Basic quasi-reductive root data and supergroups

2023/03/31 by R. Fioresi, Fioresi, Rita, Bin Shu +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2303.18065

Abstract

We investigate pairs (G,Y), where G is a reductive algebraic group and Y a purely-odd G-superscheme, asking when a pair corresponds to a quasi-reductive algebraic supergroup \mathbbG, that is, \mathbbG_ev is isomorphic to G, and the quotient \mathbbG/\mathbbG_ev is G-equivariantly isomorphic to Y. We prove that, if Y satisfies certain conditions (basic quasi-reductive root data), then the question has a positive answer given by an existence and uniqueness theorem. The corresponding supergroups are said to be basic quasi-reductive, which can be classified, up to isogeny. We then decide the structure of connected quasi-reductive algebraic supergroups provided that: (i) the root system does not contain 0; (ii) \mathfrakg:=Lie(\mathbbG) admits a non-degenerate even symmetric bilinear form. (iii) all odd reflections are invertible. Remarkably, those supergroups are exactly basic quasi-reductive supergroups of monodromy type.

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