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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.04652 |
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Table of Contents:
- In this paper we consider those involutions $θ$ of a finite-dimensional Kac-Moody Lie superalgebra $\mathfrak g$, with associated decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$, for which a Cartan subspace $\mathfrak a$ in $\mathfrak p_{\bar 0}$ is self-centralizing in $\mathfrak p$. For such $θ$ the restriction map $C_θ$ from $\mathfrak p$ to $\mathfrak a$ is injective on the algebra $P(\mathfrak p)^{\mathfrak k}$ of $\mathfrak k$-invariant polynomials on $\mathfrak p$. There are five infinite families and five exceptional cases of such involutions, and for each case we explicitly determine the structure of $P(\mathfrak p)^{\mathfrak k}$ by giving a complete set of generators for the image of $C_θ$. We also determine precisely when the restriction map $R_θ$ from $P(\mathfrak g)^{\mathfrak g}$ to $P(\mathfrak p)^{\mathfrak k}$ is surjective. Finally we introduce the notion of a generalized restricted root system, and show that in the present setting the $\mathfrak a$-roots $Δ(\mathfrak a,\mathfrak g)$ always form such a system.