Constructive noncommmutative invariant theory (Q2040589)

Let \(G\) be a subgroup of the automorphism group of a finite-dimensional Lie algebra \(L\) over a field \(K\) of characteristic zero. Then there is a natural induced action of \(G\) on the universal enveloping algebra \(U(L)\) via associative \(K\)-algebra automorphisms. The authors describe a process by which one can construct generators of the subalgebra \(U(L)^G\) of \(G\)-invariants in \(U(L)\) starting from a homogeneous generating system of \(S(L)^G\), the commutative algebra of invariants in the symmetric tensor algebra \(S(L)\) of \(L\). This process is applied to prove a constructive Hilbert-Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.