Searcharxiv⌕ Search

arXiv subjects

Ualbai Umirbaev

Publications and source records attributed to Ualbai Umirbaev.

At least 37 records · Page 2Linked to original sources
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.

math.RA↗
arXiv2020

A wild automorphism of a free Novikov algebra

An example of a non-triangulable locally nilpotent derivation and an example of a wild exponential automorphism of the free Novikov algebra $N\left\langle x,y,z\right\rangle$ in three variables $x,y,z$ over a field of characteristic zero are constructed.

math.RA↗
arXiv2020

Tame and wild automorphisms of differential polynomial algebras of rank 2

It is proved that the tame automorphism group of a differential polynomial algebra $k\{x,y\}$ over a field $k$ of characteristic $0$ in two variables $x,y$ with $m$ commuting derivations $δ_1, \ldots, δ_m$ is a free product with amalgamation. An example of a wild automorphism of the algebra $k\{x,y\}$ in the case of $m\geq 2$ derivations is constructed.

math.RA↗
arXiv2020

Differential algebraic dependence and Novikov dependence

We define an analogue of the Fox derivatives for differential polynomial algebras and give a criterion for differential algebraic dependence of a finite system of elements. In particular, we prove that differential algebraic dependence of a finite set of elements of a differential polynomial algebra over a constructive differential field $k$ of characteristic zero is algorithmically recognizable. Using a representation of free Novikov algebras by differential polynomials we also give a criterion of Novikov dependence of a finite system of elements of free Novikov algebras.

math.RA↗
arXiv2020

On the automorphisms of a free Lie algebra of rank 3 over an integral domain

We prove that the group of tame automorphisms of a free Lie algebra (as well as of a free anticommutative algebra) rank 3 over an arbitrary integral domain has the structure of an amalgamated free product. We construct an example of a wild automorphism of a free Lie algebra (as well as of a free anticommutative algebra) of rank 3 over an arbitrary Euclidean ring analogous to the Anick automorphism for free associative algebras.

math.RA↗
arXiv2015

Left-symmetric algebras of derivations of free algebras

A structure of a left-symmetric algebra on the set of all derivations of a free algebra is introduced such that its commutator algebra becomes the usual Lie algebra of derivations. Left and right nilpotent elements of left-symmetric algebras of derivations are studied. Simple left-symmetric algebras of derivations and Novikov algebras of derivations are described. It is also proved that the positive part of the left-symmetric algebra of derivations of a free nonassociative symmetric $m$-ary algebra in one free variable is generated by one derivation and some right nilpotent derivations are described.

math.RA↗
arXiv2015

Identities of the left-symmetric Witt algebras

Let $P_n=k[x_1,x_2,\ldots,x_n]$ be the polynomial algebra over a field $k$ of characteristic zero in the variables $x_1,x_2,\ldots,x_n$ and $\mathscr{L}_n$ be the left-symmetric Witt algebra of all derivations of $P_n$. We describe all right operator identities of $\mathscr{L}_n$ and prove that the set of all algebras $\mathscr{L}_n$, where $n\geq 1$, generates the variety of all left-symmetric algebras. We also describe a class of general (not only right operator) identities for $\mathscr{L}_n$.

math.RA↗
arXiv2014

Associative, Lie, and left-symmetric algebras of derivations

Let $P_n=k[x_1,x_2,\ldots,x_n]$ be the polynomial algebra over a field $k$ of characteristic zero in the variables $x_1,x_2,\ldots,x_n$ and $\mathscr{L}_n$ be the left-symmetric algebra of all derivations of $P_n$ \cite{Dzhuma99,UU2014-1}. Using the language of $\mathscr{L}_n$, for every derivation $D\in \mathscr{L}_n$ we define the associative algebra $A_D$, the Lie algebra $L_D$, and the left-symmetric algebra $\mathscr{L}_D$ related to the study of the Jacobian Conjecture. For every derivation $D\in \mathscr{L}_n$ there is a unique $n$-tuple $F=(f_1,f_2,\ldots,f_n)$ of elements of $P_n$ such that $D=D_F=f_1\partial_1+f_2\partial_2+\ldots+f_n\partial_n$. In this case, using an action of the Hopf algebra of noncommutative symmetric functions $\mathrm{NSymm}$ on $P_n$, we show that these algebras are closely related to the description of coefficients of the formal inverse to the polynomial endomorphism $X+tF$, where $X=(x_1,x_2,\ldots,x_n)$ and $t$ is an independent parameter. We prove that the Jacobian matrix $J(F)$ is nilpotent if and only if all right powers $D_F^{[r]}$ of $D_F$ in $\mathscr{L}_n$ have zero divergence. In particular, if $J(F)$ is nilpotent then $D_F$ is right nilpotent. We discuss some advantages and shortcomings of these algebras and formulate some open questions.

