2011/02/17 by Christopher P. Bendel, Daniel K. Nakano, Bendel, Christopher P. +6 · 2 citations
Mathematics · #20G10 #20G42 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:20G10 #msc:20G42
paper · pdf · doi:10.48550/arxiv.1102.3639
arxiv created 2011/02/17 · openalex publication_date 2011/02/17 · arxiv updated 2011/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let ζ be a complex ℓth root of unity for an odd integer ℓ>1. For any complex simple Lie algebra \mathfrak g, let uζ=uζ(\mathfrak g) be the associated "small" quantum enveloping algebra. In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when l (resp., p) is smaller than the Coxeter number h of the underlying root system. For example, Lusztig's conjecture concerning the characters of the rational irreducible G-modules stipulates that p ≥ h. The main result in this paper provides a surprisingly uniform answer for the cohomology algebra \opH^\bullet(uζ,\mathbb C) of the small quantum group. When ℓ>h, this cohomology algebra has been calculated by Ginzburg and Kumar \citeGK. Our result requires powerful tools from complex geometry and a detailed knowledge of the geometry of the nullcone of \mathfrak g. In this way, the methods point out difficulties present in obtaining similar results for the restricted enveloping algebra u in small characteristics, though they do provide some clarification of known results there also. Finally, we establish that if M is a finite dimensional uζ-module, then \opH^\bullet(uζ,M) is a finitely generated \opH^\bullet(uζ,\mathbb C)-module, and we obtain new results on the theory of support varieties for uζ.