Invariants of Unipotent Transformations Acting on Noetherian Relatively Free Algebras

The classical theorem of Weitzenböck states that the algebra of invariants K[X] of a single unipotent transformation g ∈ GLm(K) acting on the polynomial algebra K[X] = K[x1, . . . , xm] over a field K of characteristic 0 is finitely generated. This algebra coincides with the algebra of constants K[X] of a linear locally nilpotent derivation δ of K[X]. Recently the author and C. K. Gupta have started the study of the algebra of invariants Fm(V) where Fm(V) is the relatively free algebra of rank m in a variety V of associative algebras. They have shown that Fm(V) is not finitely generated if V contains the algebra UT2(K) of 2× 2 upper triangular matrices. The main result of the present paper is that the algebra Fm(V) is finitely generated if and only if the variety V does not contain the algebra UT2(K). As a by-product of the proof we have established also the finite generation of the algebra of invariants T g nm where Tnm is the mixed trace algebra generated by m generic n× n matrices and the traces of their products. Introduction Let K be any field of characteristic 0 and let X = {x1, . . . , xm}, where m > 1. Let g ∈ GLm = GLm(K) be a unipotent linear operator acting on the vector space KX = Kx1⊕· · ·⊕Kxm. By the classical theorem of Weitzenböck [16], the algebra of invariants K[X ] = {f ∈ K[X ] | f(g(x1), . . . , g(xm)) = f(x1, . . . , xm)} is finitely generated. A proof in modern language was given by Seshadri [12]. An elementary proof based on the ideas of [12] was presented by Tyc [14]. Since g − 1 is a nilpotent linear operator of KX , we may consider the linear locally nilpotent derivation δ = log g = ∑ i≥1 (−1) (g − 1) i called a Weitzenböck derivation. (The K-linear operator δ acting on an algebra A is called a derivation if δ(uv) = δ(u)v + uδ(v) for all u, v ∈ A.) The algebra of invariants C[X ] coincides with the algebra of constants C[X ] (= ker(δ) ). See the book by Nowicki [10] for a background on the properties of the algebras of constants of Weitzenböck derivations. Looking for noncommutative generalizations of invariant theory, see e. g. the survey by Formanek [8], let K〈X〉 = K〈x1, . . . , xm〉 be the free unitary associative 1991 Mathematics Subject Classification. 16R10; 16R30.