math.AG↗
arXiv2014

Polarization algebras and their relations

Using an approach to the Jacobian Conjecture by L.M. Drużkowski and K. Rusek 12], G. Gorni and G. Zampieri [19], and A.V. Yagzhev[27], we describe a correspondence between finite dimensional symmetric algebras and homogeneous tuples of elements of polynomial algebras. We show that this correspondence closely relates Albert's problem [10, Problem 1.1], in classical ring theory and the homogeneous dependence problem [13, page 145, Problem 7.1.5], in affine algebraic geometry related to the Jacobian Conjecture. We demonstrate these relations in concrete examples and formulate some open questions.

math.AG↗
arXiv2012

Free Poisson fields and their automorphisms

Let $k$ be an arbitrary field of characteristic 0. We prove that the group of automorphisms of a free Poisson field $P(x,y)$ in two variables $x,y$ over $k$ is isomorphic to the Cremona group $\mathrm{Cr}_2(k)$. We also prove that the universal enveloping algebra $P(x_1,...,x_n)^e$ of a free Poisson field $P(x_1,...,x_n)$ is a free ideal ring and give a characterization of the Poisson dependence of two elements of $P(x_1,...,x_n)$ via universal derivatives.

math.RA↗
arXiv2011

The Freiheitssatz for Novikov algebras

We prove the Freiheitssatz for Novikov algebras in characteristic zero. It is also proved that the variety of Novikov algebras is generated by a Novikov algebra on the space of polynomials $k[x]$ in a single variable $x$ over a field $k$ with respect to the multiplication $f\circ g=\partial(f)g$. It follows that the base rank of the variety of Novikov algebras equals 1.

math.RA↗
arXiv2011

Universal enveloping algebras and universal derivations of Poisson algebras

Let $k$ be an arbitrary field of characteristic $0$. It is shown that for any $n\geq 1$ the universal enveloping algebras of the Poisson symplectic algebra $P_n(k)$ and the Weyl algebra $A_n(k)$ are isomorphic and the canonical isomorphism between them easily leads to the Moyal product. A basis of the universal enveloping algebra $P^e$ of a free Poisson algebra $P=k\{x_1,...,x_n\}$ is constructed and proved that the left dependency of a finite number of elements of $P^e$ over $P^e$ is algorithmically recognizable. We prove that if two elements of a free Poisson algebra do not generate a free two generated subalgebra then they commute. The Fox derivatives on free Poisson algebras are defined and it is proved that an analogue of the Jacobian Conjecture for two generated free Poisson algebras is equivalent to the two-dimensional classical Jacobian Conjecture. A new proof of the tameness of automorphisms of two generated free Poisson algebras is also given.

math.RA↗
arXiv2010

The Freiheitssatz for Poisson algebras

We prove the Freiheitssatz for Poisson algebras in characteristic zero. We also give a proof of the tameness of automorphisms for two generated free Poisson algebras and prove that an analogue of the commutator test theorem is equivalent to the two dimensional classical Jacobian conjecture using the Freiheitssatz and Jung's Theorem.

math.RA↗
arXiv2010

Identities of dual Leibniz algebras

We prove that in characteristic 0 any proper subvariety of the variety of dual Leibniz algebras is nilpotent. Consequently, the variety of dual Leibniz algebras is Shpekhtian and has base rank 1.

math.RA↗
arXiv2008

The Freiheitssatz and the automorphisms of free right-symmetric algebras

We prove the Freiheitssatz for right-symmetric algebras and the decidability of the word problem for right-symmetric algebras with a single defining relation. We also prove that two generated subalgebras of free right-symmetric algebras are free and automorphisms of two generated free right-symmetric algebras are tame.

math.RA↗
arXiv2008

Automorphisms of elliptic Poisson algebras

We describe the automorphism groups of elliptic Poisson algebras on polynomial algebras in three variables and give an explicit set of generators and defining relations for this group.

math.RA↗