Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras

Abstract

In commutative algebra, a Weitzenböck derivation is a nonzero triangular linear derivation of the polynomial algebra K[x 1 , . . . , x m ] in several variables over a field K of characteristic 0. The classical theorem of Weitzenböck states that the algebra of constants is finitely generated. (This algebra coincides with the algebra of invariants of a single unipotent transformation.) In this paper we study the problem of finite generation of the algebras of constants of triangular linear derivations of finitely generated (not necessarily commutative or associative) algebras over K assuming that the algebras are free in some sense (in most of the cases relatively free algebras in varieties of associative or Lie algebras). In this case the algebra of constants also coincides with the algebra of invariants of some unipotent transformation.The main results are the following: (1) We show that the subalgebra of constants of a factor algebra can be lifted to the subalgebra of constants.(2) For all varieties of associative algebras which are not nilpotent in Lie sense the subalgebras of constants of the relatively free algebras of rank 2 are not finitely generated. (3) We describe the generators of the subalgebra of constants for all factor algebras K x, y /I modulo a GL 2 (K)-invariant ideal I . (4) Applying known results from commutative algebra, we construct classes of automorphisms of the algebra generated by two generic 2 × 2 matrices. We obtain also some partial results on relatively free Lie algebras.

Rationality of Hilbert series in noncommutative invariant theory

“…In this case G is the unipotent radical of a Borel subgroup of GL(V ), and in fact G is isomorphic to the additive group of K. By Proposition 5.3 in [14] we have that…”

Weitzenböck derivations of free metabelian associative algebras

“…Hence H GL 2 (t 1 , t 2 , z) − H GL 2 ((F ′ 3 ) δ , t 1 , t 2 , z) = (t 2 1 − t 2 2 )t 4 1 t 2 2 z 4 + · · · which suggests that there is a relation of bidegree (6, 2) and a generator of bidegree (4,4). Continuing in the same way, we have found one more generator 5 | m j , n j , p j , q j , r j ≥ 0, j = 1, 2, 3, 4} ∪{c 5 f m 1 f p 3 f q 4 f r 5 | m, p, q, r ≥ 0}.…”

The Nowicki conjecture for relatively free algebras

“…This implies also that these algebras are infinitely generated as an algebra. Also in [9] Drensky and Gupta studied Weitzenböck derivations δ acting on F m (V) proving that if the algebra U T 2 (K) of 2 × 2 upper left triangular matrices over a field K of characteristic zero belongs to a variety V, then F δ m (V) is not finitely generated whereas if U T 2 (K) does not belong to V, then F δ m (V) is finitely generated by a result of Drensky in [7].…”

“…As noticed above, both G and V have exponent two although G does not belong to the variety generated by U T 2 (K). At the light of the results [7] and [9] the algebra of constants in G is finitely generated instead of the metabelian one.…”