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V. Drensky

Publications and source records attributed to V. Drensky.

5 recordsLinked to original sources
arXiv2019

Cocharacters for the weak polynomial identities of the Lie algebra of $3\times 3$ skew-symmetric matrices

Let $so_3(K)$ be the Lie algebra of $3\times 3$ skew-symmetric matrices over a field $K$ of characteristic 0. The ideal $I(M_3(K),so_3(K))$ of the weak polynomial identities of the pair $(M_3(K),so_3(K))$ consists of the elements $f(x_1,\ldots,x_n)$ of the free associative algebra $K\langle X\rangle$ with the property that $f(a_1,\ldots,a_n)=0$ in the algebra $M_3(K)$ of all $3\times 3$ matrices for all $a_1,\ldots,a_n\in so_3(K)$. The generators of $I(M_3(K),so_3(K))$ were found by Razmyslov in the 1980's. In this paper the cocharacter sequence of $I(M_3(K),so_3(K))$ is computed. In other words, the ${\mathrm{GL}}_p(K)$-module structure of the algebra generated by $p$ generic skew-symmetric matrices is determined. Moreover, the same is done for the closely related algebra of $\mathrm{SO}_3(K)$-equivariant polynomial maps from the space of $p$-tuples of $3\times 3$ skew-symmetric matrices into $M_3(K)$ (endowed with the conjugation action). In the special case $p=3$ the latter algebra is a module over a $6$-variable polynomial subring in the algebra of $\mathrm{SO}_3(K)$-invariants of triples of $3\times 3$ skew-symmetric matrices, and a free resolution of this module is found. The proofs involve methods and results of classical invariant theory, representation theory of the general linear group and explicit computations with matrices.

math.RA↗
arXiv2018

Constructive noncommutative invariant theory

The problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert-Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.

math.RT↗
arXiv2017

Rationality of Hilbert series in noncommutative invariant theory

It is a fundamental result in commutative algebra and invariant theory that a finitely generated graded module over a commutative finitely generated graded algebra has rational Hilbert series, and consequently the Hilbert series of the algebra of polynomial invariants of a group of linear transformations is rational, whenever this algebra is finitely generated. This basic principle is applied here to prove rationality of Hilbert series of algebras of invariants that are neither commutative nor finitely generated. Our main focus is on linear groups acting on certain factor algebras of the tensor algebra that arise naturally in the theory of polynomial identities.

math.RA↗
arXiv2015

Noether bound for invariants in relatively free algebras

Let $\mathfrak{R}$ be a weakly noetherian variety of unitary associative algebras (over a field $K$ of characteristic 0), i.e., every finitely generated algebra from $\mathfrak{R}$ satisfies the ascending chain condition for two-sided ideals. For a finite group $G$ and a $d$-dimensional $G$-module $V$ denote by $F({\mathfrak R},V)$ the relatively free algebra in $\mathfrak{R}$ of rank $d$ freely generated by the vector space $V$. It is proved that the subalgebra $F({\mathfrak R},V)^G$ of $G$-invariants is generated by elements of degree at most $b(\mathfrak{R},G)$ for some explicitly given number $b(\mathfrak{R},G)$ depending only on the variety $\mathfrak{R}$ and the group $G$ (but not on $V$). This generalizes the classical result of Emmy Noether stating that the algebra of commutative polynomial invariants $K[V]^G$ is generated by invariants of degree at most $\vert G\vert$.

math.RA↗