scholarly journals CLASSIFICATION OF GRADED LEFT-SYMMETRIC ALGEBRAIC STRUCTURES ON WITT AND VIRASORO ALGEBRAS

2011 ◽  
Vol 22 (02) ◽  
pp. 201-222 ◽  
Author(s):  
XIAOLI KONG ◽  
HONGJIA CHEN ◽  
CHENGMING BAI
Keyword(s):  
Virasoro Algebra ◽  
Indecomposable Module ◽  
Multiplication Operators ◽  
Algebraic Structures ◽  
Witt Algebra ◽  
Left Multiplication ◽  
Symmetric Algebras ◽  
Nonlinear Math ◽  
Math Phys

We find that a compatible graded left-symmetric algebraic structure on the Witt algebra induces an indecomposable module V of the Witt algebra with one-dimensional weight spaces by its left-multiplication operators. From the classification of such modules of the Witt algebra, the compatible graded left-symmetric algebraic structures on the Witt algebra are classified. All of them are simple and they include the examples given by [Comm. Algebra32 (2004) 243–251; J. Nonlinear Math. Phys.6 (1999) 222–245]. Furthermore, we classify the central extensions of these graded left-symmetric algebras which give the compatible graded left-symmetric algebraic structures on the Virasoro algebra. They coincide with the examples given by [J. Nonlinear Math. Phys.6 (1999) 222–245].

scholarly journals Pre-jordan Algebras

2013 ◽  
Vol 112 (1) ◽  
pp. 19 ◽  
Author(s):  
Dongping Hou ◽  
Xiang Ni ◽  
Chengming Bai
Keyword(s):  
Lie Algebras ◽  
Jordan Algebra ◽  
Symplectic Form ◽  
Jordan Algebras ◽  
Multiplication Operators ◽  
Baxter Equation ◽  
Algebraic Structures ◽  
Left Multiplication ◽  
Close Relationship ◽  
Dendriform Algebras

The purpose of this paper is to introduce and study a notion of pre-Jordan algebra. Pre-Jordan algebras are regarded as the underlying algebraic structures of the Jordan algebras with a nondegenerate symplectic form. They are the algebraic structures behind the Jordan Yang-Baxter equation and Rota-Baxter operators in terms of $\mathcal{O}$-operators of Jordan algebras introduced in this paper. Pre-Jordan algebras are analogues for Jordan algebras of pre-Lie algebras and fit into a bigger framework with a close relationship with dendriform algebras. The anticommutator of a pre-Jordan algebra is a Jordan algebra and the left multiplication operators give a representation of the Jordan algebra, which is the beauty of such a structure. Furthermore, we introduce a notion of $\mathcal{O}$-operator of a pre-Jordan algebra which gives an analogue of the classical Yang-Baxter equation in a pre-Jordan algebra.

scholarly journals Classification of Symmetric Special Biserial Algebras With At Most One Non-Uniserial Indecomposable Projective

2015 ◽  
Vol 58 (3) ◽  
pp. 739-767 ◽  
Author(s):  
Nicole Snashall ◽  
Rachel Taillefer
Keyword(s):  
Natural Generalization ◽  
Stable Equivalence ◽  
Symmetric Algebras ◽  
Special Biserial Algebras ◽  
Weakly Symmetric ◽  
Dihedral Type ◽  
Indecomposable Projective Module ◽  
Algebras Of Dihedral Type ◽  
Euclidean Type

AbstractWe consider a natural generalization of symmetric Nakayama algebras, namely, symmetric special biserial algebras with at most one non-uniserial indecomposable projective module. We describe the basic algebras explicitly by quiver and relations, then classify them up to derived equivalence and up to stable equivalence of Morita type. This includes the weakly symmetric algebras of Euclidean type n, as studied by Bocian et al., as well as some algebras of dihedral type.

scholarly journals THE QUANDARY OF QUANDLES: A BOREL COMPLETE KNOT INVARIANT

2019 ◽  
Vol 108 (2) ◽  
pp. 262-277 ◽  
Author(s):  
ANDREW D. BROOKE-TAYLOR ◽  
SHEILA K. MILLER
Keyword(s):  
Isomorphism Problem ◽  
Algebraic Structures ◽  
Knot Invariant ◽  
Borel Reducibility ◽  
Special Case

We show that the isomorphism problems for left distributive algebras, racks, quandles and kei are as complex as possible in the sense of Borel reducibility. These algebraic structures are important for their connections with the theory of knots, links and braids. In particular, Joyce showed that a quandle can be associated with any knot, and this serves as a complete invariant for tame knots. However, such a classification of tame knots heuristically seemed to be unsatisfactory, due to the apparent difficulty of the quandle isomorphism problem. Our result confirms this view, showing that, from a set-theoretic perspective, classifying tame knots by quandles replaces one problem with (a special case of) a much harder problem.

Derivations and Automorphism Group of Original Deformative Schrödinger-Virasoro Algebra

2015 ◽  
Vol 22 (03) ◽  
pp. 517-540 ◽  
Author(s):  
Qifen Jiang ◽  
Song Wang
Keyword(s):  
Automorphism Group ◽  
Lie Algebra ◽  
Direct Product ◽  
Virasoro Algebra ◽  
Witt Algebra ◽  
Tensor Density ◽  
Derivation Algebra

In this paper, we determine the derivation algebra and the automorphism group of the original deformative Schrödinger-Virasoro algebra, which is the semi-direct product Lie algebra of the Witt algebra and its tensor density module Ig(a,b).

Mixing property of $$C_0$$ C 0 -semigroup of left multiplication operators

2015 ◽  
Vol 8 (4) ◽  
pp. 257-263 ◽  
Author(s):  
Ali Reza Sazegar ◽  
Amanollah Assadi ◽  
Omid Rabieimotlagh
Keyword(s):  
Multiplication Operators ◽  
Left Multiplication ◽  
Mixing Property

scholarly journals Classification of irreducible Harish-Chandra modules over generalized Virasoro algebras

2012 ◽  
Vol 55 (3) ◽  
pp. 697-709 ◽  
Author(s):  
Xiangqian Guo ◽  
Rencai Lu ◽  
Kaiming Zhao
Keyword(s):  
Direct Summand ◽  
Virasoro Algebra ◽  
Additive Subgroup ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Irreducible Modules ◽  
Intermediate Series ◽  
Complex Number Field ◽  
Weight Modules

AbstractLet G be an arbitrary non-zero additive subgroup of the complex number field ℂ, and let Vir[G] be the corresponding generalized Virasoro algebra over ℂ. In this paper we determine all irreducible weight modules with finite-dimensional weight spaces over Vir[G]. The classification strongly depends on the index group G. If G does not have a direct summand isomorphic to ℤ (the integers), then such irreducible modules over Vir[G] are only modules of intermediate series whose weight spaces are all one dimensional. Otherwise, there is one further class of modules that are constructed by using intermediate series modules over a generalized Virasoro subalgebra Vir[G0] of Vir[G] for a direct summand G0 of G with G = G0 ⊕ ℤb, where b ∈ G \ G0. This class of irreducible weight modules do not have corresponding weight modules for the classical Virasoro algebra.

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