Related papers: The elliptic GL(n) dynamical quantum group as an h…
The quantized universal enveloping algebra U_q(gl(n)) has two integral forms - over Z[q,q^{-1}] - the restricted (by Lusztig) and the unrestricted (by De Concini and Procesi) one. Dually, the quantum function algebra F_q[GL(n)] has two…
Notions of an (H, X)-bialgebroid and of its dynamical representation are proposed. The dynamical representations of each (H, X)-bialgebroid form a tensor category. Every dynamical Yang-Baxter map R(lambda) satisfying suitable conditions, a…
An exterior derivative, inner derivation, and Lie derivative are introduced on the quantum group $GL_{q}(N)$. $SL_{q}(N)$ is then found by constructing matrices with determinant unity, and the induced calculus is found.
We study topological Hopf algebras that are holomorphically finitely generated (HFG) as Fr\'echet Arens--Micheal algebras in the sense of Pirkovskii. Some of them, but not all, can be obtained from affine Hopf algebras by applying the…
We give presentations, in terms of the generators and relations, for the reflection equation algebras of type $GL_n$ and $SL_n$, i.e., the covariantized algebras of the dual Hopf algebras of the small quantum groups of $\mathfrak{gl}_n$ and…
Dynamical quantum groups were introduced by Etingof and Varchenko in connection with the dynamical quantum Yang-Baxter equation, and measured quantum groupoids were introduced by Enock, Lesieur and Vallin in their study of inclusions of…
We report some observations concerning two well-known approaches to construction of quantum groups. Thus, starting from a bialgebra of inhomogeneous type and imposing quadratic, cubic or quartic commutation relations on a subset of its…
We establish a bijective correspondence between gauge equivalence classes of dynamical twists in a finite-dimensional Hopf algebra $H$ based on a finite abelian group $A$ and equivalence classes of pairs $(K, \{V_{\lambda}\}_{\lambda\in…
For a finite dimensional simple Lie algebra g, the standard universal solution R(x) in $U_q(g)^{\otimes 2}$ of the Quantum Dynamical Yang--Baxter Equation can be built from the standard R--matrix and from the solution F(x) in…
We prove that the structure algebra of a Bruhat moment graph of a finite real root system is a Hopf algebroid with respect to the Hecke and the Weyl actions. We introduce new techniques (reconstruction and push-forward formula of a product,…
Let $W$ be a Coxeter group. The goal of the paper is to construct new Hopf algebras that contain Hecke algebras $H_{\bf q}(W)$ as (left) coideal subalgebras. Our Hecke-Hopf algebras ${\bf H}(W)$ have a number of applications. In particular…
We describe a Hopf algebraic approach to the Grothendieck ring of representations of subgroups $H_\pi$ of the general linear group GL(n) which stabilize a tensor of Young symmetry $\{\pi\}$. It turns out that the representation ring of the…
We expose the elliptic quantum groups in the Drinfeld realization associated with both the affine Lie algebra \g and the toroidal algebra \g_tor. There the level-0 and level \not=0 representations appear in a unified way so that one can…
After reviewing the recent results on the Drinfeld realization of the face type elliptic quantum group B_{q,lambda}(sl_N^) by the elliptic algebra U_{q,p}(sl_N^), we investigate a fusion of the vertex operators of U_{q,p}(sl_N^). The basic…
We introduce a family of dynamical Hopf algebroids $U_{q,p}(g,X_l)$ depending on a complex parameter $q$, a formal parameter $p$, a set $g$ of structure functions satisfying the so-called Ding-Iohara condition, and a finite root system of…
Starting from a recently-introduced algebraic structure on spin foam models, we define a Hopf algebra by dividing with an appropriate quotient. The structure, thus defined, naturally allows for a mirror analysis of spin foam models with…
We introduce the universal complex elliptic genus phi_ell as the ring homomorphism from the complex cobordism ring Omega^U to the polynomial ring C[A,B,C,D] associated to the characteristic power series Q(x)=x/f(x), where f is the solution…
We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra.…
A review of the multiparametric linear quantum group GL_qr(N), its real forms, its dual algebra U(gl_qr(N)) and its bicovariant differential calculus is given in the first part of the paper. We then construct the (multiparametric) linear…
Take the matrix Lie superalgebra $gl_{N|N}$ with the standard generators $E_{ij}$ where $i,j=-N,...,-1,1,...,N$. Define an involutive automorphism of $gl_{N|N}$ by sending $E_{ij}$ to $E_{-i,-j}$. Then the corresponding twisted subalgebra…