Generators and defining relations for ring of invariants of commuting locally nilpotent derivations or automorphisms

Abstract

Let A be an algebra over a field K of characteristic zero and let δ1, …, δs∈Der K(A) be commuting locally nilpotent K-derivations such that δi(xj) equals δij, the Kronecker delta, for some elements x1, …, xs∈A. A set of generators for the algebra A δ : = ⋂ i = 1 s ker ( δ i ) is found explicitly and a set of defining relations for the algebra Aδ is described. Similarly, let σ1, …, σs ∈ AutK(A) be commuting K-automorphisms of the algebra A is given such that the maps σi − idA are locally nilpotent and σi (xj) = xj + δij, for some elements x1, …, xs ∈ A. A set of generators for the algebra Aσ: = {a ∈ A | σ1(a) = … = σs(a) = a} is found explicitly and a set of defining relations for the algebra Aσ is described. In general, even for a finitely generated non-commutative algebra A the algebras of invariants Aδ and Aσ are not finitely generated, not (left or right) Noetherian and a minimal number of defining relations is infinite. However, for a finitely generated commutative algebra A the opposite is always true. The derivations (or automorphisms) just described appear often in many different situations (possibly) after localization of the algebra A.