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Journal of Lie Theory 32 (2022), No. 4, 973--996
Copyright Heldermann Verlag 2022

Iwasawa Decomposition for Lie Superalgebras

Alexander Sherman
Dept. of Mathematics, University of California, Berkeley, U.S.A.


Let $\mathfrak{g}$ be a basic simple Lie superalgebra over an algebraically closed field of characteristic zero, and $\theta$ an involution of $\mathfrak{g}$ preserving a nondegenerate invariant form. We prove that at least one of $\theta$ or $\delta\circ\theta$ admits an Iwasawa decomposition, where $\delta$ is the canonical grading automorphism $\delta(x)=(-1)^{\overline{x}}x$. The proof uses the notion of generalized root systems as developed by Serganova, and follows from a more general result on centralizers of certain tori coming from semisimple automorphisms of the Lie superalgebra $\mathfrak{g}$.

Keywords: Lie superalgebras, symmetric pairs, root systems.

MSC: 17B22, 17B20, 17B40.

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