There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

By Vyjayanthi Chari and Andrew Pressley: 651 pp., £22.95 (US$34.95), isbn 0 521 55884 0 (Cambridge University Press, 1994).

A guide to quantum groups

Quantum groups can be viewed as deformations/liberations of classical locally compact groups. Around the year 2000 a thorough definition of a locally compact quantum group was given by Kustermans and Vaes. In these lecture notes (which were used at the Będlewo summer school in 2015) we give an introduction to this Kustermans–Vaes definition and we discuss the example of SUq(1, 1) established by Koelink and Kustermans.

Locally compact quantum groups

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Book Review: Representations of *-algebras, locally compact groups, and Banach *-algebraic bundles

The Larson–Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra [15]. The result has been generalized to finite-dimensional weak Hopf algebras by Vecsernyes [44]. 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 Bohm et al. in [4]. 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 [40]. Integrals on (regular) weak multiplier Hopf algebras are treated in [43]. This result is important for the development of the theory of locally compact quantum groupoids in the operator algebra setting, see [13] and [14]. Our treatment of this material is motivated by the prospect of such a theory.

The Larson–Sweedler theorem for weak multiplier Hopf algebras

The dual Lie bialgebra of a certain quasitriangular Lie bialgebra structure on the Heisenberg 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.

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Non-compact quantum groups arising from heisenberg type lie bialgebras

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.

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Fourier transform on locally compact quantum groups

The Larson-Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra. The result has been generalized to finite-dimensional weak Hopf algebras by Vecsernyes. 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 Bohm, Gomez-Torrecillas and Lopez-Centella. 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. This result is important for the development of the theory of locally compact quantum groupoids in the operator algebra setting. Our treatment of this material is motivated by the prospect of such a theory.

The Larson-Sweedler theorem for weak multiplier Hopf algebras

As a generalization of the linear Poisson bracket on the dual space of a Lie algebra, we introduce certain nonlinear 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.

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Byung-Jay Kahng

Papers

The Larson–Sweedler theorem for weak multiplier Hopf algebras

Non-compact quantum groups arising from heisenberg type lie bialgebras

Fourier transform on locally compact quantum groups

The Larson-Sweedler theorem for weak multiplier Hopf algebras

Deformation quantization of certain nonlinear poisson structures