Misha Kornev

Misha Kornev

PhD student in mathematics

Representation Homology

5 minute read

Published:

We can think about any commutative associative algebra $A$ as a functor $\mathcal{L}(A):\mathsf{Fin}\to k\text{-}\mathsf{mod}$ from the category of finite sets and arbitrary functions to the category of $k$-modules ($k$ is a commutative ring, the letter $\mathcal{L}$ is devoted to Jean-Louis Loday). It can be constructed as follows. On objects it is defined by sending a set $\mathbf{n}$ to $A^{\otimes_k n}$. For a morphism $f: \mathbf{n}\to\mathbf{m}$, define the morphism $\mathcal{L}(A)(f)$.

\begin{equation*} a_1\otimes...\otimes a_n\mapsto b_1\otimes...\otimes b_m, \end{equation*}
\begin{equation*} b_i = \prod\limits_{j\in f^{-1}(i)}a_j, \end{equation*}

and $b_i = 1$ if $f^{-1}(i) = \varnothing$. The converse is also true, assuming $A$ as a $k$-module.

In other words, commutative $k$-algebras are nothing but algebras over PROP of finite sets in the category of $k$-modules.

In the same manner, any commutative Hopf algebra $A$ is encoded by a functor $\mathcal{L}(A):\mathfrak{G}\to k\text{-}\mathsf{mod}$ from the category of finite free groups to the category of $k$-modules, or, equivalently, they are algebras over PROP of finitely generated free groups in the category of $k$-modules.

These observations play an essential rôle in the generalizations of Hochschild homology. Given a functor $F:\mathsf{Fin}\to k\text{-}\mathsf{mod}$, define Hochschild homology of $F$ as the homology of the following simplicial object:

\begin{equation*} \Delta^{\mathrm{op}}\stackrel{\mathbb{S}^{1}}{\to}\mathfrak{G}\stackrel{F}{\to}\mathsf{ComAlg}_{k} \end{equation*}

where $\mathbb{S}^1$ denotes the simplicial circle. Taking $F$ to be the Loday functor $\mathcal{L}(A, M)$ sending $\mathbf{n}$ to $M\otimes A^{\otimes_k n}$ for some commutative algebra $A$ and $A$-bimodule $M$ (on morphisms it is defined similarly as above), we get the ordinary Hochschild homology $\mathrm{HH}_\ast(A, M)$. This approach can be generalized to the case of abelian categories.

Representation Schemes

Now, let me briefly recall some notions on Representation Theory following [1]. Consider the set $\mathrm{Rep}^A_n(B)$ of all $n$-dimensional matrix representations of a commutative $k$-algebra $A$ with coefficients in a commutative unital algebra $B$:

\begin{equation*} \mathrm{Rep}^A_n(B) = \mathrm{Hom}_{\mathrm{Alg}_k}(A, \mathrm{Mat}_n(B)), \end{equation*}

where $\mathrm{Mat}_n(B)$ denotes the algebra of matrices over $B$. Hence, we get a functor

\begin{equation*} \mathrm{Rep}^A_n: \mathrm{ComAlg}_k\to \mathrm{Set}. \end{equation*}

Definition 1

Let $n$ be a natural number and $A$ a unital associative algebra. A pair $(B,\pi)$ consisting of a commutative algebra $B$ and representation $\pi: A \to \mathrm{Mat}(B)$ with coefficients in $B$ is called universal if for any $n$-dimensional representation $\rho: A \to \mathrm{Mat}(B’)$ there is a unique morphism $f : B \to B’$ so that $\rho = f_\ast\pi$.

The following proposition is essential to introduce the notion of a representation scheme.

Proposition

Let $A$ be as above and $n \in \mathbb{N}$. Then, there is a universal representation $(A_n, \pi_n)$. It is unique in the sense that for any other universal $(A’_n,\pi’_n)$, there exists a unique isomorphism $f:A’_n\to A_n$ such that $\pi_n’ =\pi_n\circ f$.

Definition 2

The $n$-th representation scheme of $A$ is $\mathrm{Spec}(A_n)$.

