Chevalley’s theorem states that for any simple finite dimensional Lie algebra g \mathfrak {g} : (1) the restriction homomorphism of the algebra of polynomials S ( g ∗ ) ⟶ S ( h ∗ ) S(\mathfrak {g}^*)\longrightarrow S(\mathfrak {h}^*) onto the Cartan subalgebra h \mathfrak {h} induces an isomorphism S ( g ∗ ) g ≅ S ( h ∗ ) W S(\mathfrak {g}^*)^{\mathfrak {g}}\cong S(\mathfrak {h}^*)^{W} , where W W is the Weyl group of g \mathfrak {g} ; (2) each g \mathfrak {g} -invariant polynomial is a linear combination of the polynomials tr ⁡ ρ ( x ) k \operatorname {tr} \rho (x)^k , where ρ \rho is a finite dimensional representation of g \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 h \mathfrak {h} be the split Cartan subalgebra of g \mathfrak {g} ; let R = R + ∪ R − R=R_+\cup R_- be the set of nonzero roots of g \mathfrak {g} , the union of positive and negative ones. Set R ~ + = { α ∈ R + ∣ − α ∈ R − } \tilde R_+=\{\alpha \in R_+\mid -\alpha \in R_-\} . For each root α ∈

	Year	Citations

References