Related papers: Quantized function algebras at $q=0$: type $A_{n}$…
We introduce a category $\widehat{\mathcal{O}}_{\rm osc}$ of $q$-oscillator representations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_n)$. We show that $\widehat{\mathcal{O}}_{\rm osc}$ has a family of irreducible…
The phase space given by the cotangent bundle of a Lie group appears in the context of several models for physical systems. A representation for the quantum system in terms of non-commutative functions on the (dual) Lie algebra, and a…
We consider the C*-algebras O_2^q and A_2^q generated, respectively, by isometries s_1, s_2 satisfying the relation s_1^* s_2 = q s_2 s_1^* with |q| < 1 (the deformed Cuntz relation), and by isometries s_1, s_2 satisfying the relation s_2…
Quantitative algebras (QAs) are algebras over metric spaces defined by quantitative equational theories as introduced by the same authors in a related paper presented at LICS 2016. These algebras provide the mathematical foundation for…
Quantum Lie algebras $\qlie{g}$ are non-associative algebras which are embedded into the quantized enveloping algebras $U_q(g)$ of Drinfeld and Jimbo in the same way as ordinary Lie algebras are embedded into their enveloping algebras. The…
We expose a K-theoretic approach to study group C*-algebras and C*-algebraic compact quantum groups: 1. The conception of multidimensional geometric quantization and the index of group C*-algebras; 2. the entire homology of noncommutative…
Quantum groupoids are a joint generalization of groupoids and quantum groups. We propose a definition of a compact quantum groupoid that is based on the theory of C*-algebras and Hilbert bimodules. The essential point is that whenever one…
In this paper we define a new concept of quasi-convolution for analytic functions normalized by $f(0)=0$ and $f^\prime(0)=1$ in the unit disk $E=\{z\in \mathbb{C}\colon |z|<1\}$. We apply this new approach to study the closure properties of…
Aspects of the algebraic structure and representation theory of the quantum affine superalgebras with symmetrizable Cartan matrices are studied. The irreducible integrable highest weight representations are classified, and shown to be…
We introduce a new class of C^*-algebras, which is a generalization of both graph algebras and homeomorphism C^*-algebras. This class is very large and also very tractable. We prove the so-called gauge-invariant uniqueness theorem and the…
In the present paper we construct all typical finite-dimensional representations of the quantum Lie superalgebra $U_{q}[gl(2/2)]$ at generic deformation parameter $q$. As in the non-deformed case the finite-dimensional…
It is shown how to construct *-homomorphic quantum stochastic Feller cocycles for certain unbounded generators, and so obtain dilations of strongly continuous quantum dynamical semigroups on C* algebras; this generalises the construction of…
In this paper we generalize the notion of a $k$-graph into (countable) infinite rank. We then define our $C^*$-algebra in a similar way as in $k$-graph $C^*$-algebras. With this construction we are able to find analogues to the Gauge…
In this paper we define a quantum version of the ``fusion'' tensor product of two representations of an affine Kac-Moody algebra.It is replaced by what we call fusion action of the category of finite-dimensional representations of quantum…
Let p be a polynomial in one variable. It is shown that the universal C*-algebra of the relation p(x)=0, \|x\| \le C is semiprojective, residually finite-dimensional and has trivial extension group.
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…
The nonstandard q-deformation $U'_q({\rm so}_n)$ of the universal enveloping algebra $U({\rm so}_n)$ has irreducible finite dimensional representations which are a q-deformation of the well-known irreducible finite dimensional…
In the theory of C*-algebras, interesting noncommutative structures arise as deformations of the tensor product. For instance, the rotation algebra may be seen as a scalar twist deformation of the tensor product of the functions on the…
computable functions are defined by abstract finite deterministic algorithms on many-sorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable…
We introduce the notion of a generalized representation of a Jordan algebra with unit. The greneralized representation has the following properties: (1) Usual representations and Jacobson representations correspond to special cases of…