Jordan-Kronecker invariants of Lie algebra representations and degrees of invariant polynomials

TL;DR

This paper introduces Jordan-Kronecker invariants for Lie algebra representations, which serve as lower bounds for invariant polynomial degrees and are exact under specific independence and freeness conditions.

Contribution

It defines Jordan-Kronecker invariants for any Lie algebra representation and establishes their role in determining invariant polynomial degrees and algebraic independence.

Findings

Jordan-Kronecker invariants provide lower bounds for invariant degrees

Exact bounds correspond to algebraic independence of invariants

Conditions for invariants to be freely generated are characterized

Abstract

For an arbitrary representation $\rho$ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan-Kronecker invariants of $\rho$. Among other interesting properties, these numbers provide lower bounds for degrees of polynomial invariants of $\rho$. Furthermore, we prove that these lower bounds are exact if and only if the invariants are independent outside of a set of large codimension. Finally, we show that under certain additional assumptions our bounds are exact if and only if the algebra of invariants is freely generated.