Related papers: On quantization of r-matrices for Belavin-Drinfeld…
The famous Drinfeld-Kohno theorem for simple Lie algebras states that the monodromy representation of the Knizhnik-Zamolodchikov equations for these Lie algebras expresses explicitly via R-matrices of the corresponding Drinfeld-Jimbo…
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…
R-matrices are the solutions of the Yang-Baxter equation. At the origin of the quantum group theory, they may be interpreted as intertwining operators. Recent advances have been made independently in different directions. Maulik-Okounkov…
We construct twisting elements for module algebras of restricted two-parameter quantum groups from factors of their R-matrices. We generalize the theory of Giaquinto and Zhang to universal deformation formulas for categories of module…
In this paper we explicitly attach to a geometric classical r-matrix $r$ (not necessarily unitary), a geometric (i.e., set-theoretical) quantum R-matrix $R$, which is a quantization of $r$. To accomplish this, we use the language of…
Explicit expressions for three series of $R$ matrices which are related to a ``dilute'' generalisation of the Birman--Wenzl--Murakami are presented. Of those, one series is equivalent to the quantum $R$ matrices of the $D^{(2)}_{n+1}$…
The Lie bialgebras of the (1+1) extended Galilei algebra are obtained and classified into four multiparametric families. Their quantum deformations are obtained, together with the corresponding deformed Casimir operators. For the coboundary…
We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra $H$ endowed with a universal K-matrix, i.e., a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups…
Let $G$ be a simple complex classical group and $\g$ its Lie algebra. Let $\U_\hbar(\g)$ be the Drinfeld-Jimbo quantization of the universal enveloping algebra $\U(\g)$. We construct an explicit $\U_\hbar(\g)$-equivariant quantization of…
A large family of "standard" coboundary Hopf algebras is investigated. The existence of a universal R-matrix is demonstrated for the case when the parameters are in general position. Algebraic surfaces in parameter space are characterized…
Given a simple Lie algebra $\gggg$, we consider the orbits in $\gggg^*$ which are of R-matrix type, i.e., which possess a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the so-called R-matrix bracket. We call an…
A problem of constructing quantum groups from classical r-matrices is discussed.
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…
The universal $R$-matrix of the quantum affine superalgebra associated to the Lie superalgebra $\mathfrak{gl}(1,1)$ is realized as the Casimir element of certain Hopf pairing, based on the explicit coproduct formula of all the Drinfeld loop…
We propose a method of quantization of certain Lie bialgebra structures on the polynomial Lie algebras related to quasi-trigonometric solutions of the classical Yang--Baxter equation. The method is based on an affine realization of certain…
The exotic quantum double and its universal R-matrix for quantum Yang-Baxter equation are constructed in terms of Drinfeld's quantum double theory.As a new quasi-triangular Hopf algebra, it is much different from those standard quantum…
A universal R--matrix for the quantum Heisenberg algebra h(1)q is presented. Despite of the non--quasitriangularity of this Hopf algebra, the quantum group induced from it coincides with the quasitriangular deformation already known.
The quantum deformation of the Jordanian twist F_qJ for the standard quantum Borel algebra U_q(B) is constructed. It gives the family U_qJ(B) of quantum algebras depending on parameters x and h. In a generic point these algebras represent…
The notion of classical $r$-matrix is re-examined, and a definition suitable to differential (-difference) Lie algebras, -- where the standard definitions are shown to be deficient, -- is proposed, the notion of an ${\mathcal O}$-operator.…
We define cluster $R$-matrices as sequences of mutations in triangular grid quivers on a cylinder, and show that the affine geometric $R$-matrix of symmetric power representations for the quantum affine algebra…