[Paper Review] Semisimple algebraic tensor categories

Rainer Weissauer|ArXiv.org|Sep 9, 2009

Algebraic structures and combinatorial models7 references17 citations

TL;DR

This paper classifies semisimple algebraic tensor categories over an algebraically closed field of characteristic zero by proving that any such category arises as the category of finite-dimensional super-representations of a reductive supergroup scheme. The key result establishes a structural decomposition of reductive supergroups as semidirect products of orthosymplectic supergroups and reductive algebraic groups, generalizing classical results on reductive groups to the super setting via Harish-Chandra triples and Hopf algebra theory.

ABSTRACT

A semisimple algebraic tensor category over an algebraically closed field k of characteristic zero is the representation category of all finite dimensional twisted super representations of an affine reductive supergroup G over k. Such a supergroup is reductive if and only if its connected component is reductive. The connected component is reductive if and only if the Lie superalgebra divided by its center is a product of simple Lie algebras of classical type and Lie superalgebras spo(1,2r) of the orthosymplectic types BC_r.

Motivation & Objective

• To classify semisimple algebraic tensor categories over an algebraically closed field of characteristic zero.
• To determine when the category of finite-dimensional super-representations of a super-affine group scheme is semisimple.
• To establish a structural decomposition of reductive supergroup schemes in terms of orthosymplectic supergroups and reductive algebraic groups.
• To extend classical reductivity theory for Lie algebras and algebraic groups to the supergroup setting using Hopf algebra and comodule theory.

Proposed method

• Uses the equivalence between the category of super-representations of a super-affine group scheme and the category of comodules over its super-Hopf algebra.
• Applies the theory of Harish-Chandra triples (G, g₋, Q) to parametrize supergroup structures via the reduced group G, its Lie superalgebra g, and a G-equivariant symmetric bilinear map Q.
• Employs the super-radical J of a super-Hopf algebra A to define the reduced group scheme G = Spec(A/J), which is a reductive algebraic group when A is connected and char(k) = 0.
• Introduces filtrations on symmetric and exterior powers of representations to analyze irreducibility and multiplicity bounds in tensor powers.
• Uses the equivalence of categories Rep_k(G, g₋, Q) ≅ Rep_k(𝐺, ε) to reduce semisimplicity questions to Lie superalgebra reductivity.
• Applies results from Djokovic and Hochschild on reductive Lie superalgebras to characterize reductive supergroup schemes.

Experimental results

Research questions

• RQ1 When is the category of finite-dimensional super-representations of a super-affine group scheme semisimple?
• RQ2 What conditions on a supergroup scheme G ensure that its category of super-representations is semisimple?
• RQ3 How can reductive supergroup schemes be classified in terms of their underlying algebraic groups and Lie superalgebras?
• RQ4 What is the structure of a connected reductive supergroup over an algebraically closed field of characteristic zero?
• RQ5 How do automorphisms and component groups of reductive supergroups act on their orthosymplectic components?

Key findings

• Any algebraic tensor category over an algebraically closed field of characteristic zero is equivalent to the category of finite-dimensional super-representations of a super-affine group scheme.
• A super-affine group scheme G is reductive if and only if its reduced group G is reductive and its Lie superalgebra is reductive.
• A connected reductive supergroup G is isomorphic to a product G′ × H, where G′ is a product of orthosymplectic supergroups Spo(1,2r) and H is a reductive algebraic group.
• The category of super-representations of a reductive supergroup is semisimple if and only if the underlying Lie superalgebra is reductive.
• The group of components π₀(G) acts on the orthosymplectic part G′ via permutations of the factors, and this action is encoded by a canonical homomorphism p: H → ∏Σₙᵣ.
• Every reductive supergroup over an algebraically closed field of characteristic zero is isomorphic to a semidirect product G′ ⋊ H, where G′ is a product of simple BC-type supergroups and H acts via permutations on the factors.