Searcharxiv⌕ Search

arXiv subjects

Yuqun Chen

Publications and source records attributed to Yuqun Chen.

At least 37 records · Page 2Linked to original sources
arXiv2016

Extensions of associative and Lie algebras via Gröbner-Shirshov bases method

Let $\mathfrak{a},\mathfrak{b},\mathfrak{e}$ be algebras over a field $k$. Then $\mathfrak{e}$ is an extension of $\mathfrak{a}$ by $\mathfrak{b}$ if $\mathfrak{a}$ is an ideal of $\mathfrak{e}$ and $\mathfrak{b}$ is isomorphic to the quotient algebra $\mathfrak{e}/\mathfrak{a}$. In this paper, by using Gröbner-Shirshov bases theory for associative (resp. Lie) algebras, we give complete characterizations of associative (resp. Lie) algebra extensions of $\mathfrak{a}$ by $\mathfrak{b}$, where $\mathfrak{b}$ is presented by generators and relations.

math.RA↗
arXiv2016

A new Composition-Diamond lemma for associative conformal algebras

Let $C(B,N)$ be the free associative conformal algebra generated by a set $B$ with a bounded locality $N$. Let $S$ be a subset of $C(B,N)$. A Composition-Diamond lemma for associative conformal algebras is firstly established by Bokut, Fong, and Ke in 2004 \cite{BFK04} which claims that if (i) $S$ is a Gröbner-Shirshov basis in $C(B,N)$, then (ii) the set of $S$-irreducible words is a linear basis of the quotient conformal algebra $C(B,N|S)$, but not conversely. In this paper, by introducing some new definitions of normal $S$-words, compositions and compositions to be trivial, we give a new Composition-Diamond lemma for associative conformal algebras which makes the conditions (i) and (ii) equivalent. We show that for each ideal $I$ of $C(B,N)$, $I$ has a unique reduced Gröbner-Shirshov basis. As applications, we show that Loop Virasoro Lie conformal algebra and Loop Heisenberg-Virasoro Lie conformal algebra are embeddable into their universal enveloping associative conformal algebras.

math.RA↗
arXiv2014

Gröbner-Shirshov bases and PBW theorems

We review some applications of Gröbner-Shirshov bases, including PBW theorems, linear bases of free universal algebras, normal forms for groups and semigroups, extensions of groups and algebras, embedding of algebras.

math.RA↗
arXiv2014

New approaches to plactic monoid via Gröbner-Shirshov bases

We present the plactic algebra on an arbitrary alphabet set $A$ by row generators and column generators respectively. We give Gröbner-Shirshov bases for such presentations. In the case of column generators, a finite Gröbner-Shirshov basis is given if $A$ is finite. From the Composition-Diamond lemma for associative algebras, it follows that the set of Young tableaux is a linear basis of plactic algebra. As the result, it gives a new proof that Young tableaux are normal forms of elements of plactic monoid. This result was proved by D.E. Knuth \cite{Knuth} in 1970, see also Chapter 5 in \cite{M.L}.

math.RA↗
arXiv2014

Gröbner-Shirshov basis for the finitely presented algebras defined by permutation relations of symmetric type

In this paper, we give a Gröbner-Shirshov basis for the finitely presented semigroup algebra $\mathbf{k}[S_n(Sym_n)]$ defined by permutation relations of symmetric type. As an application, by the Composition-Diamond Lemma, we obtain normal forms of elements of momoid $S_n(Sym_n)$, which gives an answer to an open problem posted by F. Cedó, E. Jespers and J. Okniński [7] for the symmetric group case.

math.RA↗
arXiv2013

Groebner-Shirshov bases for free inverse semigroups

A new construction of a free inverse semigroup was obtained by Poliakova and Schein in 2005. Based on their result, we find a Groebner-Shirshov basis of a free inverse semigroup relative to the deg-lex order of words. In particular, we give the (unique and shortest) Groebner-Shirshov normal forms in the classes of equivalent words of a free inverse semigroup together with the Groebner-Shirshov algorithm to transform any word to its normal form.

