Related papers: On quantization of r-matrices for Belavin-Drinfeld…
An integrable deformation of the known integrable model of two interacting p-dimensional and q-dimensional spherical tops is considered. After reduction this system gives rise to the generalized Lagrange and the Kowalevski tops. The…
In this paper we generalize Drinfeld's twisted quantum affine algebras to construct twisted quantum algebras for all simply-laced generalized Cartan matrices and present their vertex representation realizations.
The universal R-matrix of two-parameter quantum general linear supergroups is computed explicitly based on the RTT realization of Faddeev--Reshetikhin--Takhtajan.
We show that some factors of the universal R-matrix generate a family of twistings for the standard Hopf structure of any quantized contragredient Lie (super)algebra of finite growth. As an application we prove that any two isomorphic…
For a finite dimensional simple complex Lie algebra $\mathfrak{g}$, Lie bialgebra structures on $\mathfrak{g}[[u]]$ and $\mathfrak{g}[u]$ were classified by Montaner, Stolin and Zelmanov. In our paper, we provide an explicit algorithm to…
Holonomy R-matrices parametrized by finite-dimensional representations are constructed for quantized universal enveloping algebras of simple Lie algebras at roots of 1.
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…
In this paper, we give matrix formulae for non-orientable surfaces that provide the Laurent expansion for quasi-cluster variables, generalizing the orientable surface matrix formulae by Musiker-Williams. We additionally use our matrix…
We discussed twisted quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S.Zakrzewski classification can be presented as a sum of subordinated…
We show how some classical r-matrices for the D=4 Poincare algebra can be supersymmetrized by an addition of part depending on odd supercharges. These r-matrices for D=4 super-Poincare algebra can be presented as a sum of the so-called…
We use a Riemannian (or pseudo-Riemannian) geometric framework to formulate the theory of the classical r-matrix for integrable systems. In this picture the r-matrix is related to a fourth rank tensor, named the r-tensor, on the…
We classify in this paper Poisson structures on modules over semisimple Lie algebras arising from classical r-matrices. We then study their quantizations and the relation to classical invariant theory.
We discussed quantum deformations of D=4 Lorentz and Poincare algebras. In the case of Poincare algebra it is shown that almost all classical r-matrices of S. Zakrzewski classification correspond to twisted deformations of Abelian and…
Four results are given that address the existence, ambiguities and construction of a classical R-matrix given a Lax pair. They enable the uniform construction of R-matrices in terms of any generalized inverse of $ad L$. For generic $L$ a…
For a simple Lie algebra L of type A, D, E we show that any Belavin-Drinfeld triple on the Dynkin diagram of L produces a collection of Drinfeld twists for Lusztig's small quantum group u_q(L). These twists give rise to new…
We describe a quantum Lie algebra based on the Cremmer-Gervais R-matrix. The algebra arises upon a restriction of an infinite-dimensional quantum Lie algebra.
Applications of algebras in physics are related to the connection of measurable observables to relevant elements of the algebras, usually the generators. However, in the determination of the generators in Lie algebras there is place for…
The Universal T-matrix is the capstone of the structure that consists of a quantum group and its dual, and the central object from which spring the T-matrices (monodromies) of all the associated integrable models. A closed expression is…
We discuss quantum dynamical elliptic R-matrices related to arbitrary complex simple Lie group G. They generalize the known vertex and dynamical R-matrices and play an intermediate role between these two types. The R-matrices are defined by…
We study classical twists of Lie bialgebra structures on the polynomial current algebra $\mathfrak{g}[u]$, where $\mathfrak{g}$ is a simple complex finite-dimensional Lie algebra. We focus on the structures induced by the so-called…