PlanetPhysics/Quantum Groups and Hopf Algebras 2

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Introduction

A quantum group is often defined as the dual of a Hopf algebra or coalgebra. Actually, quantum groups are constructed by employing certain Hopf algebras as "building blocks"; and in the case of finite [[../TrivialGroupoid/|groups]] they are obtained from the latter by Fourier transformation of the group elements.

Let us consider next, alternative definitions of quantum groups that indeed possess [[../TopologicalOrder2/|extended quantum symmetries]] and [[../CoIntersections/|algebraic]] properties distinct from those of Hopf algebras.

Quantum groups are defined as locally compact [[../PolishGroup/|topological groups]] endowed with a left [[../HigherDimensionalQuantumAlgebroid/|Haar measure]] [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]], and also with at least one internal [[../HilbertBundle/|quantum symmetry]], such as the intrinsic [[../QuarkAntiquarkPair/|spin]] symmetry represented by either Pauli [[../Matrix/|matrices]] or the Dirac algebra of [[../QuantumSpinNetworkFunctor2/|observable]] spin [[../QuantumOperatorAlgebra4/|operators]].

For additional examples of quantum groups the reader is referred to the last six publications listed in the bibliography.

One can also consider quantum groups as a particular case of [[../WeakHopfAlgebra/|quantum groupoids]] in the limiting case where there is only one symmetry [[../Bijective/|type]] present in the quantum groupoid.

Quantum Groups, Paragroups and Operator Algebras in Quantum Theories

[[../QuantumOperatorAlgebra5/|Quantum theories]] adopted a new lease of life post 1955 when von Neumann beautifully re-formulated [[../QuantumParadox/|quantum mechanics]] ([[../FTNIR/|QM]]) in the mathematically rigorous context of [[../NormInducedByInnerProduct/|Hilbert spaces]] and [[../Groupoid/|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 [[../Formula/|theorem]] and its applications, but also revealed the fundamental importance in quantum physics of the [[../StableAutomaton/|state space]] geometry of (quantum) operator algebras. Subsequent developments of the [[../QuantumOperatorAlgebra5/|quantum operator algebra]] were aimed at identifying more general quantum symmetries than those defined for example by [[../TopologicalOrder2/|symmetry groups]], groups of unitary operators and [[../BilinearMap/|Lie groups]]. Several fruitful quantum algebraic [[../PreciseIdea/|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 [[../WeakHopfAlgebra/|weak Hopf algebras]], and so on. The Ocneanu [[../Paragroups/|paragroups]] case is particularly interesting as it can be considered as an extension through [[../MoyalDeformation/|quantization]] of certain finite group symmetries to infinitely-dimensional von Neumann type II1 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 [[../Bijective/|relation]] to categorical aspects of paragroups (rigid monoidal [[../Tensor/|tensor]] [[../Cod/|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 [[../CategoricalGroupRepresentation/|representations]]; furthermore, if a subfactor of the [[../CoordinateSpace/|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 II1 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 [[../VonNeumannAlgebra2/|C*-algebras]]. In recent years the techniques of Hopf symmetry and those of weak Hopf C*-algebras, sometimes called quantum `groupoids' (cf B\"ohm et al.,1999), provide important tools--in addition to the paragroups-- for studying the broader relationships of the Wigner fusion rules algebra, 6j--symmetry (Rehren, 1997), as well as the study of the [[../AbelianCategory3/|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).

See also the related entry on compact quantum groups (CQGs).

All Sources

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

References

  1. ↑ M. Chaician and A. Demichev: Introduction to Quantum Groups , World Scientific (1996).
  2. ↑ V. G. Drinfel'd: Quantum groups, In \emph{Proc. Intl. Congress of Mathematicians, Berkeley 1986}, (ed. A. Gleason), Berkeley, 798-820 (1987).
  3. ↑ P.. I. Etingof and A. N. Varchenko, Solutions of the Quantum Dynamical Yang-Baxter Equation and Dynamical Quantum Groups, Comm.Math.Phys. , 196 : 591-640 (1998).
  4. ↑ P. I. Etingof and A. N. Varchenko: Exchange dynamical quantum groups, Commun. Math. Phys. 205 (1): 19-52 (1999)
  5. ↑ P. I. Etingof and O. Schiffmann: Lectures on the dynamical Yang--Baxter equations, in Quantum Groups and Lie Theory (Durham, 1999) , pp. 89-129, Cambridge University Press, Cambridge, 2001.
  6. ↑ J. M. G. Fell.: The Dual Spaces of C*--Algebras., \emph{Transactions of the American Mathematical Society}, 94 : 365--403 (1960).
  7. ↑ P. Hahn: Haar measure for measure groupoids., Trans. Amer. Math. Soc . 242 : 1--33(1978).
  8. ↑ P. Hahn: The regular representations of measure groupoids., Trans. Amer. Math. Soc . 242 :34--72(1978).
  9. ↑ C. Heunen, N. P. Landsman, B. Spitters.: A topos for algebraic quantum theory, (2008) \\ arXiv:0709.4364v2 [quant--ph]
  10. ↑ S. Majid. Quantum groups, on line

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