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Yuqun Chen

Publications and source records attributed to Yuqun Chen.

At least 19 recordsLinked to original sources
arXiv2021

Construction of free Lie Rota-Baxter superalgebra via Gröbner-Shirshov bases theory

In this paper, we construct free Lie Rota-Baxter superalgebra by using Gröbner-Shirshov bases theory. We firstly construct free operated Lie superalgebras by the operated super-Lyndon-Shirshov monomials. Secondly, we establish Gröbner-Shirshov bases theory for operated Lie superalgebras. Thirdly, we find a Gröbner-Shirshov basis of a free Lie Rota-Baxter superalgebra on a $\mathbb{Z}_2$-graded set. Consequently, we can obtain a linear basis of a free Lie Rota-Baxter superalgebra by the composition-diamond lemma for operated Lie superalgebras.

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arXiv2021

Free Products of digroups

We construct the free products of arbitrary digroups, and thus we solve an open problem of Zhuchok.

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arXiv2021

Gelfand-Kirillov dimension of bicommutative algebras

We first offer a fast method for calculating the Gelfand-Kirillov dimension of a finitely presented commutative algebra by investigating certain finite set. Then we establish a Groebner-Shirshov bases theory for bicommutative algebras, and show that every finitely generated bicommutative algebra has a finite Groebner-Shirshov basis. As an application, we show that the Gelfand-Kirillov dimension of a finitely generated bicommutative algebra is a nonnegative integer.

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arXiv2021

Free Products of Trialgebras

We apply the method of Gröbner-Shirshov bases for replicated algebras developed by Kolesnikov to offer a general approach for constructing free products of trialgebrs (resp. trioids). In particular, the open problem of Zhuchok on constructing free products of trioids is solved.

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arXiv2020

Gröbner-Shirshov bases theory and extensions of Leibniz superalgebras

In this paper, we elaborate Gröbner-Shirshov bases method for Leibniz (super)algebras. We show that there is a unique reduced Gröbner-Shirshov basis for every (graded) ideal of a free Leibniz (super)algebra. As applications, we construct linear bases of free metabelian Leibniz superalgebras and new linear bases of free metabelian Lie algebras. We present a complete characterization of extensions of a Leibniz (super)algebra by another Leibniz (super)algebra, where the former is presented by generators and relations.

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arXiv2019

Word problem for finitely presented metabelian Poisson algebras

We first construct a linear basis for a free metabelian Poisson algebra generated by an arbitrary well-ordered set. It turns out that such a linear basis depends on the characteristic of the underlying field. Then we elaborate the method of Gröbner--Shirshov bases for metabelian Poisson algebras. Finally, we show that the word problem for finitely presented metabelian Poisson algebras are solvable.

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arXiv2019

Gröbner--Shirshov bases for commutative dialgebras

We establish Gröbner--Shirshov bases theory for commutative dialgebras. We show that for any ideal $I$ of $Di[X]$, $I$ has a unique reduced Gröbner--Shirshov basis, where $Di[X]$ is the free commutative dialgebra generated by a set $X$, in particular, $I$ has a finite Gröbner--Shirshov basis if $X$ is finite. As applications, we give normal forms of elements of an arbitrary commutative disemigroup, prove that the word problem for finitely presented commutative dialgebras (disemigroups) is solvable, and show that if $X$ is finite, then the problem whether two ideals of $Di[X]$ are identical is solvable. We construct a Gröbner--Shirshov basis in associative dialgebra $Di\langle X\rangle$ by lifting a Gröbner--Shirshov basis in $Di[X]$.

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arXiv2019

No dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2

The Gelfand-Kirillov dimension measures the asymptotic growth rate of algebras. For every associative dialgebra $\mathcal{D}$, the quotient $\mathcal{A}_\mathcal{D}:=\mathcal{D}/\mathsf{Id}(S)$, where $\mathsf{Id}(S)$ is the ideal of $\mathcal{D}$ generated by the set $S:=\{x \vdash y-x\dashv y \mid x,y\in \mathcal{D}\}$, is called the associative algebra associated to $\mathcal{D}$. Here we show that the Gelfand--Kirillov dimension of $\mathcal{D}$ is bounded above by twice the Gelfand--Kirillov dimension of $\mathcal{A}_\mathcal{D}$. Moreover, we prove that no associative dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2.

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arXiv2019

A construction of free digroup

We give a construction of a free digroup $F(X)$ on a set $X$ and formulate the halo and the group parts of $F(X)$. We prove that $F(X)$ is isomorphic to $F(Y)$ if and only if $card(X)=card(Y)$.

