\scriptstyle\sim}}{{\to}}N(B_{\rm
sym}N^{\ast}A)\,,$

where the the first arrow — induced by the natural weak equivalence $A\to
NN^{\ast}A$ of DG algebras — is a weak equivalence by the Künneth Theorem, and
the second arrow is a Eilenberg-Zilber map, which is also known to be a quasi-
isomorphism by [11, Sect. A.2]. Thus, on homology, we get natural isomorphisms
of graded $k$-vector spaces

$\mathrm{HS}_{\ast}(A):=\mathrm{H}_{\ast}[k\otimes^{\bm{L}}_{\Delta S}B_{\rm
sym}A]\,\cong\,\mathrm{H}_{\ast}[k\otimes^{\bm{L}}_{\Delta S}N(B_{\rm
sym}N^{\ast}A)]=:\mathrm{HS}_{\ast}(N^{\ast}A)\ .$

proving the first statement in the desired proposition.

Next, since the abelianization functor $(\,\mbox{--}\,)_{\rm ab}$ is left
adjoint, it commutes with $N^{\ast}$: i.e., there is a commutative diagram of
functors

(3.13) $\begin{diagram}\ .$

Let $R\stackrel{{\scriptstyle\sim}}{{\to}}A$ be a semi-free resolution of $A$
in $\mathrm{dgAlg}_{k}^{+}$. There are isomorphisms (that are natural in $R$)

(3.14) $\mathrm{HR}_{\ast}(A,k)\,\cong\,\mathrm{H}_{\ast}(R_{\rm
ab})\,\cong\,\pi_{\ast}[N^{\ast}(R_{\rm
ab})]\,\cong\,\pi_{\ast}[(N^{\ast}R)_{\rm
ab}]\,\cong\,\mathrm{HR}_{\ast}(N^{\ast}A,k)\ .$

The first isomorphism above is by definition (see [7, Sect. 2]), the second
isomorphism above is by (3.13), and the last isomorphism above is by
definition (see Definition 1.8) and the fact that $N^{\ast}R\to N^{\ast}A$ is
a semi-free resolution of $A$ (by [11, Prop. A.2] and since $N^{\ast}$
preserves weak-equivalences). This shows that there is an isomorphism of
graded commutative $k$-algebras

(3.15) $\mathrm{HR}_{\ast}(A,k)\cong\mathrm{HR}_{\ast}(N^{\ast}A,k)\ .$

The second statement of the desired proposition now follows immediately from
Theorem 3.1. ∎

To illustrate the usefulness of Proposition 3.3 we give one example.

######  Example 3.1.

Let $A=k[x]/(x^{2})$ be the algebra of dual numbers defined over a field $k$
of characteristic $0$. It is known (see, e.g., [6, Sect. 6.3.1]) that $A$ has
the semi-free DG algebra resolution $\,R:=k\langle
x,t_{1},t_{2},t_{3},\ldots\rangle\,$, where $\deg(t_{i})=i$ and the
differential is given by $\,dt_{1}=x^{2}\,$ and

$dt_{i}=xt_{i-1}-t_{1}t_{i-2}+\ldots+(-1)^{i-1}t_{i-1}x\ ,\quad i\geq 2\,.$

By Proposition 3.3, ${\rm HS}_{\ast}(A)$ is the homology of the commutative DG
algebra $k[x,t_{1},t_{2},t_{3},\ldots]$ with differential given by the same
formulas as above. Using Macaulay2 computer algebra software, one can find the
groups ${\rm HS}_{i}(A)$ explicitly to sufficiently high homological degrees
(cf. [6]):

