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

Abstract

Let $k$ be an algebraically closed field of characteristic $p\ge 5$, 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 $3\le m\le p-2$, set ${\mathfrak{g}}_m:={\mathrm{Hom}}_{{\mathrm{Ver}}_p^{+}}(L_m,\mathfrak{g})$ and $E_{\mathfrak{g}}:={⨁}_{3\le m\le p-2,m\mathrm{odd}}{\mathfrak{g}}_m^{(1)}[m]$, where $(1)$ denotes Frobenius twist. Our main result is an isomorphism of graded algebras $H_{\mathrm{CE}}^{\bullet }(\mathfrak{g})\cong {\bigwedge }^{\bullet }{E}_{\mathfrak{g}}^{\ast }$. We also identify this algebra with the de Rham cohomology $H_{\mathrm{dR}}^{\bullet }(G)$ 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 $H_{\mathrm{CE}}^{\bullet }(\mathfrak{g},V)=0$. Hence for every finite-dimensional $\mathfrak{g}$-module $V$ one has $H_{\mathrm{CE}}^{\bullet }(\mathfrak{g},V)\cong {\bigwedge }^{\bullet }{E}_{\mathfrak{g}}^{\ast }\otimes V^{\mathfrak{g}}$. This recovers the theorem of Borel and Chevalley on the cohomology of complex semisimple Lie algebras and its analogue in sufficiently large positive characteristic. We also prove similar results for relative cohomology.

18 pages, latex; v2 contains a new section 6 on relative cohomology