Search results for "Quantum group"
showing 10 items of 32 documents
On the representation theory of quantum Heisenberg group and algebra
1994
We show that the quantum Heisenberg groupH q (1) and its *-Hopf algebra structure can be obtained by means of contraction from quantumSU q (2) group. Its dual Hopf algebra is the quantum Heisenberg algebraU q (h(1)). We derive left and right regular representations forU q (h(1)) as acting on its dualH q (1). Imposing conditions on the right representation, the left representation is reduced to an irreducible holomorphic representation with an associated quantum coherent state. Realized in the Bargmann-Hilbert space of analytic functions the unitarity of regular representation is also shown. By duality, left and right regular representations for quantum Heisenberg group with the quantum Heis…
The hidden group structure of quantum groups: strong duality, rigidity and preferred deformations
1994
A notion of well-behaved Hopf algebra is introduced; reflexivity (for strong duality) between Hopf algebras of Drinfeld-type and their duals, algebras of coefficients of compact semi-simple groups, is proved. A hidden classical group structure is clearly indicated for all generic models of quantum groups. Moyal-product-like deformations are naturally found for all FRT-models on coefficients andC∞-functions. Strong rigidity (H bi 2 ={0}) under deformations in the category of bialgebras is proved and consequences are deduced.
A natural and rigid model of quantum groups
1992
We introduce a natural (Frechet-Hopf) algebra A containing all generic Jimbo algebras U t (sl(2)) (as dense subalgebras). The Hopf structures on A extend (in a continuous way) the Hopf structures of generic U t (sl(2)). The Universal R-matrices converge in A\(\hat \otimes \)A. Using the (topological) dual of A, we recover the formalism of functions of noncommutative arguments. In addition, we show that all these Hopf structures on A are isomorphic (as bialgebras), and rigid in the category of bialgebras.
A candidate for a noncompact quantum group
1996
A previous letter (Bidegain, F. and Pinczon, G:Lett. Math. Phys.33 (1995), 231–240) established that the star-product approach of a quantum group introduced by Bonneau et al. can be extended to a connected locally compact semisimple real Lie group. The aim of the present Letter is to give an example of what a noncompact quantum group could be. From half of the discrete series ofSL(2,\(\mathbb{R}\)), a new type of quantum group is explicitly constructed.
The electromagnetic group: Bosonic BRST charge
1990
Abstract We give an infinite-dimensional Lie group from which a group approach to quantization (GAQ) derives a Gupta-Bleuler-like quantization for the electromagnetic field. The incorporation into the group law of the gauge transformation properties of Aμ(x), Aμ(x) → Aμ(x) + ∂μφ, requires a non-conventional generator which is related to the BRST charge.
The Bohm-Aharonov effect: A seven-dimensional structural group
1996
We realize a nonfaithful representation of a seven-dimensional Lie algebra, the extension of which to its universal enveloping algebra contains most of the observables of the scattering Aharonov-Bohm effect, as essentially self-adjoint operators: the scattering Hamiltonian, the total and kinetic angular momenta, the positions and the kinetic momenta. By restriction, we obtain the model introduced in Lett. Math. Phys.1 (1976), 155–163.
Fundamental isomorphism theorems for quantum groups
2017
The lattice of subgroups of a group is the subject of numerous results revolving around the central theme of decomposing the group into "chunks" (subquotients) that can then be compared to one another in various ways. Examples of results in this class would be the Noether isomorphism theorems, Zassenhaus' butterfly lemma, the Schreier refinement theorem for subnormal series of subgroups, the Dedekind modularity law, and last but not least the Jordan-H\"older theorem. We discuss analogues of the above-mentioned results in the context of locally compact quantum groups and linearly reductive quantum groups. The nature of the two cases is different: the former is operator algebraic and the latt…
Topological Hopf Algebras, Quantum Groups and Deformation Quantization
2019
After a presentation of the context and a brief reminder of deformation quantization, we indicate how the introduction of natural topological vector space topologi es on Hopf algebras associated with Poisson Lie groups, Lie bialgebras and their doubles explains their dualities a nd provides a comprehensive framework. Relations with deformation quantization and applications to the deformation quantization of symmetric spaces are described.
Algebra Structures on Hom(C,L)
1999
info:eu-repo/semantics/published
Quantum and Braided Integrals
2001
We give a pedagogical introduction to integration techniques appropriate for non-commutative spaces while presenting some new results as well. A rather detailed discussion outlines the motivation for adopting the Hopf algebra language. We then present some trace formulas for the integral on Hopf algebras and show how to treat the $\int 1=0$ case. We extend the discussion to braided Hopf algebras relying on diagrammatic techniques. The use of the general formulas is illustrated by explicitly worked out examples.