Chevalley-Eilenberg cohomology of linearly reductive Lie algebras in the Verlinde category

Abstract

Let $k$ be an algebraically closed field of characteristic $p\geq 5$, and let $\mathrm{Ver}_p^+$ be the even part of the Verlinde fusion category $\mathrm{Ver}_p$, the semisimplification of $\mathrm{Rep}_k(\mathbb Z/p)$. Let $\mathfrak g$ be a linearly reductive Lie algebra in $\mathrm{Ver}_p^+$, i.e., one whose finite-dimensional representations are semisimple. A basic class of examples is obtained by semisimplifying a simple Lie algebra over $k$ equipped with the action of $\mathbb Z/p$ by a principal unipotent element, when $p$ exceeds its Coxeter number. We prove that $\mathfrak g$ is invariantless, i.e., that the unit object is not a summand of $\mathfrak g$. For odd $m$ with $3\leq m\leq p-2$, set $\mathfrak{g}_m:=\operatorname{Hom}_{\mathrm{Ver}_p^+}(L_m,\mathfrak g)$ and $E_{\mathfrak g}:=\bigoplus_{3\leq m\leq p-2,\ m\ {\rm odd}}\mathfrak g_m^{(1)}[m]$, where $(1)$ denotes Frobenius twist. Our main result is an isomorphism of graded algebras $H^\bullet_{\mathrm{CE}}(\mathfrak g)\cong\bigwedge^\bullet E_{\mathfrak g}^*$. We also identify this algebra with the de Rham cohomology $H^\bullet_{\mathrm{dR}}(G)$ of the group scheme $G=\exp(\mathfrak g)$ and show that the induced graded Hopf algebra structure agrees with the standard one on the exterior algebra. Moreover, if $V$ is a simple $\mathfrak g$-module on which $\mathfrak g$ acts nontrivially, then $H^\bullet_{\mathrm{CE}}(\mathfrak g,V)=0$. Hence for every finite-dimensional $\mathfrak g$-module $V$ one has $H^\bullet_{\mathrm{CE}}(\mathfrak g,V)\cong\bigwedge^\bullet E_{\mathfrak g}^*\otimes V^{\mathfrak g}$. This recovers the theorem of Borel and Chevalley on the cohomology of complex semisimple Lie algebras and its analogue in sufficiently large positive characteristic.