$\displaystyle{\rm HS}_{0}(A)$ $\displaystyle\cong$ $\displaystyle A$
$\displaystyle{\rm HS}_{1}(A)$ $\displaystyle\cong$ $\displaystyle 0$
$\displaystyle{\rm HS}_{2}(A)$ $\displaystyle\cong$ $\displaystyle A\cdot
t_{2}$ $\displaystyle{\rm HS}_{3}(A)$ $\displaystyle\cong$ $\displaystyle
A\cdot(xt_{3}-2t_{1}t_{2})$ $\displaystyle{\rm HS}_{4}(A)$
$\displaystyle\cong$ $\displaystyle A\cdot t_{2}^{2}\oplus A\cdot t_{4}$
$\displaystyle{\rm HS}_{5}(A)$ $\displaystyle\cong$ $\displaystyle
A\cdot(-2t_{1}t_{2}^{2}+xt_{2}t_{3})\oplus
A\cdot(-t_{2}t_{3}-4t_{1}t_{4}+2xt_{5})$ $\displaystyle{\rm HS}_{6}(A)$
$\displaystyle\cong$ $\displaystyle A\cdot t_{2}t_{4}\oplus A\cdot t_{6}$
$\displaystyle{\rm HS}_{7}(A)$ $\displaystyle\cong$ $\displaystyle
A\cdot(-t_{2}^{2}t_{3}-4t_{1}t_{2}t_{4}+2xt_{2}t_{5})\oplus
A\cdot(-t_{3}t_{4}-2t_{1}t_{6}+xt_{7})$ $\displaystyle{\rm HS}_{8}(A)$
$\displaystyle\cong$ $\displaystyle A\cdot t_{2}t_{6}\oplus A\cdot
t_{4}^{2}\oplus A\cdot t_{8}$

###  3.3. Symmetric homology of universal enveloping algebras

As a main application of Proposition 3.3, we now prove the following result
that includes (as special cases) Theorem 1.2 and Theorem 1.3 stated in the
Introduction.

######  Theorem 3.2.

Let $\mathfrak{a}\in\mathrm{dgLie}^{+}_{k}$ be a chain DG Lie algebras over
$k$, and let $U\mathfrak{a}\in\mathrm{dgAlg}^{+}_{k}$ denote its universal
enveloping algebra. There is an isomorphism of graded commutative $k$-algebras

${\rm
HS}_{\ast}(U\mathfrak{a})\,\cong\,\Lambda_{k}[\overline{\mathrm{H}}_{\ast+1}(\mathfrak{a};k)]\,,$

where $\overline{\mathrm{H}}_{\ast}(\mathfrak{a};k)$ is the $($reduced$)$
Chevalley-Eilenberg homology of $\mathfrak{a}$ with trivial coefficients.

###### Proof.

Recall (see, e.g., [5, Sect. 6.2]) that $U\mathfrak{a}$ is Koszul dual to the
cocommutative DG coalgebra $C:=\mathcal{C}_{\ast}(\mathfrak{a};k)$, whose
underlying chain complex is the Chevalley-Eilenberg complex of $\mathfrak{a}$.
This means that the cobar construction
$\Omega(C)\stackrel{{\scriptstyle\sim}}{{\to}}U\mathfrak{a}$ yields a
cofibrant DG algebra resolution of $U\mathfrak{a}$ (see, e.g, [26, Chap. 2]).
Recall that $\Omega(C)$ is defined to be the tensor algebra
$T_{k}(\bar{C}[-1])$ with differential being the sum of two degree $(-1)$
derivations: $\,d=d_{1}+d_{2}$. On generators, $d_{1}$ is defined to be the
differential of $\bar{C}[-1]$, while $d_{2}$ is given on $\bar{C}[-1]$ by the
following composite map

$\begin{diagram}\,,$

where $\Delta$ is the coproduct on $\bar{C}$ and $\,\Delta_{s}:k[-1]\to
k[-1]\otimes k[-1]\,$ is the linear map taking $1_{k[-1]}$ to
$-1_{k[-1]}\otimes 1_{k[-1]}$. Since $\bar{C}$ is cocommutative, $d_{2}$, in
fact, maps $\bar{C}[-1]$ to $\wedge^{2}(\bar{C}[-1])\subset\Omega(C)$. Hence,

(3.16) $\Omega(C)_{\rm ab}\,\cong\,\Lambda_{k}(\bar{C}[-1])\,,$

with the differential in the right hand side induced by
$d_{1}=d_{\bar{C}[-1]}$.

