Representation Theory

ISSN 1088-4165

 

 

The invariant polynomials
on simple Lie superalgebras


Author: Alexander Sergeev
Journal: Represent. Theory 3 (1999), 250-280
MSC (1991): Primary 17A70; Secondary 17B35, 13A50
Published electronically: August 31, 1999
MathSciNet review: 1714627
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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 $\tr \rho(x)^k$, 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}}^*)=\mathop{\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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Additional Information

Alexander Sergeev
Affiliation: On leave of absence from the Balakovo Institute of Technique of Technology and Control, Branch of Saratov State Technical University, Russia; Correspondence: c/o D. Leites, Department of Mathematics, University of Stockholm, Roslagsv. 101, Kräftriket hus 6, S-106 91, Stockholm, Sweden
Email: mleites@matematik.su.se

DOI: http://dx.doi.org/10.1090/S1088-4165-99-00077-1
Keywords: Lie superalgebra, invariant theory.
Received by editor(s): April 22, 1999
Received by editor(s) in revised form: June 28, 1999
Published electronically: August 31, 1999
Additional Notes: I am thankful to D. Leites for help and support.
Article copyright: © Copyright 1999 American Mathematical Society