ID: math/0611200

Pairs of compatible associative algebras, classical Yang-Baxter equation and quiver representations

November 7, 2006

View on ArXiv

Similar papers 2

R-Matrices, Yetter-Drinfel$'$d Modules and Yang-Baxter Equation

August 19, 2013

84% Match
Victoria IMJ Lebed
Category Theory
K-Theory and Homology
Quantum Algebra

In the first part we recall two famous sources of solutions to the Yang-Baxter equation -- R-matrices and Yetter-Drinfel$'$d (=YD) modules -- and an interpretation of the former as a particular case of the latter. We show that this result holds true in the more general case of weak R-matrices, introduced here. In the second part we continue exploring the ''braided'' aspects of YD module structure, exhibiting a braided system encoding all the axioms from the definition of YD m...

Find SimilarView on arXiv

Affine Actions and the Yang-Baxter Equation

July 12, 2016

84% Match
Dilian Yang
Quantum Algebra

In this paper, the relations between the Yang-Baxter equation and affine actions are explored in detail. In particular, we classify solutions of the Yang-Baxter equations in two ways: (i) by their associated affine actions of their structure groups on their derived structure groups, and (ii) by the C*-dynamical systems obtained from their associated affine actions. On the way to our main results, several other useful results are also obtained.

Find SimilarView on arXiv

Analogs of q-Serre relations in the Yang-Baxter algebras

August 17, 1998

84% Match
Mirko Luedde, Alexei Vladimirov
Quantum Algebra

Yang-Baxter bialgebras, as previously introduced by the authors, are shown to arise from a double crossproduct construction applied to the bialgebra R T T = T T R, E T = T E R, \Delta(T) = T \hat{\otimes} T, \Delta(E) = E \hat{\otimes} T + 1 \hat{\otimes} E and its skew dual, with R being a numerical matrix solution of the Yang-Baxter equation. It is further shown that a set of relations generalizing q-Serre ones in the Drinfeld-Jimbo algebras U_q(g) can be naturally imposed ...

Find SimilarView on arXiv

Lectures on the dynamical Yang-Baxter equations

August 13, 1999

84% Match
Pavel Etingof, Olivier Schiffmann
Quantum Algebra

This paper contains a systematic and elementary introduction to a new area of the theory of quantum groups -- the theory of the classical and quantum dynamical Yang-Baxter equations. It arose from a minicourse given by the first author at MIT in the Spring of 1999, when the second author extended and improved his lecture notes of this minicourse. The quantum dynamical Yang-Baxter equation is a generalization of the ordinary quantum Yang-Baxter equation, considered in a phys...

Find SimilarView on arXiv

Operator forms of nonhomogeneous associative classical Yang-Baxter equation

July 21, 2020

84% Match
Chengming Bai, Xing Gao, ... , Zhang Yi
Quantum Algebra
Mathematical Physics
Rings and Algebras

This paper studies operator forms of the nonhomogeneous associative classical Yang-Baxter equation (nhacYBe), extending and generalizing such studies for the classical Yang-Baxter equation and associative Yang-Baxter equation that can be tracked back to the works of Semonov-Tian-Shansky and Kupershmidt on Rota-Baxter Lie algebras and $\mathcal{O}$-operators. In general, solutions of the nhacYBe are characterized in terms of generalized $\mathcal{O}$-operators. The characteriz...

Find SimilarView on arXiv

Trigonometric solutions of the associative Yang-Baxter equation

December 18, 2002

83% Match
Travis Schedler
Quantum Algebra

We classify trigonometric solutions to the associative Yang-Baxter equation (AYBE) for A = Mat_n, the associative algebra of n-by-n matrices. The AYBE was first presented in a 2000 article by Marcelo Aguiar and also independently by Alexandre Polishchuk. Trigonometric AYBE solutions limit to solutions of the classical Yang-Baxter equation. We find that such solutions of the AYBE are equal to special solutions of the quantum Yang-Baxter equation (QYBE) classified by Gerstenhab...

Find SimilarView on arXiv

Classical Yang-Baxter equation and the $A_{\infty}$-constraint

August 21, 2000

83% Match
Alexander Polishchuk
Algebraic Geometry

We show that elliptic solutions of the classical Yang-Baxter equation can be obtained from triple Massey products on elliptic curve. We introduce the associative version of this equation which has two spectral parameters and construct its elliptic solutions. We also study some degenerations of these solutions.

Find SimilarView on arXiv

Back to baxterisation

August 9, 2017

83% Match
N. Crampe, E. Ragoucy, M. Vanicat
Mathematical Physics
Quantum Algebra

In the continuity of our previous paper arXiv:1509.05516, we define three new algebras, $A_{n}(a,b,c)$, $B_{n}$ and $C_{n}$, that are close to the braid algebra. They allow to build solutions to the braided Yang-Baxter equation with spectral parameters. The construction is based on a baxterisation procedure, similar to the one used in the context of Hecke or BMW algebras. The $A_{n}(a,b,c)$ algebra depends on three arbitrary parameters, and when the parameter $a$ is set to ...

Find SimilarView on arXiv

Quadratic Poisson brackets and Drinfeld theory for associative algebras

March 31, 1995

83% Match
A. A. Balinsky, Yu. M. Burman
Quantum Algebra

The paper is devoted to the Poisson brackets compatible with multiplication in associative algebras. These brackets are shown to be quadratic and their relations with the classical Yang--Baxter equation are revealed. The paper also contains a description of Poisson Lie structures on Lie groups whose Lie algebras are adjacent to an associative structure.

Find SimilarView on arXiv

Algebra Structures Arising from Yang-Baxter Systems

May 6, 2010

83% Match
Barbu R. Berceanu, Florin F. Nichita, Calin Popescu
Quantum Algebra

Yang-Baxter operators from algebra structures appeared for the first time in [16], [17] and [8]. Later, Yang-Baxter systems from entwining structures were constructed in [5]. In this paper we show that an algebra factorisation can be constructed from a Yang-Baxter system.

Find SimilarView on arXiv