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

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 GL2(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.