math.GR↗
arXiv2013

Gröbner-Shirshov bases for some Lie algebras

We give Gröbner-Shirshov bases for Drinfeld-Kohno Lie algebra $\textbf{L}_{n}$ in \cite{[Et]} and Kukin Lie algebra $A_P$ in \cite{Kukin}, where $P$ is a semigroup. As applications, we show that as $\mathbb{Z}$-module $\textbf{L}_{n}$ is free and a $\mathbb{Z}$-basis of $\textbf{L}_{n}$ is given. We give another proof of Kukin Theorem: if semigroup $P$ has the undecidable word problem then the Lie algebra $A_P$ has the same property.

math.RA↗
arXiv2013

Some remarks for the Akivis algebras and the Pre-Lie algebras

In this paper, by using the Composition-Diamond lemma for non-associative algebras invented by A. I. Shirshov in 1962, we give Gröbner-Shirshov bases for free Pre-Lie algebras and the universal enveloping non-associative algebra of an Akivis algebra, respectively. As applications, we show I.P. Shestakov's result that any Akivis algebra is linear and D. Segal's result that the set of all good words in $X^{**}$ forms a linear basis of the free Pre-Lie algebra $PLie(X)$ generated by the set $X$. For completeness, we give the details of the proof of Shirshov's Composition-Diamond lemma for non-associative algebras.

math.RA↗
arXiv2012

Gröbner-Shirshov bases for semirings

In the paper, we establish Gröbner-Shirshov bases for semirings and commutative semirings. As applications, we obtain Gröbner-Shirshov bases and A. Blass's (1995) and M. Fiore -T. Leinster's (2004) normal forms of the semirings $\mathbb{N}[x]/(x=1+x+x^2)$ and $\mathbb{N}[x]/(x=1+x^2)$ with one generator $x$ and one defining relation, correspondingly.

math.RA↗
arXiv2011

Lyndon-Shirshov basis and anti-commutative algebras

Chen, Fox, Lyndon 1958 \cite{CFL58} and Shirshov 1958 \cite{Sh58} introduced non-associative Lyndon-Shirshov words and proved that they form a linear basis of a free Lie algebra, independently. In this paper we give another approach to definition of Lyndon-Shirshov basis, i.e., we find an anti-commutative Gröbner-Shirshov basis $S$ of a free Lie algebra such that $Irr(S)$ is the set of all non-associative Lyndon-Shirshov words, where $Irr(S)$ is the set of all monomials of $N(X)$, a basis of the free anti-commutative algebra on $X$, not containing maximal monomials of polynomials from $S$. Following from Shirshov's anti-commutative Gröbner-Shirshov bases theory \cite{S62a2}, the set $Irr(S)$ is a linear basis of a free Lie algebra.

math.RA↗
arXiv2011

Gröbner-Shirshov bases for metabelian Lie algebras

In this paper, we establish the Gröbner-Shirshov bases theory for metabelian Lie algebras. As applications, we find the Gröbner-Shirshov bases for partial commutative metabelian Lie algebras related to circuits, trees and some cubes.

math.RA↗
arXiv2011

Gröbner-Shirshov bases for Lie algebras over a commutative algebra

In this paper we establish a Gröbner-Shirshov bases theory for Lie algebras over commutative rings. As applications we give some new examples of special Lie algebras (those embeddable in associative algebras over the same ring) and non-special Lie algebras (following a suggestion of P.M. Cohn (1963) \cite{Conh}). In particular, Cohn's Lie algebras over the characteristic $p$ are non-special when $p=2,\ 3,\ 5$. We present an algorithm that one can check for any $p$, whether Cohn's Lie algebras is non-special. Also we prove that any finitely or countably generated Lie algebra is embeddable in a two-generated Lie algebra.

math.RA↗
arXiv2011

Gröbner-Shirshov bases for categories

In this paper we establish Composition-Diamond lemma for small categories. We give Gröbner-Shirshov bases for simplicial category and cyclic category.

math.RA↗
arXiv2011

Some new results on Gröbner-Shirshov bases

In this survey article, we report some new results of Gröbner-Shirshov bases, including new Composition-Diamond lemmas and some applications of some known Composition-Diamond lemmas.

math.RA↗
arXiv2010

Gröbner-Shirshov Bases and Hilbert Series of Free Dendriform Algebras

In this paper, we give a Gröbner-Shirshov basis of the free dendriform algebra as a quotient algebra of an $L$-algebra. As applications, we obtain a normal form of the free dendriform algebra. Moreover, Hilbert series and Gelfand-Kirillov dimension of finitely generated free dendriform algebras are obtained.

math.RA↗