Now, we have the following chain of natural isomorphisms

(3.17) ${\rm
HS}_{\ast}(U\mathfrak{a})\,\cong\,\mathrm{HR}_{\ast}(U\mathfrak{a},k)\,\cong\,\mathrm{H}_{\ast}[\Omega(C)_{\rm
ab}]\,\cong\,\mathrm{H}_{\ast}[\Lambda_{k}(\bar{C}[-1])]\,\cong\,\Lambda_{k}[\overline{\mathrm{H}}_{\ast+1}(\mathfrak{a};k)]\,,$

The first isomorphism in (3.17) follows from Proposition 3.3, the second is by
definition of representation homology for (non-negatively graded) DG algebras
(see [7, Sect. 2]), the third is induced by (3.16), and the last one is a
consequence of the well-known fact that the symmteric algebra functor
$\Lambda_{k}$ commutes with homology when $k$ is a field of characteristic $0$
(see, e.g. [28, Appendix B, Prop. 2.1]). ∎

Note that if $V$ is an $n$-dimensional vector space over $k$, we can think of
it as the Lie algebra $\mathfrak{a}=V$ with trivial bracket. Then
$U\mathfrak{a}=S_{k}(V)$ and

$\overline{\mathrm{H}}_{i}(\mathfrak{a};k)\,\cong\,\begin{cases}\wedge^{i}V&1\leqslant
i\leqslant n\\\ 0&\text{ otherwise }\end{cases}$

Hence, by Theorem 3.2, we have

(3.18) ${\rm
HS}_{\ast}[S_{k}(V)]\,\cong\,\Lambda_{k}\big{[}\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}_{i=1}^{n}\wedge^{i}V[i-1]\big{]}\,\cong\,\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigotimes$}}}}_{i=1}^{n}\Lambda_{k}(\wedge^{i}V[i-1])\,,$

which is the result of Theorem 1.2. We can now use the algebraic formula
(3.18) to verify the Ault-Fiedorowicz topological formula for symmetric
homology (see Conjecture 1 in the Introduction).

######  Corollary 3.3.

If $k$ is a field of characteristic $0$, then there is an isomorphism

(3.19)
$\mathrm{H}_{\ast}\bigg{(}\prod_{i=1}^{n}\mathcal{C}_{\infty}({\mathbb{S}}^{0})\times\prod_{i=2}^{n}\,\Omega^{\infty}\Sigma^{\infty}({\mathbb{S}}^{i-1})^{\binom{n}{i}}\,,\,k\bigg{)}\,\cong\,\mathrm{HS}_{\ast}(k[x_{1},\ldots,x_{n}])\
,\quad\forall\,n\geq 1\,.$

###### Proof.

Since $\mathcal{C}_{\infty}$ is an $E_{\infty}$-operad, and the monads
associated to different $E_{\infty}$-operads are equivalent (see [27]), we can
use the monad (3.11) to compute the homology of
$\mathcal{C}_{\infty}({\mathbb{S}}^{0})$: specifically, writing
${\mathbb{S}}^{0}=\\{\ast,x\\}$ as a discrete two-point space, we can identify

$\mathrm{H}_{\ast}(\mathcal{C}_{\infty}({\mathbb{S}}^{0}),\,k)\,\cong\,\mathrm{H}_{\ast}[\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigoplus$}}}}_{n=0}^{\infty}k[E_{\ast}\Sigma_{n}^{\rm
op}]\otimes_{k[\Sigma_{n}]}(k\cdot x)^{\otimes n}]\,\cong\,k[x]\,,$

where the last isomorphism holds thanks to the fact that $k$ is a field of
characteristic $0$. Then, by Künneth’s Theorem,

