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

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.

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...

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...

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...

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...

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...

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.

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.

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.

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...