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Ualbai Umirbaev

Publications and source records attributed to Ualbai Umirbaev.

At least 19 recordsLinked to original sources
arXiv2026

Automorphisms of free metabelian Lie algebras, I

We show that all Chein automorphisms (or one-row transformations) of lower degree $\geq 4$ of a free metabelian Lie algebra $M_n$ of rank $n\geq 4$ over an arbitrary field $K$ of characteristic $\neq 3$ are tame. We then show that all exponential automorphisms of $M_n$ of lower degree $\geq 5$ are also tame under the same conditions. The same results hold for fields of any characteristic when $n\geq 5$. These results contradict some long-standing results in the area. We also prove that a large class of automorphisms of $M_n$ of rank $n\geq 4$ that move only two variables are almost tame, that is, they can be expressed as a product of Chein automorphisms.

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arXiv2025

Chein Automorphisms of Free Metabelian Anticommutative Algebras

We describe all automorphisms of a free metabelian anticommutative algebra of rank $n\geq 3$ over a field $K$ that move only one variable while fixing the others. Such automorphisms are called Chein automorphisms in the cases of free metabelian groups and free metabelian Lie algebras. We show that all automorphisms of a free metabelian anticommutative algebra of rank $n=2$ are linear, and that the simplest non elementary Chein automorphism of degree $3$ is absolutely wild for all $n\geq 3$.

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arXiv2025

Representations of the Grassmann Poisson superalgebras

We prove that every irreducible Poisson supermodule over the Grassmann Poisson superalgebra $G_n$ over a field of characteristic different from $2$ is isomorphic to the regular Poisson supermodule $\mathrm{Reg}\,G_n$ or to its opposite supermodule. Moreover, every unital Poisson supermodule over $G_n$ is completely reducible. If $P$ is a unital Poisson superalgebra which contains $G_n$ with the same unit then $P\cong Q\otimes G_n$ for some Poisson superalgebra $Q$. Furthermore, we classify the supermodules over $G_n$ in the category of dot-bracket superalgebras with Jordan brackets, and we prove that every irreducible Jordan supermodule over the Kantor double $\mathrm{Kan}\,G_n$ is isomorphic to the supermodule $\mathrm{Kan}\,V$, where $V$ is an irreducible dot-bracket supermodule with a Jordan bracket over $G_n$.

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arXiv2025

Tangent Lie Algebras of Automorphism Groups of Free Algebras

We study an analogue of the Andreadakis-Johnson filtration for automorphism groups of free algebras and introduce the notion of tangent Lie algebras for certain automorphism groups, defined as subalgebras of the Lie algebra of derivations. We show that, for many classical varieties of algebras, the tangent Lie algebra is contained in the Lie algebra of derivations with constant divergence. We also introduce the concepts of approximately tame and absolutely wild automorphisms of free algebras in arbitrary varieties and employ tangent Lie algebras to investigate their properties. It is shown that nearly all known examples of wild automorphisms of free algebras are absolutely wild -- with the notable exceptions of the Nagata and Anick automorphisms. We show that the Bergman automorphism of free matrix algebras of order two is absolutely wild. Furthermore, we prove that free algebras in any variety of polynilpotent Lie algebras -- except for the abelian and metabelian varieties -- also possess absolutely wild automorphisms.

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arXiv2024

Erroneous proofs of the wildness of some automorphisms of free metabelian Lie algebras

The well-known Bachmuth-Mochizuki-Roman'kov Theorem \cite{BM,Romankov85} states that every automorphism of the free metabelian group of rank $\geq 4$ is tame. In 1992 Yu. Bahturin and S. Nabiyev \cite{BN} claimed that every nontrivial inner automorphism of the free metabelian Lie algebra $M_n$ of any rank $n\geq 2$ over a field of characteristic zero is wild. More examples of wild automorphisms of $M_n$ of rank $n\geq 4$ were given in 2008 by Z. Özcurt and N. Ekici \cite{OE}. The main goal of this note is to show that both articles contain uncorrectable errors and to draw the attention of specialists to the fact that the question of tame and wild automorphisms for free metabelian Lie algebras $M_n$ of rank $n\geq 4$ is still widely open.

