ID: math/9808075

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

August 17, 1998

View on ArXiv
Mirko Luedde, Alexei Vladimirov
Mathematics
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 on Yang-Baxter algebras from the requirement of non-degeneracy of the pairing.

Similar papers 1

Exotic bialgebras : non-deformation quantum groups

July 6, 2004

86% Match
D. Arnaudon, A. Chakrabarti, ... , Mihov S. G.
Quantum Algebra

In the classification of solutions of the Yang--Baxter equation, there are solutions that are not deformations of the trivial solution (essentially the identity). We consider the algebras defined by these solutions, and the corresponding dual algebras. We then study the representations of the latter. We are also interested in the Baxterisation of these $R$-matrices and in the corresponding quantum planes.

Find SimilarView on arXiv

Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation

March 7, 2022

86% Match
Anastasia Doikou, Alexandros Ghionis, Bart Vlaar
Quantum Algebra
Mathematical Physics

We examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation and their q-analogues. After providing some universal results on quasi-bialgebras and admissible Drinfeld twists we show that the quantum algebras produced from set-theoretic solutions and their q-analogues are in fact quasi-triangular quasi-bialgebras. Specific illustrative examples compatible with our generic findings are worked out. In the q-d...

Find SimilarView on arXiv

Bialgebras, Frobenius algebras and associative Yang-Baxter equations for Rota-Baxter algebras

December 21, 2021

85% Match
Chengming Bai, Li Guo, Tianshui Ma
Quantum Algebra
Rings and Algebras
Representation Theory

Rota-Baxter operators and bialgebras go hand in hand in their applications, such as in the Connes-Kreimer approach to renormalization and the operator approach to the classical Yang-Baxter equation. We establish a bialgebra structure that is compatible with the Rota-Baxter operator, called the Rota-Baxter antisymmetric infinitesimal (ASI) bialgebra. This bialgebra is characterized by generalizations of matched pairs of algebras and double constructions of Frobenius algebras t...

Find SimilarView on arXiv

Yang-Baxter equations and relative Rota-Baxter operators for left-Alia algebras associated to invariant theory

June 27, 2024

85% Match
Kang Chuangchuang, Liu Guilai, Shizhuo Yu
Rings and Algebras

Left-Alia algebras are a class of algebras with symmetric Jacobi identities. They contain several typical types of algebras as subclasses, and are closely related to the invariant theory. In this paper, we study the construction theory of left-Alia bialgebras. We introduce the notion of the left-Alia Yang-Baxter equation. We show that an antisymmetric solution of the left-Alia Yang-Baxter equation gives rise to a left-Alia bialgebra that we call triangular. The notions of rel...

Find SimilarView on arXiv

Non-associative algebras, Yang-Baxter equations and quantum computers

August 16, 2014

85% Match
Radu Iordanescu, Florin F. Nichita, Ion M. Nichita
Differential Geometry

Non-associtive algebras is a research direction gaining much attention these days. New developments show that associative algebras and some not-associative structures can be unified at the level of Yang-Baxter structures. In this paper, we present a unification for associative algebras, Jordan algebras and Lie algebras. The (quantum) Yang-Baxter equation and related structures are interesting topics, because they have applications in many areas of mathematics, physics and com...

Find SimilarView on arXiv

The classical Lie-Yamaguti Yang-Baxter equation and Lie-Yamaguti bialgebras

October 31, 2022

85% Match
Jia Zhao, Yu Qiao
Rings and Algebras
Quantum Algebra

In this paper, we develop the bialgebra theory for Lie-Yamaguti algebras. For this purpose, we exploit two types of compatibility conditions: local cocycle condition and double construction. We define the classical Yang-Baxter equation in Lie-Yamaguti algebras and show that a solution to the classical Yang-Baxter equation corresponds to a relative Rota-Baxter operator with respect to the coadjoint representation. Furthermore, we generalize some results by Bai in [1] and Semon...

Find SimilarView on arXiv
Jiakang Bao
Algebraic Geometry
Mathematical Physics
Representation Theory

In this note, we study possible $\mathcal{R}$-matrix constructions in the context of quiver Yangians and Yang-Baxter algebras. For generalized conifolds, we also discuss the relations between the quiver Yangians and some other Yangian algebras (and $\mathcal{W}$-algebras) in literature.

On twisting solutions to the Yang-Baxter equation

November 8, 1998

85% Match
P. P. Kulish, A. I. Mudrov
Quantum Algebra

Sufficient conditions for an invertible two-tensor $F$ to relate two solutions to the Yang-Baxter equation via the transformation $R\to F^{-1}_{21} R F$ are formulated. Those conditions include relations arising from twisting of certain quasitriangular bialgebras.

Find SimilarView on arXiv

On the Hopf algebras generated by the Yang-Baxter R-matrices

February 11, 1993

84% Match
A. A. Vladimirov
Quantum Algebra

We reformulate the method recently proposed for constructing quasitriangular Hopf algebras of the quantum-double type from the R-matrices obeying the Yang-Baxter equations. Underlying algebraic structures of the method are elucidated and an illustration of its facilities is given. The latter produces an example of a new quasitriangular Hopf algebra. The corresponding universal R-matrix is presented as a formal power series.

Find SimilarView on arXiv

Lie algebras and Yang-Baxter equations

July 5, 2011

84% Match
Florin F. Nichita
Quantum Algebra

At the previous congress (CRM 6), we reviewed the construction of Yang-Baxter operators from associative algebras, and presented some (colored) bialgebras and Yang-Baxter systems related to them. The current talk deals with Yang-Baxter operators from (G, \theta)-Lie algebras (structures which unify the Lie algebras and the Lie superalgebras). Thus, we produce solutions for the constant and the spectral-parameter Yang-Baxter equations, Yang-Baxter systems, etc. Attempting to p...

Find SimilarView on arXiv