Related papers: Reynolds Lie bialgebras
Representations and relative Rota-Baxter operators with respect to representations of Hom-Leibniz Poisson algebras are introduced and studied. Some characterizations of these operators are obtained. The notion of matched pair and Nijenhuis…
The representation theory of the Drinfeld doubles of dihedral groups is used to solve the Yang-Baxter equation. Use of the 2-dimensional representations recovers the six-vertex model solution. Solutions in arbitrary dimensions, which are…
This paper studies super $r$-matrices and operator forms of the super classical Yang-Baxter equation. First by a unified treatment, the classical correspondence between $r$-matrices and $\mathcal{O}$-operators is generalized to a…
All possible Lie bialgebra structures on the harmonic oscillator algebra are explicitly derived and it is shown that all of them are of the coboundary type. A non-standard quantum oscillator is introduced as a quantization of a triangular…
In this paper, first we introduce the concept of modified Rota-Baxter Lie-Yamaguti algebras. Then the cohomology of a modified Rota-Baxter Lie-Yamaguti algebra with coefficients in a suitable representation is established. As applications,…
The goal of this paper is to present an algebraic approach to the basic results of the theory of linear recurrence relations. This approach is based on the ideas from the theory of representations of one endomorphisms (a special case of…
We introduce the notion of quasi-triangular anti-dendriform bialgebras, which can be induced by the solutions of the AD-YBE whose symmetric parts are invariant. A factorizable anti-dendriform bialgebra leads to a factorization of the…
We introduce a bicomplex which computes the triple cohomology of Lie--Rinehart algebras. We prove that the triple cohomology is isomorphic to the Rinehart cohomology \cite{Ri} provided the Lie--Rinehart algebra is projective over the…
We consider some algebraic and geometric aspects of the theory of integrable systems in finite dimensions, associated with the existence of a classical $r$-matrix, first introduced by Sklyanin as the classical analogue of the quantum…
We introduce the notion of associative (BiHom-)Yang-Baxter pair of weight $(\lambda,\gamma)$ which can provide the solution to the double curved Rota-Baxter (BiHom-)system. Equivalent characterizations of (quasitriangular) covariant…
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 study Lie-Rinehart algebra structures in the framework provided by a duality pairing of modules over a unital commutative associative algebra. Thus, we construct examples of Lie brackets corresponding to a fixed anchor map whose image is…
A Malcev-Poisson algebra is a Malcev algebra together with a commutative associative algebra structure related by a Leibniz rule. In this paper, we introduce the notion of Malcev-Poisson bialgebra as an analogue of a Malcev bialgebra and…
We give a new type of Schur-Weyl duality for the representations of a family of quantum subgroups and their centralizer algebra. We define and classify singly-generated, Yang-Baxter relation planar algebras. We present the skein theoretic…
The notion of a matched pair of Lie algebras was introduced in the study of Lie bialgebras and Poisson-Lie groups. In this paper, we introduce representations and cohomology of a matched pair of Lie algebras. We show that there is a…
The purpose of the present paper is to investigate cohomologies and deformations of weighted Rota-Baxter Lie algebras as well as weighted Rota-Baxter associative algebras with derivations. First we introduce a notion of weighted Rota-Baxter…
The standard Lie bialgebra structure on an affine Kac-Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. In this paper we classify all classical twists of the induced Lie bialgebra…
Xu introduced a family of root-tree-diagram nilpotent Lie algebras of differential operators, in connection with evolution partial differential equations. We generalized his notion to more general oriented tree diagrams. These algebras are…
This paper addresses the representation theory of the insertion-elimination Lie algebra, a Lie algebra that can be naturally realized in terms of tree-inserting and tree-eliminating operations on rooted trees. The insertion-elimination…
The Yang-Mills equations are formulated in the form of generalized Maurer-Cartan equations, such that the corresponding algebraic operations are shown to satisfy the defining relations of homotopy Lie superalgebra.