2020/07/28 by Phommady, Kenny
#FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2007.14185
Let K be an algebraically closed field with characteristic zero, and \mathfrakg a Lie algebra. Let Y(\mathfrakg) be the subalgebra of the symmetric algebra S(\mathfrakg)=K[\mathfrakg^*] made of the polynomials which are invariant under the adjoint action. Also define Sy(\mathfrakg) as the algebra generated by elements of S(\mathfrakg) for which the adjoint action acts homothetically. When \mathfrakq is a parabolic contraction in type A or C, and in some cases in type B, Panyushev and Yakimova showed that the algebra of invariants Y(\mathfrakq) is an algebra of polynomials. Using Panyushev's and Yakimova's result, we show the polynomiality of Sy(\mathfrakq) by constructing an algebraically free set of generators in type A and in some cases in type C. We also study an example in type C where Sy(\mathfrakq) is not polynomial.