Related papers: Lectures on Quantum Groups and Yang-Baxter Equatio…
We define generalized bialgebras and Hopf algebras and on this basis we introduce quantum categories and quantum groupoids. The quantization of the category of linear (super)spaces is constructed. We establish a criterion for the classical…
The quantum dynamical Yang-Baxter (or Gervais-Neveu-Felder) equation defines an R-matrix R(p), where $p$ stands for a set of mutually commuting variables. A family of SL(n)-type solutions of this equation provides a new realization of the…
Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of "degroupoidification": a…
Multiparametric quantum deformations of $gl(2)$ are studied through a complete classification of $gl(2)$ Lie bialgebra structures. From them, the non-relativistic limit leading to harmonic oscillator Lie bialgebras is implemented by means…
Recently V.Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested to consider ``set-theoretical'' solutions of the quantum Yang-Baxter equation, i.e. solutions given by a permutation $R$ of the set…
This is a review article in which we will introduce, in a unifying fashion and with more intermediate steps in some difficult calculations, two infinite-dimensional Lie algebras of quantum matrix models, one for the open string sector and…
We quantize the regularity properties of classical graphs that determine spin models for singly-generated Yang-Baxter planar algebras, including the Kauffman polynomial, and construct explicit examples. A source of examples comes from…
The Gauss decomposition of quantum groups and supergroups are considered. The main attention is paid to the R-matrix formulation of the Gauss decomposition and its properties as well as its relation to the contraction procedure. Duality…
The meaning of quantum group transformation properties is discussed in some detail by comparing the (co)actions of the quantum group with those of the corresponding Lie group, both of which have the same algebraic (matrix) form of the…
We introduce a new concept of quasi-Yang-Baxter algebras. The quantum quasi-Yang-Baxter algebras being simple but non-trivial deformations of ordinary algebras of monodromy matrices realize a new type of quantum dynamical symmetries and…
Quantum integrable spin chains are known to possess a large family of long-range deformations generated by the local, boost and bilocal operators. Although these deformations are well-understood on the level of the pairwise commuting…
In this paper it is shown that a quantum observable algebra, the Heisenberg-Weyl algebra, is just given as the Hopf algebraic dual to the classical observable algebra over classical phase space and the Plank constant is included in this…
In this article we use a parametrized version of the FRT construction to construct two new coquasitriangular Hopf algebras. The first one, $\widehat{SL_q(2)}$, is a quantization of the coordinate ring on affine $SL(2)$. We show that there…
A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from \fun\ to \uqg\ , given by elements of the pure braid group. These operators --- the `reflection matrix' $Y \equiv…
We introduce the notion of a braided Lie algebra consisting of a finite-dimensional vector space $\CL$ equipped with a bracket $[\ ,\ ]:\CL\tens\CL\to \CL$ and a Yang-Baxter operator $\Psi:\CL\tens\CL\to \CL\tens\CL$ obeying some axioms. We…
We introduce and study the Koszul complex for a Hecke $R$-matrix. Its cohomologies, called the Berezinian, are used to define quantum superdeterminant for a Hecke $R$-matrix. Their behaviour with respect to Hecke sum of $R$-matrices is…
We give a pedagogical introduction to the differential calculus on quantum groups by stressing at all stages the connection with the classical case ($q \rightarrow 1$ limit). The Lie derivative and the contraction operator on forms and…
This paper is a short account of the construction of a new class of the infinite-dimensional representations of the quantum groups. The examples include finite-dimensional quantum groups $U_q(\mathfrak{g})$, Yangian $Y(\mathfrak{g})$ and…
There is a unique finite group that lies inside the 2-dimensional unitary group but not in the special unitary group, and maps by the symmetric square to an irreducible subgroup of the 3-dimensional real special orthogonal group. In an…
We formulate quantum group Riemannian geometry as a gauge theory of quantum differential forms. We first develop (and slightly generalise) classical Riemannian geometry in a self-dual manner as a principal bundle frame resolution and a dual…