(3.20)
$\mathrm{H}_{\ast}(\mathcal{C}_{\infty}({\mathbb{S}}^{0})\,\times\,\stackrel{{\scriptstyle
n}}{{\ldots}}\,\times\,\mathcal{C}_{\infty}({\mathbb{S}}^{0}),\
k)\,\cong\,S_{k}(V)\,=\,\Lambda_{k}(V)\,.$

where $V$ is the $k$-vector space spanned by the set
$\\{x_{1},x_{2},\ldots,x_{n}\\}$ of zero-dimensional cycles corresponding to
$n$ different copies of ${\mathbb{S}}^{0}$. On the other hand,

(3.21)
$\mathrm{H}_{\ast}(\Omega^{\infty}\Sigma^{\infty}({\mathbb{S}}^{i-1}),\,k)\,\cong\,\mathrm{H}_{\ast}(\mathrm{SP}^{\infty}({\mathbb{S}}^{i-1}),\,k)\,\cong\,\Lambda_{k}(k[i-1])\,,$

where again the first isomorphism holds thanks to the fact that $k$ is a field
of characteristic $0$ (see, e.g., [13, Sect. 7.3, Remark, p. 565] and the
second because ${\rm SP}^{\infty}({\mathbb{S}}^{i-1})$ is a
$K(i-1,\mathbb{Z})$-space. It follows from (3.21) that there is a (non-
canonical) isomorphism

(3.22)
$\mathrm{H}_{\ast}[\Omega^{\infty}\Sigma^{\infty}({\mathbb{S}}^{i-1})\,\times\stackrel{{\scriptstyle{\binom{n}{i}}}}{{\ldots}}\,\times\,\Omega^{\infty}\Sigma^{\infty}({\mathbb{S}}^{i-1}),\,k]\,\cong\,\Lambda_{k}(\wedge^{i}V[i-1])$

Now, combining (3.20) and (3.22), we conclude by Künneth’s Theorem:

$\mathrm{H}_{\ast}\bigg{(}\prod_{i=1}^{n}\mathcal{C}_{\infty}({\mathbb{S}}^{0})\times\prod_{i=2}^{n}\,\Omega^{\infty}\Sigma^{\infty}({\mathbb{S}}^{i-1})^{\binom{n}{i}}\,,\,k\bigg{)}\
\cong\
\Lambda_{k}(V)\otimes\,\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigotimes$}}}}_{i=2}^{n}\Lambda_{k}(\wedge^{i}V[i-1])\
\cong\
\operatorname*{\text{\raisebox{1.07639pt}{\scalebox{0.8}{$\bigotimes$}}}}_{i=1}^{n}\Lambda_{k}(\wedge^{i}V[i-1])\,,$

Comparing with formula (3.18) shows that this last homology is indeed
isomorphic to the symmetric homology of the polynomial algebra in $n$
variables. This completes the proof of the corollary. ∎

### Acknowledgments

The work of the first author was partially supported by NSF grant DMS 1702372
and the Simons Collaboration Grant 712995. The second author was partially
supported by NSF grant DMS 1702323.