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arXiv2018

On free Gelfand--Dorfman--Novikov superalgebras and a PBW type theorem

We construct a linear basis of a free GDN superalgebra over a field of characteristic $\neq 2$. As applications, we prove a PBW theorem, that is, any GDN superalgebra can be embedded into its universal enveloping commutative associative differential superalgebra. An Engel theorem under some assumptions is given.

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arXiv2017

Gröbner-Shirshov bases for associative conformal modules

We construct free modules over an associative conformal algebra. We establish Composition-Diamond lemma for associative conformal modules. As applications, Gröbner-Shirshov bases of the Virasoro conformal module and module over the semidirect product of Virasoro conformal algebra and current algebra are given respectively.

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arXiv2017

On free Gelfand-Dorfman-Novikov-Poisson algebras and a PBW theorem

In 1997, X. Xu \cite{Xiaoping Xu Poisson} invented a concept of Novikov-Poisson algebras (we call them Gelfand-Dorfman-Novikov-Poisson (GDN-Poisson) algebras). We construct a linear basis of a free GDN-Poisson algebra. We define a notion of a special GDN-Poisson admissible algebra, based on X. Xu's definition and an S.I. Gelfand's observation (see \cite{Gelfand}). It is a differential algebra with two commutative associative products and some extra identities. We prove that any GDN-Poisson algebra is embeddable into its universal enveloping special GDN-Poisson admissible algebra. Also we prove that any GDN-Poisson algebra with the identity $x\circ(y\cdot z)=(x\circ y )\cdot z +(x\circ z) \cdot y$ is isomorphic to a commutative associative differential algebra.

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arXiv2017

On formulas and some combinatorial properties of Schubert Polynomials

By applying a Gröbner-Shirshov basis of the symmetric group $S_{n}$, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schubert polynomials. As applications, we give two algorithms to calculate the structure constants for Schubert polynomials, one of which depends on Monk's formula.

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arXiv2017

Gröbner-Shirshov bases method for Gelfand-Dorfman-Novikov algebras

We establish Gröbner-Shirshov bases theory for Gelfand-Dorfman-Novikov algebras over a field of characteristic $0$. As applications, a PBW type theorem in Shirshov form is given and we provide an algorithm for solving the word problem of Gelfand-Dorfman-Novikov algebras with finite homogeneous relations. We also construct a subalgebra of one generated free Gelfand-Dorfman-Novikov algebra which is not free.

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arXiv2017

Automaticity of one-relator semigroups with length less than or equal to three

The main results of this paper is to give a complete characterization of the automaticity of one-relator semigroups with length less than or equal to three. Let $S=sgp\langle A|u=v\rangle$ be a semigroup generated by a set $A=\{a_1,a_2,\dots,a_n\},\ n\in \mathbb{N}$ with defining relation $u=v$, where $u,v\in A^*$ and $A^*$ is the free monoid generated by $A$. Such a semigroup is called a one-relator semigroup. Suppose that $|v|\leq|u|\leq3$, where $|u|$ is the length of the word $u$. Suppose that $a,b\in A,\ a\neq b$. Then we have the following: (1) $S$ is prefix-automatic if $u=v\not\in \{aba=ba,\ aab=ba,\ abb=bb\}$. Moreover, if $u=v\in \{aba=ba,\ aab=ba,\ abb=bb\}$ then $S$ is not automatic. (2) $S$ is biautomatic if one of the following holds: (i) $|u|=3,\ |v|=0$, (ii) $|u|=|v|=3$, (iii) $|u|=2$ and $u=v\not\in \{ab=a,\ ab=b\}$. Moreover, if $u=v\in \{ab=a,\ ab=b\}$ then $S$ is not biautomatic.

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arXiv2016

Gröbner-Shirshov bases for Lie $Ω$-algebras and free Rota-Baxter Lie algebras

In this paper, we generalize the Lyndon-Shirshov words to Lyndon-Shirshov $Ω$-words on a set $X$ and prove that the set of all non-associative Lyndon-Shirshov $Ω$-words forms a linear basis of the free Lie $Ω$-algebra on the set $X$. From this, we establish Gröbner-Shirshov bases theory for Lie $Ω$-algebras. As applications, we give Gröbner-Shirshov bases for free $λ$-Rota-Baxter Lie algebras, free modified $λ$-Rota-Baxter Lie algebras and free Nijenhuis Lie algebras and then linear bases of such three free algebras are obtained.

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