Hence, for any affine algebraic group $G$, the functor

\begin{equation*} \begin{array}{cc} \mathrm{Rep}_{G}(A): & \mathsf{ComAlg}_{k}\to\mathsf{Set}\\ & B\mapsto\mathrm{Hom}_{\mathsf{Gr}}(A,G(B)) \end{array} \end{equation*}

can be viewed as the family of representations of $A$ in $G(A)$ parametrized by points of $k$-scheme $\mathrm{Spec}(A)$. The Proposition says that this functor is representable by some coordinate ring $\mathcal{O}[\mathrm{Rep}_G A]$ of the corresponding affine scheme. Varying $A$, we get the functor

\begin{equation*} (-)_G:\mathsf{Gr}\to \mathsf{ComAlg}_k \end{equation*}

Two Views On Representation Homology

In this section, we will follow [2]. Proceed the discussion of the previous section by considering the degreewise prolongation of the functor $(-)_G$ between simplicial groups and simplicial commutative algebras:

\begin{equation*} (-)_G: s\mathsf{Gr}\to s\mathsf{ComAlg}_k. \end{equation*}

It turns out that this new functor does have the left derived version:

\begin{equation*} \mathbb{L}(-)_G: \mathsf{Ho}(s\mathsf{Gr})\to \mathsf{Ho}(s\mathsf{ComAlg}_k). \end{equation*}

The latter functor is representable by some derived scheme, i.e., the simplicial algebra $\mathbb{L}(A)_G$ considered as an object of the opposite category $\mathsf{Ho}(s\mathsf{ComAlg})^{\mathrm{op}}$.

Recall the Quillen equivalence between the category of reduced simplicial sets and the category of simplicial groups (both of them endowed with standard model structures):

\begin{equation*} \mathbb{G}:s\mathsf{Set}_0\rightleftarrows s\mathsf{Gr}:\overline{W}, \end{equation*}

where $\mathbb{G}$ stands for the Kan loop group functor and $\overline{W}$ for the simplicial delooping functor. We are ready to give the following

Definition 3

For $G$ being an affine algebraic group, the representation homology of $X\in s\mathsf{Set}_0$ is defined to be

\begin{equation*} \mathrm{HR}_\ast(X, G) := \pi_\ast\mathbb{L}(\mathbb{G}X)_G. \end{equation*}

Another definition can be done in the spirit of the discussion on the Hochschild homology of functors above. For a commutative algebra $\mathcal{H}$, consider the corresponding Loday functor

\begin{equation*} \underline{\mathcal{H}}:\mathfrak{G}\to \mathsf{Vect}_k. \end{equation*}

Now, suppose we are given $X\in s\mathsf{Set}_0$. Applying the left Kan extension to the functor $\underline{\mathcal{H}}$ along the inclusion $\mathfrak{G}\hookrightarrow\mathsf{Free}$ to the category of all free groups and precomposing it with $\mathbb{G}X$, we get the following simplicial object:

\begin{equation*} \Delta^{\mathrm{op}}\stackrel{\mathbb{G}X}{\to}\mathrm{Free}\stackrel{\underline{\mathcal{\widetilde{H}}}}{\to}\mathsf{ComAlg}_{k} \end{equation*}

Definition 4

Given $X\in s\mathsf{Set}_0$ and a commutative Hopf algebra $\mathcal{H}$. Representation homology is defined to be

\begin{equation*} \mathrm{HR}_\ast(X,\mathcal{H}) := \pi_\ast\underline{\mathcal{H}}(\mathbb{G}X) = H_\ast(N(\underline{\mathcal{H}}(\mathbb{G}X))). \end{equation*}

Applications

In [2], you can find some computations of representation homology for well-known low-dimensional manifolds, including Riemann surfaces, link complements in $\mathbb{S}^3$, lense spaces. The representation homology is also an invariant of rational homotopy equivalences. There is a spectral sequence relating the Pontryagin algebra and representation homology of a pointed simplicial set. Furthermore, representation homology generalizes topological Hochschild homology.

References

  1. Felder, Giovanni, ‌Derived representation schemes and supersymmetric gauge theory. Institute of Mathematics, Polish Academy of Sciences, 123, 9-9 - 35, 2021
  2. Yuri Berest, Ajay C Ramadoss, Wai-Kit Yeung, Representation Homology of Topological Spaces, International Mathematics Research Notices, Volume 2022, Issue 6, March 2022, Pages 4093–4180
  3. Birgit Richter, From Categories to Homotopy Theory, Cambridge University Press, 2020

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