JORDAN–KRONECKER INVARIANTS OF LIE ALGEBRA REPRESENTATIONS AND DEGREES OF INVARIANT POLYNOMIALS

For an arbitrary representation ρ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan–Kronecker invariants of ρ . Among other interesting properties, these numbers provide lower bounds for degrees of polynomial invariants of ρ . 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.