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arXiv2023

Automorphisms of Veronese subalgebras of polynomial algebras and free Poisson algebras

The Veronese subalgebra $A_0$ of degree $d\geq 2$ of the polynomial algebra $A=K[x_1,x_2,\ldots,x_n]$ over a field $K$ in the variables $x_1,x_2,\ldots,x_n$ is the subalgebra of $A$ generated by all monomials of degree $d$ and the Veronese subalgebra $P_0$ of degree $d\geq 2$ of the free Poisson algebra $P=P\langle x_1,x_2,\ldots,x_n\rangle$ is the subalgebra spanned by all homogeneous elements of degree $kd$, where $k\geq 0$. If $n\geq 2$ then every derivation and every locally nilpotent derivation of $A_0$ and $P_0$ over a field $K$ of characteristic zero is induced by a derivation and a locally nilpotent derivation of $A$ and $P$, respectively. Moreover, we prove that every automorphism of $A_0$ and $P_0$ over a field $K$ closed with respect to taking all $d$-roots of elements is induced by an automorphism of $A$ and $P$, respectively.

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arXiv2023

An effective criterion for Nielsen-Schreier varieties

All algebras of a certain type are said to form a Nielsen-Schreier variety if every subalgebra of a free algebra is free. This property has been perceived as extremely rare; in particular, only six Nielsen-Schreier varieties of algebras with one binary operation have been discovered in prior work on this topic. We propose an effective combinatorial criterion for the Nielsen-Schreier property in the case of algebras over a field of zero characteristic; in our approach, operads play a crucial role. Using this criterion, we show that the well known varieties of all pre-Lie algebras and of all Lie-admissible algebras are Nielsen-Schreier, and, quite surprisingly, that there are already infinitely many non-equivalent Nielsen-Schreier varieties of algebras with one binary operation and identities of degree four.

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arXiv2023

Polynomial identities in Novikov algebras

In this paper, we study Novikov algebras satisfying nontrivial identities. We show that a Novikov algebra over a field of zero characteristic that satisfies a nontrivial identity satisfies some unexpected "universal" identities, in particular, right associator nilpotence, and right nilpotence of the commutator ideal. This, in particular, implies that a Novikov algebra over a field of zero characteristic satisfies a nontrivial identity if and only if it is Lie-solvable. We also establish that any system of identities of Novikov algebras over a field of zero characteristic follows from finitely many of them, and that the same holds over any field for multilinear Novikov identities. Some analogous simpler statements are also proved for commutative differential algebras.

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arXiv2023

Representations via differential algebras and equationally Noetherian algebras

We show that free algebras of the variety of algebras generated by the Witt algebra $W_n$, the left-symmetric Witt algebra $L_n$, and the symplectic Poisson algebra $P_n$ can be described as subalgebras of differential polynomial algebras with respect to appropriately defined products. Using these representations, we prove that $W_n$, $L_n$, $P_n$, and the free algebras of the varieties of algebras generated by these algebras are equationally Noetherian.

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arXiv2022

Automorphisms of affine Veronese surfaces

We prove that every derivation and every locally nilpotent derivation of the subalgebra $K[x^n, x^{n-1}y,\ldots,xy^{n-1}, y^n]$, where $n\geq 2$, of the polynomial algebra $K[x,y]$ in two variables over a field $K$ of characteristic zero is induced by a derivation and a locally nilpotent derivation of $K[x,y]$, respectively. Moreover, we prove that every automorphism of $K[x^n, x^{n-1}y,\ldots,xy^{n-1}, y^n]$ over an algebraically closed field $K$ of characteristic zero is induced by an automorphism of $K[x,y]$. We also show that the group of automorphisms of $K[x^n, x^{n-1}y,\ldots,xy^{n-1}, y^n]$ admits an amalgamated free product structure.

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arXiv2021

On the Lie-solvability of Novikov algebras

We prove that any Novikov algebra over a field of characteristic $\neq 2$ is Lie-solvable if and only if its commutator ideal $[N,N]$ is right nilpotent. We also construct examples of infinite-dimensional Lie-solvable Novikov algebras $N$ with non nilpotent commutator ideal $[N,N]$.

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arXiv2021

Some examples of nonassociative coalgebras and supercoalgebras

Locally finiteness of some varieties of nonassociative coalgebras is studied and the Gelfand-Dorfman construction for Novikov coalgebras and the Kantor construction for Jordan super-coalgebras are given. We give examples of a non-locally finite differential coalgebra, Novikov coalgebra, Lie coalgebra, Jordan super-coalgebra, and right-alternative coalgebra. The dual algebra of each of these examples satisfies very strong additional identities. We also constructed examples of an infinite dimensional simple differential coalgebra, Novikov coalgebra, Lie coalgebra, and Jordan super-coalgebra over a field of characteristic zero.

