arXiv subjects
Publications and source records attributed to Yuqun Chen.
In this survey article, we report some new results of Groebner-Shirshov bases, including new Composition-Diamond lemmas, applications of some known Composition-Diamond lemmas and content of some expository papers.
Let $U_q(A_N)$ be the Drinfeld-Jimbo quantum group of type $A_N$. In this paper, by using Groebner-Shirshov bases, we give a simple (but not short) proof of the Rosso-Yamane Theorem on PBW basis of $U_q(A_N)$.
In this paper we give some relationships among the Groebner-Shirshov bases in free associative algebras, free left modules and "double-free" left modules (free modules over a free algebra). We give the Chibrikov's Composition-Diamond lemma for modules and show that Kang-Lee's Composition-Diamond lemma follows from this lemma. As applications, we also deal with highest weight module over the Lie algebra $sl_2$, Verma module over a Kac-Moody algebra, Verma module over Lie algebra of coefficients of a free conformal algebra and the universal enveloping module for a Sabinin algebra.
One of the natural ways to prove that the Hall words (Philip Hall, 1933) consist of a basis of a free Lie algebra is a direct construction: to start with a linear space spanned by Hall words, to define the Lie product of Hall words, and then to check that the product yields the Lie identities (Marshall Hall, 1950). Here we suggest another way using the Composition-Diamond lemma for free anti-commutative (non-associative) algebras (A.I. Shirshov, 1962).
An algebra $\cal{R}$ is called an extension of the algebra $M$ by $B$ if $M^2=0$, $M$ is an ideal of $\cal{R}$ and $\cal{R}$$/M\cong B$ as algebras. In this paper, by using the Gröbner-Shirshov bases, we characterize completely the extensions of $M$ by $B$. An algorithm to find the conditions of an algebra $A$ to be an extension of $M$ by $B$ is obtained.
In this paper, we generalize the Shirshov's Composition Lemma by replacing the monomial order for others. By using Groebner-Shirshov bases, the normal forms of HNN extension of a group and the alternating group are obtained.
In this paper, by using the Groebner-Shirshov bases, we give characterizations of the Schreier extensions of groups when the group is presented by generators and relations. An algorithm to find the conditions of a group to be a Schreier extension is obtained. By introducing a special total order, we obtain the structure of the Schreier extension by an HNN group.
We introduce the notion of idempotent radical class of module coalgebras over a bialgebra B. We prove that if R is an idempotent radical class of B-module coalgebras, then every B-module coalgebra contains a unique maximal B-submodule coalgebra in R. Moreover, a B-module coalgebra C is a member of R if, and only if, DB is in R for every simple subcoalgebra D of C. The collection of B-cocleft coalgebras, and the collection of H-projective module coalgebras over a Hopf algebra H are idempotent radical classes. As applications, we use these idempotent radical classes to give another proofs for a projectivity theorem and a normal basis theorem of Schneider without assuming bijective antipode.