Chevalley-Eilenberg cohomology of linearly reductive Lie algebras in the Verlinde category
Abstract
Let k be an algebraically closed field of characteristic p≥ 5, and let Verp+ be the even part of the Verlinde fusion category Verp, the semisimplification of Repk(ℤ/p). Let g be a linearly reductive Lie algebra in Verp+, 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 ℤ/p by a principal unipotent element, when p exceeds its Coxeter number. We prove that g is invariantless, i.e., that the unit object is not a summand of g. For odd m with 3≤ m≤ p−2, set gm:=HomVerp+(Lm,g) and Eg:=⨁3≤ m≤ p−2, m oddgm(1)[m], where (1) denotes Frobenius twist. Our main result is an isomorphism of graded algebras H•CE(g)≅bigwedge• Eg*. We also identify this algebra with the de Rham cohomology H•dR(G) of the group scheme G=exp(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 g-module on which g acts nontrivially, then H•CE(g,V)=0. Hence for every finite-dimensional g-module V one has H•CE(g,V)≅bigwedge• Eg*⊗ Vg. 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.
Related
- Steinberg module and top-dimensional cohomology for arithmetic Chevalley groups
- 2-generation of Chevalley Lie algebras over finite fields
- Cohomology of Linearly Constrained Kloosterman Families: Newton Polytopes, Weights, and Boundary Monodromy
- Automorphic C*-algebras of reductive groups
- BV structure on the Hochschild cohomology of Liu-Schulz algebras in characteristic two
BibTeX
@misc{etingof2026chevalleyeilenberg,
title = {{Chevalley-Eilenberg cohomology of linearly reductive Lie algebras in the Verlinde category}},
author = {Pavel Etingof and Serina Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.01576},
eprint = {2608.01576},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.01576},
}
RIS
TY - UNPB
TI - Chevalley-Eilenberg cohomology of linearly reductive Lie algebras in the Verlinde category
AU - Pavel Etingof
AU - Serina Hu
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2608.01576
UR - https://arxiv.org/abs/2608.01576
AB - Let k be an algebraically closed field of characteristic p≥ 5, and let Ver_p^+ be the even part of the Verlinde fusion category Ver_p, the semisimplification of Rep_k(ℤ/p). Let g be a linearly reductive Lie algebra in 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 ℤ/p by a principal unipotent element, when p exceeds its Coxeter number. We prove that g is invariantless, i.e., that the unit object is not a summand of g. For odd m with 3≤ m≤ p−2, set g_m:=Hom_Ver_p^+(L_m,g) and E_g:=⨁_3≤ m≤ p−2, m oddg_m^(1)[m], where (1) denotes Frobenius twist. Our main result is an isomorphism of graded algebras H^•_CE(g)≅bigwedge^• E_g^*. We also identify this algebra with the de Rham cohomology H^•_dR(G) of the group scheme G=exp(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 g-module on which g acts nontrivially, then H^•_CE(g,V)=0. Hence for every finite-dimensional g-module V one has H^•_CE(g,V)≅bigwedge^• E_g^*⊗ V^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.
ER -