Skew-commutator relations and Gröbner-Shirshov basis of quantum group of type F 4

We give a Grobner-Shirshov basis of quantum group of type F4 by using the Ringel-Hall algebra approach. We compute all skew-commutator relations between the isoclasses of indecomposable representations of Ringel- Hall algebras of type F4 by using an ＇inductive＇ method. Precisely, we do not use the traditional way of computing the skew-commutative relations, that is first compute all Hall polynomials then compute the corresponding skew- commutator relations; instead, we compute the ＇easier＇ skew-commutator relations which correspond to those exact sequences with middle term indecomposable or the split exact sequences first, then ＇deduce＇ others from these ＇easier＇ ones and this in turn gives Hall polynomials as a byproduct. Then using the composition-diamond lemma prove that the set of these relations constitute a minimal CrSbner-Shirshov basis of the positive part of the quantum group of type F4. Dually, we get a Grobner-Shirshov basis of the negative part of the quantum group of type F4. And finally, we give a Gr6bner-Shirshov basis for the whole quantum group of type F4.     

The Elements in Crystal Bases Corresponding to Exceptional Modules

According to the Ringel-Green theorem, the generic composition algebra of the Hall algebra provides a realization of the positive part of the quantum group. Furthermore, its Drinfeld double can be identified with the whole quantum group, in which the BGP- reflection functors coincide with Lusztig＇s symmetries. It is first asserted that the elements corresponding to exceptional modules lie in the integral generic composition algebra, hence in the integral form of the quantum group. Then it is proved that these elements lie in the crystal basis up to a sign. Eventually, it is shown that the sign can be removed by the geometric method. The results hold for any type of Cartan datum.     

A Parameterization of the Canonical Bases of Affine Modified Quantized Enveloping Algebras

For a symmetrizable Kac-Moody Lie algebra g, Lusztig introduced the corresponding modified quantized enveloping algebra˙U and its canonical basis˙B given by Lusztig in 1992. In this paper, in the case that g is a symmetric Kac-Moody Lie algebra of finite or affine type, the authors define a set M which depends only on the root category R and prove that there is a bijection between M and ˙B, where R is the T~2-orbit category of the bounded derived category of the corresponding Dynkin or tame quiver. The method in this paper is based on a result of Lin, Xiao and Zhang in 2011, which gives a PBW-type basis of U~+.     

Gr?bner–Shirshov Basis of Derived Hall Algebra of Type A_n

We know that in Ringel–Hall algebra of Dynkin type, the set of all skew commutator relations between the iso-classes of indecomposable modules forms a minimal Gr?bner–Shirshov basis,and the corresponding irreducible elements forms a PBW type basis of the Ringel–Hall algebra. We aim to generalize this result to the derived Hall algebra DH(A_n) of type A_n. First, we compute all skew commutator relations between the iso-classes of indecomposable objects in the bounded derived category D~b(A_n) using the Auslander–Reiten quiver of D~b(A_n), and then we prove that all possible compositions between these skew commutator relations are trivial. As an application, we give a PBW type basis of DH(A_n).     

Totally compatible associative and Lie dialgebras, tridendriform algebras and PostLie algebras

This paper studies the concepts of a totally compatible dialgebra and a totally compatible Lie dialgebra,defined to be a vector space with two binary operations that satisfy individual and mixed associativity conditions and Lie algebra conditions respectively.We show that totally compatible dialgebras are closely related to bimodule algebras and semi-homomorphisms.More significantly,Rota-Baxter operators on totally compatible dialgebras provide a uniform framework to generalize known results that Rota-Baxter related operators give tridendriform algebras.Free totally compatible dialgebras are constructed.We also show that a Rota-Baxter operator on a totally compatible Lie dialgebra gives rise to a PostLie algebra,generalizing the fact that a Rota-Baxter operator on a Lie algebra gives rise to a PostLie algebra.     

Gröbner-Shirshov Bases of Irreducible Modules of the Quantum Group of Type G2

First, the authors give a Grbner-Shirshov basis of the finite-dimensional irreducible module Vq(λ) of the Drinfeld-Jimbo quantum group U_q(G_2) by using the double free module method and the known Grbner-Shirshov basis of U_q(G_2). Then, by specializing a suitable version of U_q(G_2) at q = 1, they get a Grbner-Shirshov basis of the universal enveloping algebra U(G_2) of the simple Lie algebra of type G_2 and the finite-dimensional irreducible U(G_2)-module V(λ).     

A direct product decomposition of QMV algebras

We study the direct product decomposition of quantum many-valued algebras (QMV algebras) which generalizes the decomposition theorem of ortholattices (orthomodular lattices).In detail,for an idempo- tent element of a given QMV algebra,if it commutes with every element of the QMV algebra,it can induce a direct product decomposition of the QMV algebra.At the same time,we introduce the commutant C(S) of a set S in a QMV algebra,and prove that when S consists of idempotent elements,C(S) is a subalgebra of the QMV algebra.This also generalizes the cases of orthomodular lattices.     

The Lie Algebras in which Every Subspace Is Its Subalgebra

In this paper, we study the Lie algebras in which every subspace is its subalgebra （denoted by HB Lie algebras）. We get that a nonabelian Lie algebra is an HB Lie algebra if and only if it is isomorphic to g＋Cidg, where g is an abelian Lie algebra. Moreover we show that the derivation algebra and the holomorph of a nonabelian HB Lie algebra are complete.     

