相关论文: Grobner-Shirshov Basis for affine Weyl Group $\wid…
We give Gr\"obner-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…
In this paper, we elaborate Gr\"obner-Shirshov bases method for Leibniz (super)algebras. We show that there is a unique reduced Gr\"obner-Shirshov basis for every (graded) ideal of a free Leibniz (super)algebra. As applications, we…
In this paper, we establish the Composition-Diamond lemma for right-symmetric algebras. As an application, we give a Gr\"{o}bner-Shirshov basis for universal enveloping right--symmetric algebra of a Lie algebra.
We develop Groebner---Shirshov bases technique for pre-associative algebras also known as dendriform (di-)algebras.
The generalized Drinfel'd-Sokolov hierarchies studied by de Groot-Hollowood-Miramontes are extended from the viewpoint of Sato-Wilson dressing method. In the A_1^(1) case, we obtain the hierarchy that include the derivative nonlinear…
We present a new class of extended affine Weyl groups $\widetilde{W}^{(k,k+1)}(A_l)$ for $1\leq k <l$ and obtain an analogue of Chevalley-type theorem for their invariants. We further show the existence of Frobenius manifold structures on…
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.
This is the second of two papers where we study polytopes arising from affine Coxeter arrangements. Our results include a formula for their volumes, and also compatible definitions of hypersimplices, descent numbers and major index for all…
In this paper, we generalize the Lyndon-Shirshov words to Lyndon-Shirshov $\Omega$-words on a set $X$ and prove that the set of all non-associative Lyndon-Shirshov $\Omega$-words forms a linear basis of the free Lie $\Omega$-algebra on the…
We construct a Gr\"obner-Shirshov basis of the Temperley-Lieb algebra $\mathfrak{T}(d,n)$ of the complex reflection group $G(d,1,n)$, inducing the standard monomials expressed by the generators $\{E_i\}$ of $\mathfrak{T}(d,n)$. This result…
The author is mainly interest in the Gr\"{o}bner-Shirshov bases of finite Coxeter groups. It is known that the finite Coxeter groups are classified in terms of Coxeter-Dynkin diagrams. Under the fixed order, it is worth mention that the…
We find a generalization of the restricted PBW basis for pointed Hopf algebras over abelian groups constructed by Kharchenko. We obtain a factorization of the Hilbert series for a wide class of graded Hopf algebras. These factors are…
We consider a new version of Composition-Diamond Lemma for dialgebras in order to obtain an explicit Groebner-Shirshov basis for HNN-extension of dialgebras and determine a normal form for that.
We apply crystal theory to affine Schubert calculus, Gromov-Witten invariants for the complete flag manifold, and the positroid stratification of the positive Grassmannian. We introduce operators on decompositions of elements in the…
We present algorithms to compute generalized root systems of Nichols algebras of diagonal type and of contragredient Lie superalgebras. As a consequence, we obtain an algorithm to compute the Lyndon words in the Kharchenko PBW basis…
We are interested in cell properties of translations in affine Weyl groups. We find out some translations in the second lowest two-sided cell of an affine Weyl group and use the translations to formulate a refinement of Jianyi Shi's…
Given $\mathfrak{F}$ a coherent sheaf on a Noetherian integral algebraic stack $\mathfrak{P}$, we give two constructions of stacks $\widetilde{\mathfrak{P}}$, equipped with birational morphisms $p:\widetilde{\mathfrak{P}}\to \mathfrak{P}$…
We give a bijection between certain colored partitions and the elements in the quotient of an affine Weyl group modulo its Weyl group. By Bott's formula these colored partitions give rise to some partition identities. In certain types,…
Let $A$ be a brace algebra. This structure implies that $A$ is also a pre-Lie algebra. In this paper, we establish Composition-Diamond lemma for brace algebras. Using this Composition-Diamond lemma we prove that each pre-Lie algebra $L$ can…
In this paper we prove Lusztig's conjecture on based ring for an affine Weyl group of type $\tilde A_{n-1}$.