arXiv subjects
Publications and source records attributed to Yuqun Chen.
We establish the Composition-Diamond lemma for non-associative algebras over a free commutative algebra. As an application, we prove that every countably generated non-associative algebra over an arbitrary commutative algebra $K$ can be embedded into a two-generated non-associative algebra over $K$.
In this paper, by using Gröbner-Shirshov bases for Rota-Baxter algebras, we prove that every dendriform dialgebra over a field of characteristic 0 can be embedded into its universal enveloping Rota-Baxter algebra of weight 0.
In this paper, we define the Gröbner-Shirshov basis for a dialgebra. The Composition-Diamond lemma for dialgebras is given then. As results, we give Gröbner-Shirshov bases for the universal enveloping algebra of a Leibniz algebra, the bar extension of a dialgebra, the free product of two dialgebras, and Clifford dialgebra. We obtain some normal forms for algebras mentioned the above.
In this paper, we firstly establish Composition-Diamond lemma for $Ω$-algebras. We give a Gröbner-Shirshov basis of the free $L$-algebra as a quotient algebra of a free $Ω$-algebra, and then the normal form of the free $L$-algebra is obtained. We secondly establish Composition-Diamond lemma for $L$-algebras. As applications, we give Gröbner-Shirshov bases of the free dialgebra and the free product of two $L$-algebras, and then we show four embedding theorems of $L$-algebras: 1) Every countably generated $L$-algebra can be embedded into a two-generated $L$-algebra. 2) Every $L$-algebra can be embedded into a simple $L$-algebra. 3) Every countably generated $L$-algebra over a countable field can be embedded into a simple two-generated $L$-algebra. 4) Three arbitrary $L$-algebras $A$, $B$, $C$ over a field $k$ can be embedded into a simple $L$-algebra generated by $B$ and $C$ if $|k|\leq \dim(B*C)$ and $|A|\leq|B*C|$, where $B*C$ is the free product of $B$ and $C$.
In this paper, we establish the Composition-Diamond lemma for associative nonunitary Rota-Baxter algebras with weight $λ$. As applications, we obtain a linear basis of a free commutative Rota-Baxter algebra without unity and show that every countably generated Rota-Baxter algebra with weight 0 can be embedded into a two-generated Rota-Baxter algebra.
A conjecture of Gröbner-Shirshov basis of any Coxeter group has proposed by L.A. Bokut and L.-S. Shiao \cite{bs01}. In this paper, we give an example to show that the conjecture is not true in general. We list all possible nontrivial inclusion compositions when we deal with the general cases of the Coxeter groups. We give a Gröbner-Shirshov basis of a Coxeter group which is without nontrivial inclusion compositions mentioned the above.
In this paper, we give a Gröbner-Shirshov basis of the braid group $B_{n+1}$ in Adyan-Thurston generators. We also deal with the braid group of type $\bf{B}_{n}$. As results, we obtain a new algorithm for getting the Adyan-Thurston normal form, and a new proof that the braid semigroup $B^+_{n+1}$ is the subsemigroup in $B_{n+1}$.
In this paper, by using Gröbner-Shirshov bases, we show that in the following classes, each (resp. countably generated) algebra can be embedded into a simple (resp. two-generated) algebra: associative differential algebras, associative $Ω$-algebras, associative $λ$-differential algebras. We show that in the following classes, each countably generated algebra over a countable field $k$ can be embedded into a simple two-generated algebra: associative algebras, semigroups, Lie algebras, associative differential algebras, associative $Ω$-algebras, associative $λ$-differential algebras. Also we prove that any countably generated module over a free associative algebra $k< X>$ can be embedded into a cyclic $k< X>$-module, where $|X|>1$. We give another proofs of the well known theorems: each countably generated group (resp. associative algebra, semigroup, Lie algebra) can be embedded into a two-generated group (resp. associative algebra, semigroup, Lie algebra).
In this paper, we establish the Composition-Diamond lemma for $λ$-differential associative algebras over a field $K $ with multiple operators. As applications, we obtain Gröbner-Shirshov bases of free $λ$-differential Rota-Baxter algebras. In particular, linear bases of free $λ$-differential Rota-Baxter algebras are obtained and consequently, the free $λ$-differential Rota-Baxter algebras are constructed by words.
In this paper, we establish the Composition-Diamond lemma for free differential algebras. As applications, we give Groebner-Shirshov bases for free Lie-differential algebra and free commutative-differential algebra, respectively.
In this paper, we study the concept of associative $n$-conformal algebra over a field of characteristic 0 and establish Composition-Diamond lemma for a free associative $n$-conformal algebra. As an application, we construct Gröbner-Shirshov bases for Lie $n$-conformal algebras presented by generators and defining relations.
In this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Groebner-Shirshov bases of free Rota-Baxter algebra, $λ$-differential algebra and $λ$-differential Rota-Baxter algebra, respectively. In particular, linear bases of these three free algebras are respectively obtained, which are essentially the same or similar to those obtained by Ebrahimi-Fard and Guo, and Guo and Keigher recently by using other methods.
In this paper, we establish the Composition-Diamond lemma for right-symmetric algebras. As an application, we give a Gröbner-Shirshov basis for universal enveloping right--symmetric algebra of a Lie algebra.
In a recent paper by L. A. Bokut, V. V. Chaynikov and K. P. Shum in 2007, Braid group $B_n$ is represented by Artin-Burau's relations. For such a representation, it is told that all other compositions can be checked in the same way. In this note, we support this claim and check all compositions.
In this paper, we prove that two-generator one-relator groups with depth less than or equal to 3 can be effectively embedded into a tower of HNN-extensions in which each group has the effective standard normal form. We give an example to show how to deal with some general cases for one-relator groups. By using the Magnus method and Composition-Diamond Lemma, we reprove the G. Higman, B. H. Neumann and H. Neumann's embedding theorem.
In this paper, a Groebner-Shirshov basis for the Chinese monoid is obtained and an algorithm for the normal form of the Chinese monoid is given.
In this paper, we establish Composition-Diamond lemma for tensor product $k< X> \otimes k< Y>$ of two free algebras over a field. As an application, we construct a Groebner-Shirshov basis in $k< X> \otimes k< Y>$ by lifting a Groebner-Shirshov basis in $k[X] \otimes k< Y>$, where $k[X]$ is a commutative algebra.
In this paper, we review Shirshov's method for free Lie algebras invented by him in 1962 which is now called the Groebner-Shirshov bases theory.