Invariants of unipotent transformations acting on noetherian relatively free algebras

The classical theorem of Weitzenboeck states that the algebra of invariants of a single unipotent transformation $g$ in $GL_m(K)$ acting on the polynomial algebra $K[x_1,...,x_m]$ over a field $K$ of characteristic 0 is finitely generated. Recently the author and C.K. Gupta have started the study of the algebra of $g$-invariants of relatively free algebras of rank $m$ in varieties of associative algebras. They have shown that the algebra of invariants is not finitely generated if the variety contains the algebra $UT_2(K)$ of $2\times 2$ upper triangular matrices. The main result of the present paper is that the algebra of invariants is finitely generated if and only if the variety does not contain the algebra $UT_2(K)$. As a by-product of the proof we have established also the finite generation of the algebra of $g$-invariants of the mixed trace algebra generated by $m$ generic $n\times n$ matrices and the traces of their products.