Classification of crossed $G$-algebras

Question

Added later: As Viktor Ostrik points out in a comment, what I'm looking for is a classification of so-called crossed $G$-algebras corresponding to homotopy TQFTs with homotopy target space $K(G, 1)$ and functor target category $\operatorname{Vec}(\mathbb{C})$, which are defined on the n-cat lab, as well as in Turaev's preprint from 1999 on page 14. As far as I can tell, those references do not provide a full classification though. Below the original question. I also added two axioms I originally had missed (thanks to Kevin Walker for pointing that out), and changed the title.

Consider a finite-dimensional (complex or real) algebra $A$ graded by a (finite) group $G$, i.e., a finite-dimensional vector space $V_g$ for every $g\in G$, and a linear map $$ A_{g,h}: V_g\otimes V_h\rightarrow V_{gh}\;,$$ such that the overall algebra on $\bigoplus_g V_g$ is a symmetric Frobenius algebra. Furthermore, let there be a representation $$\rho^h_g: V_h\rightarrow V_{g^{-1}hg}$$ of $G$ acting on $\bigoplus_h V_h$, such that $$A_{h,i}\circ (\rho^{ghg^{-1}}_g\otimes \rho^{gig^{-1}}_g) = \rho^{hi}_g \circ A_{h,i}\;.$$ Also, we want $A$ to be graded-commutative, i.e., $$A_{g,h}\circ \tau = A_{h,h^{-1}gh}\circ (1\otimes \rho_h^g)\;,$$ where $\tau$ is the permutation map $$\tau: V_h\otimes V_g\rightarrow V_g\otimes V_h\;.$$ Furthermore, we impose $$\rho_g^g=1\;,$$ as well as the torus condition $$\epsilon_g\circ(A_{ghg^{-1}h^{-1},hgh^{-1}}\otimes 1)\circ(1\otimes \rho^g_{h^{-1}}\otimes 1)\circ(1\otimes \eta_g) =\\ \epsilon_h\circ(\rho^{ghg^{-1}}_g\otimes 1)\circ(A_{ghg^{-1}h^{-1}, h}\otimes 1)\circ(1\otimes \eta_h)\;, $$ where $\epsilon$ is the Frobenius form and $\eta$ is its dual.

Also, for the instances I'm interested in, we can assume $$A_{g,h} = \hat A_{g,h}^\dagger\;,$$ where $\hat A$ is the co-product of the Frobenius algebra, $$ \hat A_{g,h}: V_{gh}\rightarrow V_g\otimes V_h\;.$$

What is the classification of those structures up to isomorphism? Have these structures or anything related been studied in the literature? If yes, can you give me the reference? Is there any established name for these structures?

The reason I'm interested in those structures is that they should correspond to 2-dimensional TQFT with group symmetries. Therefore, I expect the sub-classification where $\operatorname{dim}(A_h)=1$ for all $h$ to be via the second group cohomology $H^2(BG,U(1))$.

Answer 1

The classification of 2-dimensional TQFTs with G-action that I'm familiar with goes as follows. Such TQFTs are equivalent to (1) a module category for the tensor category $Vec_G$; or, equivalently, (2) a 2-functor from $Vec_G$ to [1-categories, functors, natural transformations] (or to [1-categories, bimodules, bimodule maps]).

According to a theorem of Ostrik from around 2000, the above are classified (up to isomorphism) by pairs $(H, \omega)$, where $H$ is a subgroup of $G$ and $\omega$ is a 2-cocycle on $H$. (Two such pairs $(H, \omega)$ and $(H', \omega')$ are considered equivalent if $H'$ is conjugate to $H$, and this conjugation takes $\omega'$ to a cocycle cohomologous to $\omega$.)

The simple objects of the 1-category being acted upon correspond to cosets $G/H$. In particular, actions on the trivial 1-category correspond to 2-cocycles on $G$, in accordance with your expectation.

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Added later: The above is for fully extended TQFTs. Perhaps you were asking about 2-level, not-fully-extended TQFTs?

Answer 2

This is an answer to a question posed in the comments, not an answer to the original question. (Too long to fit comfortably as a comment.)

Given a surface $Y$ (e.g. annulus or pair-of-pants), let $R(Y)$ denote the set of homomorphisms from the fundamental groupoid of $Y$ (with one basepoint in each boundary component) to $G$.

$R(S^1\times I)$ has the structure of a groupoid. (Objects biject with $G$, morphisms from $g$ to $g'$ biject with $h$ such that $g' = h^{-1}gh$.) Your representation $\rho$ captures this.

Let $P$ be the pair of pants. The set $R(P)$ can be parameterized by $G^4$. $R(P)$ affords an action of $C\times C\times C$, where $C$ is the groupoid $R(S^1\times I)$. Your axiom for the compatibility of $\rho$ and $A$ captures only part of this trimodule structure on $R(P)$.

To reconstruct a general $G$-TQFT, I think you need to treat $R(P)$ is its full trimodule generality. One way of saying this in your framework is to posit $$ A_{g,h}^{a,b}:V_g\otimes V_h\to V_{a^{-1}gab^{-1}hb} $$ for all $g,h,a,b\in G$. These $G\times G$-indexed tensor products need to intertwine with $\rho$ in three different places (trimodule). Furthermore, the associativity constraint becomes more complicated to state.

(I see you have edited the $\rho$/$A$ compatibility axiom, but I still think you need more conditions on $\rho$ and $A$.)