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.Comment: 8 page

Classical invariant theory for free metabelian Lie algebras

Let $KX_d$ be a vector space with basis $X_d=\{x_1,\ldots,x_d\}$ over a field $K$ of characteristic 0. One of the main topics of classical invariant theory is the study of the algebra of invariants $K[X_d]^{SL_2(K)}$, where $KX_d$ is a module of the special linear group $SL_2(K)$ isomorphic to a direct sum $V_{k_1}\oplus\cdots\oplus V_{k_r}$ and $V_k$ is the $SL_2(K)$-module of binary forms of degree $k$. Noncommutative invariant theory deals with the algebra of invariants $F_d({\mathfrak V})^G$ of the group $G<GL_d(K)$ acting on the relatively free algebra $F_d({\mathfrak V})$ of a variety of $K$-algebras $\mathfrak V$. In this paper we consider the free metabelian Lie algebra $F_d({\mathfrak A}^2)$ which is the relatively free algebra in the variety ${\mathfrak A}^2$ of metabelian (solvable of class 2) Lie algebras. We study the algebra $F_d({\mathfrak A}^2)^{SL_2(K)}$ of $SL_2(K)$-invariants of $F_d({\mathfrak A}^2)$. We describe the cases when this algebra is finitely generated. This happens if and only if $KX_d\cong V_1\oplus V_0\oplus\cdots\oplus V_0$ or $KX_d\cong V_2$ as an $SL_2(K)$-module (and in the trivial case $KX_d\cong V_0\oplus\cdots\oplus V_0$). For small $d$ we give a list of generators even when $F_d({\mathfrak A}^2)^{SL_2(K)}$ is not finitely generated. The methods for establishing that the algebra $F_d({\mathfrak A}^2)^{SL_2(K)}$ is not finitely generated work also for other relatively free algebras $F_d({\mathfrak V})$ and for other groups $G$.Comment: Revised version of the preprint posted in Dec. 201

Coordinates and Automorphisms of Polynomial and Free Associative Algebras of Rank Three

We study z-automorphisms of the polynomial algebra K[x,y,z] and the free associative algebra K over a field K, i.e., automorphisms which fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K we include also results about the structure of the z-tame automorphisms and algorithms which recognize z-tame automorphisms and z-tame coordinates