## References

  * [1] S. Ault.  Symmetric homology of algebras.  Algebr. Geom. Topol., 10(4):2343–2408, 2010.
  * [2] S. Ault.  Homology operations in symmetric homology.  Homology Homotopy Appl., 16(2):239–261, 2014.
  * [3] S. Ault and Z Fiedorowicz.  Symmetric homology of algebras, (2007).  preprint arXiv:0708.1575.
  * [4] M. G. Barratt and P. J. Eccles.  $\Gamma^{+}$-structures. III. The stable structure of $\Omega^{\infty}\Sigma^{\infty}A$.  Topology, 13:199–207, 1974.
  * [5] Yu. Berest, G. Felder, S. Patotski, A. C. Ramadoss, and T. Willwacher.  Representation homology, Lie algebra cohomology and the derived Harish-Chandra homomorphism.  J. Eur. Math. Soc. (JEMS), 19(9):2811–2893, 2017.
  * [6] Yu. Berest, G. Felder, and A. Ramadoss.  Derived representation schemes and noncommutative geometry.  In Expository lectures on representation theory, volume 607 of Contemp. Math., pages 113–162. A. M. S., Providence, RI, 2014.
  * [7] Yu. Berest, G. Khachatryan, and A. Ramadoss.  Derived representation schemes and cyclic homology.  Adv. Math., 245:625–689, 2013.
  * [8] Yu. Berest and A. Ramadoss.  Stable representation homology and Koszul duality.  J. Reine Angew. Math., 715:143–187, 2016.
  * [9] Yu. Berest and A. C. Ramadoss.  Derived character maps of group representations, (2022).  preprint arXiv:2210.01304.
  * [10] Yu. Berest and A. C. Ramadoss.  Homotopy algebras and functor homology, (2022).  in preparation.
  * [11] Yu. Berest, A. C. Ramadoss, and W.-K. Yeung.  Representation homology of simply connected spaces.  J. Topol., 15(2):692–744, 2022.
  * [12] Yu. Berest, A. C. Ramadoss, and W.-K. Yeung.  Representation homology of topological spaces.  Int. Math. Res. Not. IMRN, (6):4093–4180, 2022.
  * [13] G. Carlsson and R. J. Milgram.  Stable homotopy and iterated loop spaces.  In Handbook of algebraic topology, pages 505–583. North-Holland, Amsterdam, 1995.
  * [14] P. M. Cohn.  The affine scheme of a general ring.  Lecture Notes in Math., 753:197–211, 1979.
  * [15] A. Connes.  Cohomologie cyclique et foncteurs ${\rm Ext}^{n}$.  C. R. Acad. Sci. Paris Sér. I Math., 296(23):953–958, 1983\.
  * [16] J. Cuntz and D. Quillen.  Algebra extensions and nonsingularity.  J. Amer. Math. Soc., 8(2):251–289, 1995.
  * [17] A. Djament.  Décomposition de Hodge pour l’homologie stable des groupes d’automorphismes des groupes libres.  Compos. Math., 155(9):1794–1844, 2019.
  * [18] B. L. Feĭgin and B. L. Tsygan.  Additive $K$-theory.  Lecture Notes in Math., 1289:67–209, 1987.
  * [19] Z. Fiedorowicz.  Symmetric bar construction, (1991).  http://www.math.ohio-state.edu/~fiedorow/symbar.ps.gz.
  * [20] Z. Fiedorowicz and J.-L. Loday.  Crossed simplicial groups and their associated homology.  Trans. Amer. Math. Soc., 326(1):57–87, 1991.
  * [21] P. Goerss and K. Schemmerhorn.  Model categories and simplicial methods.  Contemp. Math., 436:3–49, 2007.
  * [22] P. G. Goerss and J. F. Jardine.  Simplicial homotopy theory.  Birkhäuser Verlag, Basel, 2009.
  * [23] D. M. Kan.  On homotopy theory and c.s.s. groups.  Ann. of Math. (2), 68:38–53, 1958.
  * [24] R. Krasauskas.  Skew-simplicial groups.  Litovsk. Mat. Sb., 27(1):89–99, 1987.
  * [25] J.-L. Loday.  Cyclic homology.  Springer-Verlag, Berlin, second edition, 1998.
  * [26] J.-L. Loday and B. Vallette.  Algebraic operads.  Springer, Heidelberg, 2012.
  * [27] J. P. May.  The geometry of iterated loop spaces.  Lecture Notes in Mathematics, Vol. 271. Springer-Verlag, 1972.
  * [28] D. Quillen.  Rational homotopy theory.  Ann. of Math. (2), 90:205–295, 1969.
  * [29] D. G. Quillen.  Homotopical algebra.  Lecture Notes in Mathematics, No. 43. Springer-Verlag, Berlin-New York, 1967.
  * [30] B. Richter.  From categories to homotopy theory.  Cambridge University Press, Cambridge, 2020.
  * [31] B. L. Tsygan.  Homology of matrix Lie algebras over rings and the Hochschild homology.  Uspekhi Mat. Nauk, 38(2(230)):217–218, 1983.