ABSTRACT. As a generalization of the
linear Poisson bracket on the dual space of a Lie algebra, we introduce
certain non-linear Poisson brackets which are ``cocycle perturbations''
of the linear Poisson bracket. We show that these special Poisson brackets
are equivalent to Poisson brackets of central extension type, which
resemble the central extensions of an ordinary Lie bracket via Lie algebra
cocycles. We are able to formulate (strict) deformation quantizations
of these Poisson brackets by means of twisted group C*-algebras.
We also indicate that these deformation quantizations can be used to
construct some specific non-compact quantum groups.
ABSTRACT. The dual Lie bialgebra of a certain quasitriangular Lie bialgebra structure on the Heisenberg Lie algebra determines a (non-compact) Poisson-Lie group G. The compatible Poisson bracket on G is non-linear, but it can still be realized as a ``cocycle perturbation'' of the linear Poisson bracket. We construct a certain twisted group C*-algebra A, which is shown to be a strict deformation quantization of G. Motivated by the data at the Poisson (classical) level, we then construct on A its locally compact quantum group structures: comultiplication, counit, antipode and Haar weight, as well as its associated multiplicative unitary operator. We also find a quasitriangular ``quantum universal R-matrix'' type operator for A, which agrees well with the quasitriangularity at\ the Lie bialgebra level.
ABSTRACT. In our earlier work, we constructed a specific non-compact quantum group whose quantum group structures have been constructed on a certain twisted group C*-algebra. In a sense, it may be considered as a ``quantum Heisenberg group C*-algebra''. In this paper, we will find, up to equivalence, all of its irreducible *-representations. We will point out the Kirillov type correspondence between the irreducible representations and the so-called dressing orbits. By taking advantage of its comultiplication, we will then introduce and study the notion of inner tensor product representations. We will show that the representation theory satisfies a ``quasitriangular'' type property, which does not appear in ordinary group representation theory.
ABSTRACT. In the general theory of locally compact quantum groups, the notion of Haar measure (Haar weight) plays the most significant role. The aim of this paper is to carry out a careful analysis regarding Haar weight, in relation to general theory, for the specific non-compact quantum group (A,Δ) constructed earlier by the author. In this way, one can show that (A,Δ) is indeed a ``(C*-algebraic) locally compact quantum group'' in the sense of the recently developed definition given by Kustermans and Vaes. Attention will be given to pointing out the relationship between the original construction (obtained by deformation quantization) and the structure maps suggested by general theory.
ABSTRACT. In this paper, we give a construction of a (C*-algebraic) quantum Heisenberg group. This is done by viewing it as the dual quantum group of the specific non-compact quantum group (A,Δ) constructed earlier by the author. Our definition of the quantum Heisenberg group is different from the one considered earlier by Van Daele. To establish our object of study as a locally compact quantum group, we also give a discussion on its Haar weight, which is no longer a trace. In the latter part of the paper, we give some additional discussion on the duality mentioned above.
ABSTRACT. In this paper, as a
generalization of Kirillov's orbit theory, we explore the relationship
between the dressing orbits and irreducible *-representations of the Hopf
C*-algebras (A,Δ) and (Ã,Δ˜) we
constructed earlier. We discuss the one-to-one correspondence between
them, including their topological aspects.
On each dressing orbit (which are symplectic leaves of the underlying
Poisson structure), one can define a Moyal-type deformed product at
the function level. The deformation is more or less modeled by the
irreducible representation corresponding to the orbit. We point out
that the problem of finding a direct integral decomposition of the
regular representation into irreducibles (Plancherel theorem) has an
interesting interpretation in terms of these deformed products.
ABSTRACT. By working with several specific Poisson-Lie groups arising from Heisenberg Lie bialgebras and by carrying out their quantizations, a case is made for a useful but simple method of constructing locally compact quantum groups. The strategy is to analyze and collect enough information from a Poisson-Lie group, and using it to carry out a ``cocycle bicrossed product construction''. Constructions are done using multiplicative unitary operators, obtaining C*-algebraic, locally compact quantum (semi-)groups.
ABSTRACT. We carry out the quantum double construction of the specific quantum groups we constructed earlier, namely, the ``quantum Heisenberg group algebra'' (A,Δ) and its dual (Â,Δ^). Our approach is by constructing a suitable multiplicative unitary operator, retaining the C*-algebra framework of locally compact quantum groups. We also discuss the dual of the quantum double and the Haar weights on both of these double objects. Towards the end, a construction of a (quasitriangular) ``quantum universal R-matrix'' is given.
ABSTRACT. Quantum double construction, originally due to Drinfeld and has been since generalized even to the operator algebra framework, is naturally associated with a certain (quasitriangular) R-matrix R. It turns out that R determines a twisting of the comultiplication on the quantum double. It then suggests a twisting of the algebra structure on the dual of the quantum double. For the case of D(G), the C*-algebraic quantum double of an ordinary group G, the ``twisted D(G)^'' turns out to be the Weyl algebra C0(G)×τ G ≅ K(H). This is the C*-algebraic counterpart to an earlier (finite-dimensional) result by Lu. It is not so easy technically to extend this program to the general locally compact quantum group case, but we propose here some possible approaches, using the notation of the Fourier transform.
