Quantum SL(3,C)'s with Classical Representation Theory

Ohn, Christian

doi:10.1006/jabr.1998.7658

Cited by 18 publications

(13 citation statements)

References 17 publications

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“…Woronowicz [16], a complete proof being given in P. Podleś and E. Müller's notes [13]. The SL(3)-case has been done by C. Ohn [12] with a constraint on the dimension of the fundamental comodule. Finally the compact case SU (N ) was done in [2], without any dimension constraint but without an isomorphic classification.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The Representation Category of the Quantum Group of a Non-degenerate Bilinear Form

Bichon

2003

Communications in Algebra

130

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We show that the representation category of the quantum group of a non-degenerate bilinear form is monoidally equivalent to the representation category of the quantum group SL q (2) for a well-chosen non-zero parameter q. The key ingredient for the proof of this result is the direct and explicit construction of an appropriate Hopf bigalois extension. Then we get, when the base field is of characteristic zero, a full description of cosemisimple Hopf algebras whose representation semi-ring is isomorphic to the one of SL(2).

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Section: Introduction and Main Resultsmentioning

confidence: 99%

The Representation Category of the Quantum Group of a Non-degenerate Bilinear Form

Bichon

2003

Communications in Algebra

130

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“…Furthermore, the structure of non-unitary cocycles is much more complicated. For example, as we mentioned above, any unitary dual cocycle on SU (3) is cohomologous to one induced from a maximal torus, while the computations of Ohn [40] show that there are non-unitary cocycles that cannot be obtained this way. This is apparently related to the fact that the complexification G C of a compact Lie group G has many more Poisson-Lie structures than the group G itself.…”

Section: Dual Cocycles and Ergodic Actionsmentioning

confidence: 99%

On Second Cohomology of Duals of Compact Groups

Neshveyev

Tuset

2011

Int. J. Math.

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We show that for any compact connected group G the second cohomology group defined by unitary invariant 2-cocycles onĜ is canonically isomorphic to H 2 ( Z(G); T). This implies that the group of autoequivalences of the C * -tensor category Rep G is isomorphic to H 2 ( Z(G); T) ⋊ Out(G). We also show that a compact connected group G is completely determined by Rep G. More generally, extending a result of Etingof-Gelaki and Izumi-Kosaki we describe all pairs of compact separable monoidally equivalent groups. The proofs rely on the theory of ergodic actions of compact groups developed by Landstad and Wassermann and on its algebraic counterpart developed by Etingof and Gelaki for the classification of triangular semisimple Hopf algebras.In two appendices we give a self-contained account of amenability of tensor categories, fusion rings and discrete quantum groups, and prove an analogue of Radford's theorem on minimal Hopf subalgebras of quasitriangular Hopf algebras for compact quantum groups.

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“…See [1] for a detailed construction of SU(N) R . See also [37] and [64] for related results, obtained via different methods.…”

Section: Theorem 24 ([50]mentioning

confidence: 99%

Representations of compact quantum groups and subfactors

Banica

1999

Journal f�r die reine und angewandte Mathematik (Crelles Journ

152

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IntroductionThe compact quantum groups are objects which generalise at the same time the compact groups, the duals of discrete groups and the q−deformations (with q > 0) of classical compact Lie groups. A compact quantum group is an abstract object which may be described by (is by definition the dual of) the algebra of "continuous functions on it", which is a Hopf C * -algebra. A system of axioms for Hopf C * -algebras which leads to a satisfactory theory of compact quantum groups (e.g. a theorem stating the existence of the Haar measure) was found by Woronowicz at the end of the 80's.The representation theory of compact quantum groups gives rise to rich combinatorial structures. By Woronowicz's analogue of the Peter-Weyl theory, each (finite dimensional unitary) representation of a compact quantum group G is completely reducible. In particular given two irreducible representations a and b, their tensor product decomposes in a unique way (up to equivalence) as a sum of irreducible representationsThese formulae are called fusion rules for irreducible representations of G. The fusion semiring R + (G) is by definition the set of equivalence classes of finite dimensional continuous representations of G, endowed with the binary operations + (the sum of classes of corepresentations) and ⊗ (the tensor product of classes of corepresentations). It is the algebraic structure describing the collection of all fusion rules. There are two basic examples: -if G is a compact group then R + (G) is the usual fusion semiring of G.-if G is the dual of a discrete group Γ then R + (G) is the convolution semiring of Γ.More generally, we have the following construction of fusion rules and semirings: in a semisimple monoidal category C, formulae of the form a ⊗ b ≃ c + d + e + · · · with a, b, c, d, e, . . . simple objects of C, may be called fusion rules for simple objects of C. The algebraic structure describing the collection of all fusion rules is the Grothendieck semiring ([C], +, ⊗) of C. Fusion rules -and related algebraic objects, such as fusion semirings, rings, algebras and principal graphs -appear in this way in many recent theories arising from mathematics and physics, such as conformal field theory, subfactors, quantum groups at roots of unity. They can be thought of as being a common language for these theories. The above fusion semiring R + (G) arises also in this way: it is isomorphic to the Grothendieck semiring of the semisimple monoidal category Rep(G) of finite dimensional continuous representations of G.In this paper we give a survey of some recent results on the fusion semirings of compact quantum groups (computations of and applications to discrete quantum groups) by using the following simplifying terminology: we say that a compact quantum group 1 2 TEODOR BANICA G is an R + -deformation of a compact quantum group H if their fusion semirings are isomorphic. The paper contains also some easy related results (with proofs), two conjectures and many remarks and comments, some of them concerning classification by invar...

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Quantum SL(3,C)'s with Classical Representation Theory

Abstract: Ž. We study and classify almost all quantum SL 3, C 's whose representation theory Ž . Ž . is ''similar'' to that of the ordinary group SL 3, C . Only one case, related to smooth elliptic curves, could not be treated completely. ᮊ

Cited by 18 publications

References 17 publications

The Representation Category of the Quantum Group of a Non-degenerate Bilinear Form

The Representation Category of the Quantum Group of a Non-degenerate Bilinear Form

On Second Cohomology of Duals of Compact Groups

Representations of compact quantum groups and subfactors

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