Solvable, reductive and ⋊reductive supergroups

A. N. Grishkov, A. N. Zubkov

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


It is well known that if the ground field K has characteristic zero and G is a connected algebraic group, defined over K, then the Lie algebra Lie(G') of the commutant G' of G coincides with the commutant Lie(G)' of Lie(G). We show that this result is no longer true in the category of algebraic supergroups. We also construct a reductive supergroup H=X⋊G, where X and G are connected, reduced and abelian supergroups, such that Xu≠1 and (Hev)u is non-trivial connected (super)group. ⋊-reductive supergroups have been introduced in [10]. We prove that a supergroup H is ⋊-reductive if and only if the largest even (super)subgroup of the solvable radical R(H) is a torus, H~=H/R(H) contains a normal supersubgroup U, which is ⋊-isomorphic to a direct product of normal supersubgroups Ui, and H~/U is a triangulizable supergroup with odd unipotent radical. Moreover, for every i, Lie(Ui)=Ui⊗Sym(ni) are such that either ni=0 and Ui is a classical simple Lie superalgebra, or ni=1 and Ui is a simple Lie algebra.

Original languageEnglish
Pages (from-to)448-473
Number of pages26
JournalJournal of Algebra
Publication statusPublished - Apr 15 2016
Externally publishedYes


  • Algebraic supergroup
  • Lie superalgebra
  • Solvable radical
  • Solvable supergroup
  • Unipotent radical
  • ⋊-reductive supergroup

ASJC Scopus subject areas

  • Algebra and Number Theory


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