Let us consider next, alternative definitions of quantum groups that indeed possess extended quantum symmetries and algebraic properties distinct from those of Hopf algebras.
For additional examples of quantum groups the reader is referred to the last six publications listed in the bibliography.
Quantum theories
adopted a new lease of life post 1955 when von Neumann beautifully re-formulated quantum mechanics
(QM) in the mathematically rigorous context of Hilbert spaces
and operator algebras. From a current physics perspective, von Neumann's approach to quantum mechanics has done however much more: it has not only paved the way to expanding the role of symmetry in
physics, as for example with the Wigner-Eckhart theorem
and its
applications, but also revealed the fundamental importance in
quantum physics of the state space
geometry of (quantum) operator
algebras. Subsequent developments of the quantum operator algebra
were aimed at identifying more general quantum symmetries than those defined for example by symmetry groups, groups of unitary operators and Lie groups. Several fruitful quantum algebraic concepts
were developed, such as: the Ocneanu paragroups-later found to be represented by Kac-Moody algebras, quantum `groups' represented either as Hopf algebras or locally compact groups with Haar measure, `quantum' groupoids represented as weak Hopf algebras, and so on. The Ocneanu paragroups case is particularly interesting as it can be considered as an extension through quantization
of certain finite group symmetries to infinitely-dimensional von Neumann type
factors (subalgebras), and are, in effect, `quantized groups' that can be nicely constructed as Kac algebras; in fact, it was recently shown that a paragroup can be constructed from a crossed product by an outer action of a Kac algebra. This suggests a relation
to categorical aspects of paragroups (rigid monoidal tensor
categories
previously reported in the literature). The strict symmetry of the group of (quantum) unitary operators is thus naturally extended through paragroups to the symmetry of the latter structure's unitary representations; furthermore, if a subfactor of the von Neumann algebra
arises as a crossed product by a finite group action, the paragroup for this subfactor contains a very similar group structure to that of the original finite group, and also has a unitary representation theory similar to that of the original finite group. Last-but-not least, a paragroup yields a complete invariant for irreducible inclusions of AFD von Neumannn type
factors with finite index and finite depth (Theorem 2.6. of Sato, 2001). This can be considered as a kind of internal, `hidden' quantum symmetry of von Neumann algebras.
On the other hand, unlike paragroups, (quantum) locally compact groups are not readily constructed as either Kac or Hopf C*-algebras. In recent years the techniques of Hopf symmetry and those of weak Hopf C*-algebras, sometimes called quantum `groupoids' (cf Böhm et al.,1999),
provide important tools-in addition to the paragroups- for studying the broader relationships of the Wigner fusion rules algebra,
-symmetry (Rehren, 1997), as well as the study of the noncommutative
symmetries of subfactors within the Jones tower constructed from finite index depth 2 inclusion of factors, also recently considered from the viewpoint of related Galois correspondences (Nikshych and Vainerman, 2000).
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