Related papers: Flag varieties and the Yang-Baxter equation
It is shown how Yang-Baxter maps may be directly obtained from classical counterparts of the star-triangle relations and quantum Yang-Baxter equations. This is based on reinterpreting the latter equation and its solutions which are given in…
This work is intended as an attempt to extend the notion of bialgebra for Lie algebras to Leibniz algebras and also, the correspondence between the Leibniz bialgebras and its dual is investigated. Moreover, the coboundary Leibniz…
Diagram algebras (e.g. graded braid groups, Hecke algebras, Brauer algebras) arise as tensor power centralizer algebras, algebras of commuting operators for a Lie algebra action on a tensor space. This work explores centralizers of the…
The Yang-Baxter Equation (YBE) plays a crucial role for studying integrable many-body quantum systems. Many known YBE solutions provide various examples ranging from quantum spin chains to superconducting systems. Models of solvable…
Braided algebras are algebraic structures consisting of an algebra endowed with a Yang-Baxter operator, satisfying some compatibility conditions.Yang-Baxter Hochschild cohomology was introduced by the authors to classify infinitesimal…
A new type of algebras that represent a generalization of both quantum groups and braided groups is defined. These algebras are given by a pair of solutions of the Yang--Baxter equation that satisfy some additional conditions. Several…
Quite recently, a ``coloured'' extension of the Yang-Baxter equation has appeared in the literature and various solutions of it have been proposed. In the present contribution, we introduce a generalization of Hopf algebras, to be referred…
In this paper we introduce the notion of a geometric associative r-matrix attached to a genus one fibration with a section and irreducible fibres. It allows us to study degenerations of solutions of the classical Yang-Baxter equation using…
The theory of Lie algebras can be categorified starting from a new notion of "2-vector space", which we define as an internal category in Vect. There is a 2-category 2Vect having these 2-vector spaces as objects, "linear functors" as…
We introduce some basic concepts for Jacobi-Jordan algebras such as: representations, crossed products or Frobenius/metabelian/co-flag objects. A new family of solutions for the quantum Yang-Baxter equation is constructed arising from any…
In this paper, we estimate the Hilbert-Kunz multiplicity of the (extended) Rees algebras in terms of some invariants of the base ring. Also, we give an explicit formula for the Hilbert-Kunz multiplicities of Rees algebras over Veronese…
This paper gives an algebraic presentation of an algebra called the fused permutations algebra in the one-boundary case. It is obtained through a detailed study of the degenerate cyclotomic Hecke algebra. In particular, we prove that the…
By applying a recent construction of free Baxter algebras, we obtain a new class of Hopf algebras that generalizes the classical divided power Hopf algebra. We also study conditions under which these Hopf algebras are isomorphic.
Some new algebraic structures related to the coloured Yang-Baxter equation, and termed coloured Hopf algebras, are reviewed. Coloured quantum universal enveloping algebras of Lie algebras are defined in this context. An extension to the…
We investigate the possible regular solutions of the boundary Yang-Baxter equation for the vertex models associated to the D_{n+1}^(2) affine Lie algebra. We have classified them in terms of three types of K-matrices. The first one have n+2…
In this paper, we initiate the study of the interplay between $k$-graphs and the Yang-Baxter equation. For this, we provide two very different perspectives. One one hand, we show that the set of all set-theoretic solutions of the…
We introduce a notion of a para-K\"{a}hler strict Lie 2-algebra, which can be viewed as a categorification of a para-K\"{a}hler Lie algebra. In order to study para-K\"{a}hler strict Lie 2-algebra in terms of strict pre-Lie 2-algebras, we…
A cohomology theory for multiplications and comultiplications of Frobenius algebras is developed in low dimensions in analogy with Hochschild cohomology of bialgebras based on deformation theory. Concrete computations are provided for key…
We introduce some braided varieties -- braided orbits -- by considering quotients of the so-called Reflection Equation Algebras associated with Hecke symmetries (i.e. special type solutions of the quantum Yang-Baxter equation). Such a…
We consider some bases in the Hecke algebra and exhibit certain dualities between them.