From multiplicative unitaries to quantum groups II

Sołtan, Piotr M.; Woronowicz, S. L.

doi:10.1016/j.jfa.2007.07.006

Cited by 51 publications

(57 citation statements)

References 18 publications

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“…In particular, representations of C*-algebra A are in one-to-one correspondence with corepresentations of the quantum group y G. This follows from Theorem 5.4 of [9].…”

Section: A/ and There Exists A Positive Constant C Such Thatmentioning

confidence: 77%

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The Heisenberg–Lorentz quantum group

Kasprzak¹

2010

J. Noncommut. Geom.

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Abstract. In this article we present a new C -algebraic deformation of the Lorentz group. It is obtained by means of the Rieffel deformation applied to SL.2; C/. We give a detailed description of the resulting quantum group G D .A; / in terms of generators Ǫ; Ǒ ; O ; O ı 2 A Á -the quantum counterparts of the matrix coefficients˛,ˇ, , ı of the fundamental representation of SL.2; C/. In order to construct Ǒ -the most involved of the four generators -we first define it on the quantum Borel subgroup G 0 G, then on the quantum complement of the Borel subgroup and finally we perform the gluing procedure. In order to classify representations of the C -algebra A and to analyze the action of the comultiplication on the generators Ǫ, Ǒ , O , O ı we employ the duality in the theory of locally compact quantum groups.Mathematics Subject Classification (2010). 46L89; 58B32, 22D25.

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“…In particular, representations of C*-algebra A are in one-to-one correspondence with corepresentations of the quantum group y G. This follows from Theorem 5.4 of [9].…”

Section: A/ and There Exists A Positive Constant C Such Thatmentioning

confidence: 77%

“…Let us now draw an important conclusion from the existence of the counit for G. Using Proposition 5.16 of [9] we can see that the universal dual quantum group of…”

Section: A/ and There Exists A Positive Constant C Such Thatmentioning

confidence: 95%

The Heisenberg–Lorentz quantum group

Kasprzak¹

2010

J. Noncommut. Geom.

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“…In order to introduce a concept of a quantum group homomorphism π : H → G we need the universal objects H u = (C u 0 (H), u H ) and G u = (C u 0 (G), u G ) assigned to H and G respectively. For the definition of the universal version of a quantum group given by a multiplicative unitary W we refer to [14,Definition 5.1]. Let us emphasize that the universal version of a quantum group is not a quantum group in the sense of Definition 4.1.…”

Section: Rieffel Deformation and Braided Quantum Groupsmentioning

confidence: 99%

“…For the axiomatic formulations of LCQG with the existence of Haar measure postulated, we refer the reader to [5] or [8]. For the theory with the multiplicative unitary playing a central role we refer to [14]. Roughly speaking, a quantum group is a pair G = (A, ) where A is a C * -algebra and ∈ Mor(A, A ⊗ A).…”

Section: Introductionmentioning

confidence: 99%

Rieffel Deformation of Tensor Functor and Braided Quantum Groups

Kasprzak

2015

Commun. Math. Phys.

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We apply Rieffel deformation to C * -tensor product viewed as a functor on the category of C * -algebras with an abelian group action. In the case of the Rieffel deformation of a quantum group with the action by automorphisms the deformed tensor product enables us to view the deformed object as a braided quantum group. We construct a bicharacter for a braided quantum group and the dual braided quantum group. We employ our method to get a braided quantum Minkowski space. Its description in terms of the deformed space-time coordinates is provided.

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“…On the other hand, our construction works for general bisimplifiable Hopf C * -algebras, not only for locally compact quantum groups. In particular we can use universal versions of quantum groups ( [9], [20,Section 5]). …”

Section: Introductionmentioning

confidence: 99%

Examples of non-compact quantum group actions

Sołtan

2010

Journal of Mathematical Analysis and Applications

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We present two examples of actions of non-regular locally compact quantum groups on their homogeneous spaces. The homogeneous spaces are defined in a way specific to these examples, but the definitions we use have the advantage of being expressed in purely C * -algebraic language. We also discuss continuity of the obtained actions. Finally we describe in detail a general construction of quantum homogeneous spaces obtained as quotients by compact quantum subgroups.

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From multiplicative unitaries to quantum groups II

Cited by 51 publications

References 18 publications

The Heisenberg–Lorentz quantum group

The Heisenberg–Lorentz quantum group

Rieffel Deformation of Tensor Functor and Braided Quantum Groups

Examples of non-compact quantum group actions

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