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arXiv2021

Automorphisms of simple quotients of the Poisson and universal enveloping algebras of $\mathrm{sl}_2$

Let $P(\mathrm{sl}_2(K))$ be the Poisson enveloping algebra of the Lie algebra $\mathrm{sl}_2(K)$ over an algebraically closed field $K$ of characteristic zero. The quotient algebras $ $ $P(\mathrm{sl}_2(K))/(C_P-λ)$, where $C_P$ is the standard Casimir element of $\mathrm{sl}_2(K)$ in $P(\mathrm{sl}_2(K))$ and $0\neq λ\in K$, are proven to be simple in \cite{UZh}. Using a result by L. Makar-Limanov \cite{ML90}, we describe generators of the automorphism group of $P(\mathrm{sl}_2(K))/(C_P-λ)$ and represent this group as an amalgamated product of its subgroups. Moreover, using similar results by J. Dixmier \cite{Dixmier73} and O. Fleury \cite{Fleury} for the quotient algebras $U(\mathrm{sl}_2(K))/(C_U-λ)$, where $C_U$ is the standard Casimir element of $\mathrm{sl}_2(K)$ in the universal enveloping algebra $U(\mathrm{sl}_2(K))$, we prove that the automorphism groups of $P(\mathrm{sl}_2(K))/(C_P-λ)$ and $U(\mathrm{sl}_2(K))/(C_U-λ)$ are isomorphic.

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arXiv2021

On the solvability of graded Novikov algebras

We show that the right ideal of a Novikov algebra generated by the square of a right nilpotent subalgebra is nilpotent. We also prove that a $G$-graded Novikov algebra $N$ over a field $K$ with solvable $0$-component $N_0$ is solvable, where $G$ is a finite additive abelean group and the characteristic of $K$ does not divide the order of the group $G$. We also show that any Novikov algebra $N$ with a finite solvable group of automorphisms $G$ is solvable if the algebra of invariants $N^G$ is solvable.

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arXiv2021

Automorphisms of free braided associative algebras in two variables

We describe the groups of automorphisms of two generated free braided associative algebras with involutive diagonal braidings over a field of characteristic $\neq 2$. Depending on the form of the diagonal involutive braiding, five different automorphism groups arise as automorphism groups of two generated free braided associative algebras. This list covers all three automorphism groups that arise in the case of quantum planes \cite{AC}.

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arXiv2020

A Dixmier theorem for Poisson enveloping algebras

We consider a skew-symmetric $n$-ary bracket on the polynomial algebra $K[x_1,\ldots,x_n,x_{n+1}]$ ($n\geq 2$) over a field $K$ of characteristic zero defined by $\{a_1,\ldots,a_n\}=J(a_1,\ldots,a_n,C)$, where $C$ is a fixed element of $K[x_1,\ldots,x_n,x_{n+1}]$ and $J$ is the Jacobian. If $n=2$ then this bracket is a Poisson bracket and if $n\geq 3$ then it is an $n$-Lie-Poisson bracket on $K[x_1,\ldots,x_n,x_{n+1}]$. We describe the center of the corresponding $n$-Lie-Poisson algebra and show that the quotient algebra $K[x_1,\ldots,x_n,x_{n+1}]/(C-λ)$, where $(C-λ)$ is the ideal generated by $C-λ$, $0\neq λ\in K$, is a simple central $n$-Lie-Poisson algebra if $C$ is a homogeneous polynomial that is not a proper power of any nonzero polynomial. This construction includes the quotients $P(\mathrm{sl}_2(K))/(C-λ)$ of the Poisson enveloping algebra $P(\mathrm{sl}_2(K))$ of the simple Lie algebra $\mathrm{sl}_2(K)$, where $C$ is the standard Casimir element of $\mathrm{sl}_2(K)$ in $P(\mathrm{sl}_2(K))$. It is also proven that the quotients $P(\mathbb{M})/(C-λ)$ of the Poisson enveloping algebra $P(\mathbb{M})$ of the exceptional simple seven dimensional Malcev algebra $\mathbb{M}$ are central simple.

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arXiv2020

Free braided nonassociative Hopf algebras and Sabinin $τ$-algebras

Let $V$ be a linear space over a field ${\bf k}$ with a braiding $τ: V\otimes V\rightarrow V\otimes V.$ We prove that the braiding $τ$ has a unique extension on the free nonassociative algebra ${\bf k}\{V\}$ freely generated by $V$ so that ${\bf k}\{V\}$ is a braided algebra. Moreover, we prove that the free braided algebra ${\bf k}\{V\}$ has a natural structure of a braided nonassociative Hopf algebra such that every element of the space of generators $V$ is primitive. In the case of involutive braidings, $τ^2={\rm id}$, we describe braided analogues of Shestakov-Umirbaev operations and prove that these operations are primitive operations. We introduce a braided version of Sabinin algebras and prove that the set of all primitive elements of a nonassociative $τ$-algebra is a Sabinin $τ$-algebra.

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