# Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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## The invariant polynomials on simple Lie superalgebrasHTML articles powered by AMS MathViewer

by Alexander Sergeev
Represent. Theory 3 (1999), 250-280 Request permission

## Abstract:

Chevalley’s theorem states that for any simple finite dimensional Lie algebra $\mathfrak {g}$: (1) the restriction homomorphism of the algebra of polynomials $S(\mathfrak {g}^*)\longrightarrow S(\mathfrak {h}^*)$ onto the Cartan subalgebra $\mathfrak {h}$ induces an isomorphism $S(\mathfrak {g}^*)^{\mathfrak {g}}\cong S(\mathfrak {h}^*)^{W}$, where $W$ is the Weyl group of $\mathfrak {g}$; (2) each $\mathfrak {g}$-invariant polynomial is a linear combination of the polynomials $\operatorname {tr} \rho (x)^k$, where $\rho$ is a finite dimensional representation of $\mathfrak {g}$. None of these facts is necessarily true for simple Lie superalgebras. We reformulate Chevalley’s theorem as formula $(*)$ below to include Lie superalgebras. Let $\mathfrak {h}$ be the split Cartan subalgebra of $\mathfrak {g}$; let $R=R_+\cup R_-$ be the set of nonzero roots of $\mathfrak {g}$, the union of positive and negative ones. Set $\tilde R_+=\{\alpha \in R_+\mid -\alpha \in R_-\}$. For each root $\alpha \in \tilde R_+$ denote by $\mathfrak {g}(\alpha )$ the Lie superalgebra generated by $\mathfrak {h}$ and the root superspaces $\mathfrak {g}_\alpha$ and $\mathfrak {g}_{-\alpha }$. Let the image of $S(\mathfrak {g}(\alpha )^*)^{\mathfrak {g}(\alpha )}$ under the restriction homomorphism $S(\mathfrak {g}(\alpha )^*)\longrightarrow S(\mathfrak {h}^*)$ be denoted by $I^{\alpha }(\mathfrak {h}^*)$ and the image of $S(\mathfrak {g}^*)^{\mathfrak {g}}$ by $I(\mathfrak {h}^*)$. Then \begin{equation*} I(\mathfrak {h}^*)=\bigcap \limits _{\alpha \in \tilde R_+}I^{\alpha }(\mathfrak {h}^*).\tag *{(*)} \end{equation*} Chevalley’s theorem for anti-invariant polynomials is also presented.
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