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Projective representations are homotopy fixed points

Yesterday we described how a (finite-dimensional) projective representation $\rho : G \to PGL_n(k)$ of a group $G$ functorially gives rise to a $k$-linear action of $G$ on $\text{Mod}(M_n(k)) \cong \text{Mod}(k)$ such that the Schur class $s(\rho) \in H^2(BG, k^{\times})$ classifies this action.

Today we’ll go in the other direction. Given an action of $G$ on $\text{Mod}(k)$ explicitly described by a 2-cocycle $\eta \in Z^2(BG, k^{\times})$, we’ll recover the category of $\eta$-projective representations, or equivalently the category of modules over the twisted group algebra $k \rtimes_{\eta} G$, by taking the homotopy fixed points of this action. We’ll end with another puzzle.

Spelling it out

It’s straightforward to verify this. Given a 2-cocycle $\eta \in Z^2(BG, k^{\times})$, which we can normalize to be unital, the corresponding action of $G$ on $\text{Mod}(k)$ is given by taking each $F(g) : \text{Mod}(k) \to \text{Mod}(k)$ to be the identity and taking multiplication by $\eta(g, h) \in k^{\times}$ to be the natural isomorphism $F(g) F(h) \cong F(gh)$. Then a homotopy fixed point for this action is a $k$-vector space $V$ together with isomorphisms

$\displaystyle \alpha(g) : V \to F(g) V = V$

such that the composites

$\displaystyle V \xrightarrow{\alpha(g)} V \xrightarrow{\alpha(h)} V \xrightarrow{\eta(g, h)} V$

and

$\displaystyle V \xrightarrow{\alpha(gh)} V$

agree (and $\alpha(e) = 1$), and up to matching our conventions (we again need to interpret compositions in diagrammatic order) this is precisely the condition that

$\displaystyle \alpha(gh) = \eta(g, h) \alpha(g) \alpha(h)$

so we get that homotopy fixed point data for $V$ is precisely the data of an $\eta$-projective representation $\alpha : G \to GL(V)$. Moreover, the definition we gave of a morphism between $\eta$-projective representations generalizes to a definition of morphism between homotopy fixed points, so we really get an identification between the category of homotopy fixed points of the $G$-action on $\text{Mod}(k)$ and the category of $\eta$-projective representations.

The claim that if $\eta, \eta'$ are cohomologous then the categories of $\eta$-projective and $\eta'$-projective representations are equivalent is then subsumed in the claim that taking homotopy fixed points is functorial in the appropriate sense (there is really a 2-category of actions of $G$ on categories which we haven’t described, and taking homotopy fixed points is a 2-functor from this 2-category to $\text{Cat}$).

The observation that the Schur class is multiplicative with respect to tensor products of projective representations also has an interpretation here. It reflects the fact that $\text{Mod}(k)$ is an object of a symmetric monoidal 2-category, namely the Morita 2-category $\text{Mor}(k)$, and that taking homotopy fixed points of group actions is lax monoidal: it gives rise to functors

$\displaystyle \text{Mod}(A)^G \otimes_k \text{Mod}(B)^G \to \text{Mod}(A \otimes_k B)^G$

where $(-)^G$ denotes taking homotopy fixed points. This is a natural categorification of the corresponding fact for ordinary linear representations of finite groups, where taking fixed points in the usual sense is also lax monoidal in the sense that we get natural maps $V^G \otimes W^G \to (V \otimes W)^G$, provided you believe that the categories $\text{Mod}(A)$ are natural categorifications of vector spaces (sometimes called 2-vector spaces).

What’s the topological picture?

As usual, by passing through the homotopy hypothesis, we can get a topological picture of what’s going on here. The 2-groupoids $BGL_n(k), BPGL_n(k), B^2 k^{\times}$ fit into a fiber sequence

$\displaystyle BGL_n(k) \to BPGL_n(k) \xrightarrow{s} B^2 k^{\times}$

(where $s$ denotes the universal Schur class) which expresses a generalization of the interpretation of the Schur class in terms of an obstruction to lifting: this fiber sequence says that $BGL_n(k)$ is the homotopy fiber of the Schur class, which means that if

$\displaystyle f : X \to BPGL_n(k)$

is any map of spaces (where, again, by “spaces” I mean weak homotopy types, or equivalently $\infty$-groupoids, or equivalently spaces with the homotopy type of a CW complex), which can be interpreted as classifying a “projective bundle” on $X$, then the space of lifts of this projective bundle to a vector bundle $f' : X \to BGL_n(k)$ is the space of nullhomotopies of the composite map

$\displaystyle X \xrightarrow{f} BPGL_n(k) \xrightarrow{s} B^2 k^{\times}$.

In particular, any projective bundle on $X$ has a Schur class in $H^2(X, k^{\times})$ which vanishes iff it lifts to a vector bundle.

How can we think about the Schur class in this generality? A map $X \to BPGL_n(k)$ can also be thought of as classifying a bundle of matrix algebras $M_n(k)$ over $X$ (while a lift of this map to a map $X \to BGL_n(k)$ exhibits a bundle of Morita equivalences between these matrix algebras and $k$, or equivalently a $k$-vector bundle $V$ and an isomorphism between this bundle of matrix algebras and $\text{End}(V)$). We might call such a thing an Azumaya algebra over $X$.

When we fiberwise take module categories over these matrix algebras, we get a “2-line bundle,” namely a bundle of categories equivalent to $\text{Mod}(k)$, thought of as a “free module of rank $1$” over $\text{Mod}(k)$. Taking global sections of this 2-line bundle in an appropriate sense is a generalization of taking homotopy fixed points, which is the special case where $X = BG$. This interpretation exhibits $H^2(X, k^{\times})$ as a topological analogue of the Brauer group.

We can also use this topological picture to describe in exactly what sense an $\eta$-projective representation has more structure than a projective representation with Schur class $[\eta]$. Thinking of $\eta$ as a map $X \to B^2 k^{\times}$ (which can be represented by a 2-cocycle on $G$ for $X = BG$), the space of $n$-dimensional “$\eta$-projective bundles” on $X$ (which reduces to $\eta$-projective representations of $G$ for $X = BG$) can be identified with the homotopy pullback of the diagram

$\displaystyle X \to B^2 k^{\times} \xleftarrow{s} BPGL_n(k)$.

In other words, an $\eta$-projective bundle is a map $f : X \to BPGL_n(k)$ together with a choice of homotopy between the Schur class $f \circ s : X \to B^2 k^{\times}$ of this map and another fixed map $X \to B^2 k^{\times}$. It’s this structure of a choice of homotopy, rather than the property that such a homotopy exists, that is the extra structure on an $\eta$-projective representation. This extra structure is hard to see if we only think about $H^2(X, k^{\times})$ as a group rather than, according to taste, either the set of connected components of a space (namely the space of maps $X \to B^2 k^{\times}$) or the set of isomorphism classes of objects in a 2-groupoid (namely the 2-groupoid of 2-line bundles on $X$).

The puzzle

Nothing we’ve said above has explicitly mentioned the twisted group algebra $k \rtimes_{\eta} G$.

Puzzle: How can we fit the twisted group algebra into this story?

I like this puzzle because I know of two good answers to it.