## The invariant polynomials on simple Lie superalgebras

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- by Alexander Sergeev PDF
- 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.## References

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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
- 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.
- © Copyright 1999 American Mathematical Society
- Journal: Represent. Theory
**3**(1999), 250-280 - MSC (1991): Primary 17A70; Secondary 17B35, 13A50
- DOI: https://doi.org/10.1090/S1088-4165-99-00077-1
- MathSciNet review: 1714627