ABSTRACT. The notion of Fourier transform is among the more important tools in analysis, which has been generalized in abstract harmonic analysis to the level of abelian locally compact groups. The aim of this paper is to further generalize the Fourier transform: Motivated by some recent works by Van Daele in the multiplier Hopf algebra framework, and by using the Haar weights, we define here the (generalized) Fourier transform and the inverse Fourier transform, at the level of locally compact quantum groups. We will then consider the analogues of the Fourier inversion theorem, Plancherel theorem, and the convolution product. Along the way, we also obtain an alternative description of the dual pairing map between a quantum group and its dual.
ABSTRACT. Beginning with a skew-symmetric matrix, we define a certain Poisson--Lie group. Its Poisson bracket can be viewed as a cocycle perturbation of the linear (or ``Lie--Poisson'') Poisson bracket. By analyzing this Poisson structure, we gather enough information to construct a C*-algebraic locally compact quantum group, via the ``cocycle bicrossed product construction'' method. The quantum group thus obtained is shown to be a deformation quantization of the Poisson--Lie group, in the sense of Rieffel.
CONTENTS.
- Preliminaries
- 1. Deformation quantization and noncommutative manifolds
- 2. Locally compact quantum groups
- 3. Beyond quantum groups: Duality, Quantum groupoids
- References
ABSTRACT. The Larson-Sweedler theorem says that
a finite-dimensional bialgebra with a faithful integral is a Hopf algebra [L-S]. The result has been
generalized to finite-dimensional weak Hopf algebras by Vecsernyés [Ve]. In this paper,
we show that the result is still true for weak multiplier Hopf algebras.
The notion of a weak multiplier bialgebra was introduced by Böhm, Gómez-Torrecillas
and Lopez-Centella [B-G-L]. In this note it is shown that a weak multiplier bialgebra with a regular
and full coproduct is a regular weak multiplier Hopf algebra if there is a faithful set of
integrals. Weak multiplier Hopf algebras are introduced and studied in [VD-W3].
Integrals on regular weak multiplier Hopf algebras are treated in [VD-W5].
This result is important for the development of the theory of locally compact quantum groupoids
in the operator algebra setting [K-VD]. Our treatment of this material is motivated by the prospect of such a theory.
ABSTRACT. In this series of papers, we develop the theory of a class of locally compact quantum groupoids, which is motivated by the purely algebraic notion of weak multiplier Hopf algebras. In this Part I, we provide motivation and formulate the definition in the C$-algebra framework. Existence of a certain canonical idempotent element is required and it plays a fundamental role, including the establishment of the coassociativity of the comultiplication. This class contains locally compact quantum groups as a subclass.
ABSTRACT. In this paper, we study the notion of a separability idempotentin the C*-algebra framework. This is analogous to the notion in the purely algebraic setting, typically considered in the case of (finite-dimensional) algebras with identity, then later also considered in the multiplier algebra framework by the second-named author. The current work was motivated by the appearance of such objects in the authors' ongoing work on locally compact quantum groupoids.
ABSTRACT. This is Part II in our multi-part series of papers developing the theory of a subclass of locally compact quantum groupoids (quantum groupoids of separable type), based on the purely algebraic notion of weak multiplier Hopf algebras. The definition was given in Part I. The existence of a certain canonical idempotent element E plays a central role. In this Part II, we develop the main theory, discussing the structure of our quantum groupoids. We will construct from the defining axioms the right/left regular representations and the antipode map.
ABSTRACT. Generalizing the notion of a multiplicative unitary (in the sense of Baaj-Skandalis), which plays a fundamental role in the theory of locally compact quantum groups, we develop in this paper the notion of a multiplicative partial isometry. The axioms include the pentagon equation, but more is needed. Under suitable conditions (such as the ``manageability''), it is possible to construct from it a pair of C*-algebras having the structure of a C*-algebraic quantum groupoid of separable type.
ABSTRACT. Van Daele and Wang developed a purely algebraic notion of weak multiplier Hopf algebras, which extends the notions of Hopf algebras, multiplier Hopf algebras, and weak Hopf algebras. With an additional requirement of an existence of left or right integrals, this framework provides a self-dual class of algebraic quantum groupoids. The aim of this paper is to show that from this purely algebraic data, with only a minimal additional requirement (``quasi-invariance''), one can construct a $C*-algebraic quantum groupoid of separable type, recently defined by the author, with Van Daele. The C*-algebraic quantum groupoid is represented as an operator algebra on the Hilbert space constructed from the left integral, and the comultiplication is determined by means of a certain multiplicative partial isometry W, which is no longer unitary. In the last section (Appendix), we obtain some results in the purely algebraic setting, which have not appeared elsewhere.
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