与一类unit型相联系的有限维单李代数的结构

Let n ≥ 4. The complex Lie algebra, which is attached to the unit form q(x1, x2,..., xn)■ and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type Dn, and realized by the Ringel-Hall Lie algebra of a Nakayama algebra. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.     

q-Deformation of W（2, 2） Lie Algebra Associated with Quantum Groups

The q-deformation of W (2, 2) Lie algebra is well defined based on a realization of this Lie algebra by using the famous bosonic and fermionic oscillators in physics. Furthermore, the quantum group structures on the q-deformation of W (2, 2) Lie algebra are completely determined. Finally, the 1-dimensional central extension of the q-deformed W (2, 2) Lie algebra is studied, which turns out to be coincided with the conventional W (2, 2) Lie algebra in the q → 1 limit.     

Derivations of the Even Part of Finite-Dimensional Simple Modular Lie Superalgebra M

Let F be the underlying base field of characteristic p > 3 and denote by M the even part of the finite-dimensional simple modular Lie superalgebra M. In this paper, the generator sets of the Lie algebra M which will be heavily used to consider the derivation algebra Der(M) are given. Furthermore, the derivation algebra of M is determined by reducing derivations and a torus of M, i.e.,As a result, the derivation algebra of the even part of M does not equal the even part of the derivation superalgebra of M.     

Graded Hochschild cohomology of a path algebra with oriented cycles

The aim of this paper is to characterize the first graded Hochschild cohomology of a hereditary algebra whose Gabriel quiver is admitted to have oriented cycles. The interesting conclusion we have obtained shows that the standard basis of the first graded Hochschild cohomology depends on the genus of a quiver as a topological object. In this paper, we overcome the limitation of the classical Hochschild cohomology for hereditary algebra where the Gabriel quiver is assumed to be acyclic. As preparation, we first investigate the graded differential operators on a path algebra and the associated graded Lie algebra.     

B型和C型Weyl模的一个新重数公式

A monomial basis and a filtration of subalgebras for the universal enveloping algebra U(gl) of a complex simple Lie algebra gl of type Bl and Cl are given, and the decomposition of the Weyl module V (λ) as a U(gl)-module into a direct sum of Weyl modules V (μ)’s as U(gl-1)modules is described. In particular, a new multiplicity formula for the Weyl module V (λ) is obtained in this note.     

A class of irreducible modules for the extended affne Lie algebra ■

We construct a class of modules for extended affne Lie algebra ■ by using the free fields. A necessary and sufficient condition is given for those modules being irreducible.     

TWO CLASSES OF INTEGRABLE COUPLING FOR AKNS HIERARCHY

A set of new matrix Lie algebra is constructed, which is devoted to obtaining a new loop algebra A 2M . Then we use the idea of enlarging spectral problems to make an enlarged spectral problems. It follows that the multi-component AKNS hierarchy is presented. Further, two classes of integrable coupling of the AKNS hierarchy are obtained by enlarging spectral problems.     

On two Hall algebra approaches to odd periodic triangulated categories

Given an odd-periodic algebraic triangulated category, we compare Bridgeland's Hall algebra in the sense of Bridgeland(2013) and Gorsky(2014), and the derived Hall algebra in the sense of Ten(2006), Xiao and Xu(2008) and Xu and Chen(2013), and show that the former one is the twisted form of the tensor product of the latter one and a suitable group algebra.     

Generations of integrable hierarchies and exact solutions of related evolution equations with variable coefficients

We first propose a way for generating Lie algebras from which we get a few kinds of reduced 6 6 Lie algebras, denoted by R6, R8 and R1,R6/2, respectively. As for applications of some of them, a Lax pair is introduced by using the Lie algebra R6 whose compatibility gives rise to an integrable hierarchy with 4- potential functions and two arbitrary parameters whose corresponding Hamiltonian structure is obtained by the variational identity. Then we make use of the Lie algebra R6 to deduce a nonlinear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is also obtained. Again,via using the Lie algebra R62, we introduce a Lax pair and work out a linear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is obtained. Finally, we get some reduced linear and nonlinear equations with variable coefficients and work out the elliptic coordinate solutions, exact traveling wave solutions, respectively.     

以幂零李代数R_n为幂零根基的可解李代数(英文)

In this paper we explicitly determine automorphism group of filiform Lie algebra Rn to find the indecomposable solvable Lie algebras with filiform Lie algebra Rn nilradicals.We also prove that the indecomposable solvable Lie algebras with filiform Rn nilradicals is complete.     

On U-orthodox semigroups

As a generalization of an orthodox semigroup in the class of regular semigroups, a type W semigroup was first investigated by El-Qallali and Fountain. As an analogy of the type W semi-groups in the class of abundant semigroups, we introduce the U-orthodox semigroups. It is shown that the maximum congruence μ contained in HU on U-orthodox semigroups can be determined. As a consequence, we show that a U-orthodox semigroup S can be expressed by the spined product of a Hall semigroup WU and a V-ample semigroup ...     

On maximal injective subalgebras in a wΓ factor

Let L(FQ) ×α Z be the crossed product von Neumann algebra of the free group factor L(FQ), associated with the left regular representation λ of the free group FQ with the set {ur : r ∈ Q} of generators, by an automorphism α defined by α(λ(ur)) = exp(2πri)λ(ur), where Q is the rational number set. We show that L(FQ) ×α Z is a wΓ factor, and for each r ∈ Q, the von Neumann subalgebra Ar generated in L(FQ) ×α Z by λ(ur) and v is maximal injective, where v is the unitary implementing the automorphism α. In parti...