Journal of Lie Theory

no. 4
Iwasawa Decomposition for Lie Superalgebras
Journal of Lie Theory, Volume 32 (2022) no. 4, pp. 973-996
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 $\mathfrak{g}$.
DOI: 10.5802/jolt.1261
Classification: 17B22, 17B20, 17B40
Keywords: Lie superalgebras, symmetric pairs, root systems 
@article{JOLT_2022_32_4_a3,
     author = {A. Sherman},
     title = {Iwasawa {Decomposition} for {Lie} {Superalgebras}},
     journal = {Journal of Lie Theory},
     pages = {973--996},
     year = {2022},
     volume = {32},
     number = {4},
     doi = {10.5802/jolt.1261},
     zbl = {1521.17020},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1261/}
}
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A. Sherman. Iwasawa Decomposition for Lie Superalgebras. Journal of Lie Theory, Volume 32 (2022) no. 4, pp. 973-996. doi: 10.5802/jolt.1261

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