Point Group Invariants in the $U_{qp}(u(2))$ Quantum Algebra Picture

In this paper, we study and classify Hilbert space representations of cross product *-algebras of the quantized enveloping algebra $U_q(e_2)$ with the coordinate algebras $O(E_q(2))$ of the quantum motion group and $O(\C_q)$ of the complex plane, and of the quantized enveloping algebra $U_q(su_{1,1})$ with the coordinate algebras $O(SU_q(1,1))$ of the quantum group $SU_q(1,1)$ and $O(U_q)$ of the quantum disc. Invariant positive functionals and the corresponding Heisenberg re...

This is an introduction to Quantum Integrability and Quantum Groups, a special issue collection of articles published in Journal of Physics A in memory of Petr P. Kulish. A list of Kulish's publications is included.

We address the study of multiparameter quamtum groups (=MpQG's) at roots of unity, namely quantum universal enveloping algebras $ U_{\boldsymbol{\rm q}}(\mathfrak{g}) $ depending on a matrix of parameters $ \boldsymbol{\rm q} = {\big( q_{ij} \big)}_{i, j \in I} \, $. This is performed via the construction of quantum root vectors and suitable "integral forms" of $ U_{\boldsymbol{\rm q}}(\mathfrak{g}) \, $, a \textsl{restricted one} - generated by quantum divided powers and qua...

These notes contain an introduction to the theory of complex semisimple quantum groups. Our main aim is to discuss the classification of irreducible Harish-Chandra modules for these quantum groups, following Joseph and Letzter. Along the way we cover extensive background material on quantized universal enveloping algebras and explain connections to the analytical theory in the setting of locally compact quantum groups.

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 earlier paper I showed how the representation theory of this group over the real numbers gives rise to much of the structure of the standard model of particle physics, but with a number of added twists. In this theory the group is quantised, bu...

Quantum Lie algebras are generalizations of Lie algebras whose structure constants are power series in $h$. They are derived from the quantized enveloping algebras $\uqg$. The quantum Lie bracket satisfies a generalization of antisymmetry. Representations of quantum Lie algebras are defined in terms of a generalized commutator. In this paper the recent general results about quantum Lie algebras are introduced with the help of the explicit example of $(sl_2)_h$.

In the Hopf algebra GL_{p,q}(2) the determinant is central iff p=q. In this case we put determinant to be equal to 1 to get SL_q(2). In this paper I consider the case when p/q is a root of unity; and, consequently, a power of the determinant is central.

We define for real $q$ a unital $*$-algebra $U_q(\mathfrak{sl}(2,\mathbb{R}))$ quantizing the universal enveloping $*$-algebra of $\mathfrak{sl}(2,\mathbb{R})$. The $*$-algebra $U_q(\mathfrak{sl}(2,\mathbb{R}))$ is realized as a $*$-subalgebra of the Drinfeld double of $U_q(\mathfrak{su}(2))$ and its dual Hopf $*$-algebra $\mathcal{O}_q(SU(2))$, generated by the equatorial Podle\'s sphere coideal $*$-subalgebra $\mathcal{O}_q(K\backslash SU(2))$ of $\mathcal{O}_q(SU(2))$ and ...

In this article, we obtain a complete list of inequivalent irreducible representations of the compact quantum group $U_q(2)$ for non-zero complex deformation parameters $q$, which are not roots of unity. The matrix coefficients of these representations are described in terms of the little $q$-Jacobi polynomials. The Haar state is shown to be faithful and an orthonormal basis of $L^2(U_q(2))$ is obtained. Thus, we have an explicit description of the Peter-Weyl decomposition of...

In this paper we describe certain homological properties and representations of a two-parameter quantum enveloping algebra $U_{g,h}$ of ${\frak {sl}}(2)$, where $g,h$ are group-like elements.