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Artem Lopatin

Publications and source records attributed to Artem Lopatin.

27 records · Page 2Linked to original sources
arXiv2020

Separating invariants for multisymmetric polynomials

This article studies separating invariants for the ring of multisymmetric polynomials in $m$ sets of $n$ variables over an arbitrary field $\mathbb{K}$. We prove that in order to obtain separating sets it is enough to consider polynomials that depend only on $\lfloor \frac{n}{2} \rfloor + 1$ sets of these variables. This improves a general result by Domokos about separating invariants. In addition, for $n \leq 4$ we explicitly give minimal separating sets (with respect to inclusion) for all $m$ in case $\text{char}(\mathbb{K}) = 0$ or $\text{char}(\mathbb{K}) > n$.

math.RT↗
arXiv2020

Identities of sum of two PI-algebras in the case of positive characteristic

We consider the following question posted by K.I. Beidar and A.V. Mikhalev in 1995 for an associative ring $R=R_1+R_2$: is it true that if the subrings $R_1$ and $R_2$ satisfy polynomial identities, then $R$ also satisfies a polynomial identity? Over a field of positive characteristic we establish new conditions on $R_1$ and $R_2$ that guarantee a positive answer to the question. We find upper and low bounds on the degrees of identities of $R$.

math.RA↗
arXiv2020

Separating invariants of three nilpotent 3x3 matrices

The algebra of GL_n-invariants of d-tuple of n x n nilpotent matrices with respect to the action by simultaneous conjugation is generated by the traces of products of nilpotent generic matrices in the case of an algebraically closed field of characteristic zero. We described a minimal separating set for this algebra in case n=d=3.

math.RA↗
arXiv2020

Jordan algebras admitting derivations with invertible values

The notion of derivation with invertible values as a derivation of ring with unity that only takes multiplicatively invertible or zero values appeared in a paper of Bergen, Herstein and Lanski, in which they determined the structure of associative rings that admit derivations with invertible values. Later, the results of this paper were generalized in many cases, for example, for generalized derivations, associative super-algebras, alternative algebras and many others. The present work is dedicated to description of all Jordan algebras admitting derivations with invertible values.

math.RA↗
arXiv2017

The structure of simple noncommutative Jordan superalgebras

In this paper we describe all subalgebras and automorphisms of simple noncommutative Jordan superalgebras $K_3(α,β,γ)$ and $D_t(α,β,γ)$ and compute the derivations of the nontrivial simple finite-dimensional noncommutative Jordan superalgebras.

math.RA↗
arXiv2016

Identities for mixed representations of quiver are finitely based

The classical construction of representations of quivers enables us to consider linear maps between several vector spaces. The mixed representations of quivers helps us to work with linear maps as well as bilinear forms on several vector spaces. We construct a finite generating set for the T-ideal of identities of the polynomial invariants of mixed representations of a quiver.

math.RA↗
arXiv2016

Identities for the Lie algebra gl(2) over an infinite field of characteristic two

In 1970 Vaughan-Lee established that over an infinite field of characteristic two the ideal T[gl(2)] of all polynomial identities for the Lie algebra gl(2) is not finitely generated as a T-ideal. But a generating set for this ideal of polynomial identities was not found. We establish some generating set for the T-ideal T[gl(2)].

math.RA↗