Related papers: Generalization of the linear r-matrix formulation …
The aim of the paper is to build a universal R-matrix for the multiparameter deformation of any reductive Lie algebra. Such deformations, formulated in the recent past by Truini and Varadarajan, have the property of universality in a…
Using a geometrical approach to the quantum Yang-Baxter equation, the quantum algebra ${\cal U}_{\hbar}(sl_{2})$ and its universal quantum $R$-matrix are explicitely constructed as functionals of the associated classical $r$-matrix. In this…
This is a review/announcement of results concerning the connection between certain exactly solvable two-dimensional models of statistical mechanics, namely loop models, and the equivariant $K$-theory of the cotangent bundle of the…
In this paper, we define and study the classical $R$-matrix for vertex Lie algebra, based on which we propose to construct a new vertex Lie algebra. We give a systematic way to construct the $R$-matrix for affine Kac-Moody vertex Lie…
We describe classical top-like integrable systems arising from the quantum exchange relations and corresponding Sklyanin algebras. The Lax operator is expressed in terms of the quantum non-dynamical $R$-matrix even at the classical level,…
R-matrix is explicitly constructed for simplest representations of the Ding-Iohara-Miki algebra. The calculation is straightforward and significantly simpler than the one through the universal R-matrix used for a similar calculation in the…
Let R be a ring. A construction method for flexible quadratic algebras with scalar involution over R is presented which unifies various classical constructions in the literature, in particular those to construct composition algebras.
By applying the Hamiltonian reduction scheme we recover the R-matrix of the trigonometric and elliptic Calogero-Moser system.
We construct integrable Hamiltonian systems on $G/K$, where $G$ is a quasitriangular Poisson Lie group and $K$ is a Lie subgroup arising as the fixed point set of a group automorphism $\sigma$ of $G$ satisfying the classical reflection…
This is the central article of a series of three papers on cross product bialgebras. We present a universal theory of bialgebra factorizations (or cross product bialgebras) with cocycles and dual cocycles. We also provide an equivalent…
The present paper deals with the representation theory of the reflection equation algebra, connected with a Hecke type R-matrix. Up to some reasonable additional conditions the R-matrix is arbitrary (not necessary originated from quantum…
Quantum loop algebras generalize $U_q(\widehat{\mathfrak{g}})$ for simple Lie algebras $\mathfrak{g}$, and they include examples such as quantum affinizations of Kac-Moody Lie algebras, K-theoretic Hall algebras of quivers, and BPS algebras…
We study the interplay between double cross sum decompositions of a given Lie algebra and classical r-matrices for its semidual. For a class of Lie algebras which can be obtained by a process of generalised complexification we derive an…
The classical r-matrix structure for the generic elliptic Ruijsenaars-Schneider model is presented. It makes the integrability of this model as well as of its discrete-time version that was constructed in a recent paper manifest.
A method to construct both classical and quantum completely integrable systems from (Jordan-Lie) comodule algebras is introduced. Several integrable models based on a so(2,1) comodule algebra, two non-standard Schrodinger comodule algebras,…
In this paper we use some basic facts from the theory of (matrix) Lie groups and algebras to show that many of the classical matrix splittings used to construct stationary iterative methods and preconditioniers for Krylov subspace methods…
We consider a hierarchy of the natural type Hamiltonian systems of $n$ degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates. We give a Lax representation in terms of $2\times 2$…
Given an associative, not necessarily commutative, ring R with identity, a formal matrix calculus is introduced and developed for pairs of matrices over R. This calculus subsumes the theory of homogeneous systems of linear equations with…
A number of models of linear logic are based on or closely related to linear algebra, in the sense that morphisms are "matrices" over appropriate coefficient sets. Examples include models based on coherence spaces, finiteness spaces and…
We find the general solution to the twisting equation in the tensor bialgebra $T({\bf R})$ of an associative unital ring ${\bf R}$ viewed as that of fundamental representation for a universal enveloping Lie algebra and its quantum…