Examining the dual of an algebraic quantum group

Johan Kustermans

Home/Archive/arXiv

arXiv · funct-an/9704004

Examining the dual of an algebraic quantum group

Johan Kustermans

View original source ↗Read source document ↗Export metadata ↓

Abstract

In the first part of this paper, we implement the multiplier algebra of the dual of an algebraic quantum group (A,Delta) as a space of linear functionals on A. In the second part, we construct the universal corepresentation of (A,Delta) and use it to prove a bijective correspondence between corepresentations of (A,Delta) and homomorphisms on the dual.

Explore related subjects

funct-an math.OA math.QA q-alg

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Johan Kustermans. 1997-04-21. Examining the dual of an algebraic quantum group. https://arxiv.org/abs/funct-an/9704004

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2020

Isomorphism classes for quantum Heisenberg manifolds

We embed the quantum Heisenberg manifold in a crossed product algebra. This enables us to show that, in the irrational case, all tracial states on $\dc$ induce the same homomorphism on the K_0-group. We conclude that two irrational quantum Heisenberg manifolds $\dc$ and $D^c_{μ' ν'}$ are isomorphic if and only if the parameters $(μ,ν)$ and $(μ',ν')$ belong to the same orbit under the usual action of $GL_2(\ZZ)$ on the torus.

funct-an↗

arXiv2007

Quantum Mechanics and Operator algebras on the Hilbert ball

Cirelli, Manià and Pizzocchero generalized quantum mechanics by Kähler geometry. Furthermore they proved that any unital C$^{*}$-algebra is represented as a function algebra on the set of pure states with a noncommutative $*$-product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space ${\cal P}({\cal H})$ of a Hilbert space ${\cal H}$. The space ${\cal P}({\cal H})$ is an infinite dimensional Kähler manifold of positive constant holomorphic sectional curvature. In general, such dynamical system is constructed for a general Kähler manifold of nonzero constant holomorphic sectional curvature $c$. The Hilbert ball $B_{\cal H}$ is defined by the open unit ball in ${\cal H}$ and it is a Kähler manifold with $c<0$. We introduce the quantum mechanics on $B_{\cal H}$. As an application, we show the structure of the noncommutative function algebra on $B_{\cal H}$.

funct-an↗

arXiv2004

No More Than Mechanics. I

One can introduce so-called {\em Plain Mechanics} having an {\bf operator realization}. Then the set of one-dimension representations of this operator realization may be identified with the Classical Mechanics. Different irreducible infinite-dimension representations may be recognized as Quantum Mechanics for different $\hbar$ (the Planck constant). It can be done in the such manner that the following diagram will be commutative. Plain Mechanics / \ / \ / \ Quantum Mechanics --> Classical Mechanics h->0 Here the horizontal arrow is well known correspondence between Quantum and Classical Mechanics if Planck constant tensing to zero. A {\em realization} of this scheme for a particle in $n$-dimensional space by two-sided convolutions on the Heisenberg group is constructed. We also introduce the {\em motion equations} for observables in this realization. The left arrow of the given diagram carries this equation to the Heisenberg one and the right arrow maps it to the Hamilton equation.

funct-an↗