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Generators and relations are given for the subalgebra of cocommutative elements in the quantized coordinate rings of the classical groups, where the deformation parameter q is transcendental. This is a ring theoretic formulation of the well…

Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \u its Lie algebra. We prove the separability of orbit maps and the…

The category of all $k$-algebras with a bilinear form, whose objects are all pairs $(R,b)$ where $R$ is a $k$-algebra and $b\colon R\times R\to k$ is a bilinear mapping, is equivalent to the category of unital $k$-algebras $A$ for which the…

Let $\mathcal U_\hbar(\hat{\mathfrak g})$ be the untwisted quantum affinization of a symmetrizable quantum Kac-Moody algebra $\mathcal U_\hbar({\mathfrak g})$. For $\ell\in\mathbb C$, we construct an $\hbar$-adic quantum vertex algebra…

We provide an alternative approach to the Faddeev-Reshetikhin-Takhtajan presentation of the quantum group U_q(g), with L-operators as generators and relations ruled by an R-matrix. We look at U_q(g) as being generated by the quantum Borel…

By means of graphical calculus we prove that, over fields of characteristic zero, any bialgebra deformation of the universal enveloping algebra of the algebra of traceless octonions satisfying the dual of the left and right Moufang…

We first prove that the K-theoretic Hall algebra of a preprojective algebra of affine type is isomorphic to the positive half of a quantum toroidal quantum group. An essential step consists to deform the K-theoretic Hall algebra so that the…

Multi-parameter versions U_p(g) and C_p[G] of the standard quantum groups U_q(g) and C_q[G] are considered where G is a semi-simple connected complex algebraic group and g is the Lie algebra of G. The primitive spectrum of C_p[G] is…

We study the algebra $U_{\zeta}$ obtained via Lusztig's `integral' form [Lu 1, 2] of the generic quantum algebra for the Lie algebra $\frak {g=sl}_2$ modulo the two-sided ideal generated by $K^l-1$. We show that $U_{\zeta}$ is a smash…

Let $A_n$ be an $n$-dimensional algebra with zero multiplication over a field $K$ of characteristic $0$. Then its universal (multiplicative) enveloping algebra $U_n$ in the variety of left-symmetric algebras is a homogeneous quadratic…

A $C^*$-algebra satisfies the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet if it is equivalent in Kasparov's $KK$-theory to a commutative $C^*$-algebra. This paper is motivated by the problem of establishing the range of…

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…

The two-parametric quantum superalgebra $U_{p,q}[gl(2/1)]$ is consistently defined. A construction procedure for induced representations of $U_{p,q}[gl(2/1)]$ is described and allows us to construct explicitly all (typical and nontypical)…

Let $G$ be the Lie group $SO_e(4,1)$, with maximal compact subgroup $K = S(O(4) \times O(1))_e\cong SO(4)$. Let $\mathfrak{g}=\mathfrak{so}(5,\mathbb{C})$ be the complexification of the Lie algebra $\mathfrak{g}_0 = \mathfrak{so}(4,1)$ 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. Special values of the parameters are characterized by the…

Let $\frak{g}$ be a finite dimensional simple complex Lie algebra and $U=U_q(\frak{g})$ the quantized enveloping algebra (in the sense of Jantzen) with $q$ being generic. In this paper, we show that the center $Z(U_q(\frak{g}))$ of the…

We revisit the procedure of deformation of $C^*$-algebras via coactions of locally compact groups and extend the methods to cover deformations for maximal, reduced, and exotic coactions for a given group $G$ and circle-valued Borel…

Let $L$ denote a right-invariant sub-Laplacian on an exponential, hence solvable Lie group $G$, endowed with a left-invariant Haar measure. Depending on the structure of $G$, and possibly also that of $L$, $L$ may admit differentiable…

A triangular quantum deformation of $ osp(2/1) $ from the classical $r$-matrix including an odd generator is presented with its full Hopf algebra structure. The deformation maps, twisting element and tensor operators are considered for the…

Let $\mathfrak g$ be a complex simple Lie algebra and let $U_{\zeta}({\mathfrak g})$ be the corresponding Lusztig ${\mathbb Z}[q,q^{-1}]$-form of the quantized enveloping algebra specialized to an $\ell$th root of unity. Moreover, let…