The invariant polynomials on simple Lie superalgebras

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.