Krein duality, positive 2-algebras, and dilation of comultiplications

Vershik, A. M.

doi:10.1007/s10688-007-0010-2

Cited by 9 publications

(3 citation statements)

References 20 publications

(64 reference statements)

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“…Conversely, the product (in the above sense) of central ergodic measures M Γ ∈ E (Γ) and M P ∈ E (P) is a central ergodic measure M e Γ ∈ E ( Γ). It is central by Lemma 2.7, and it is ergodic by relation (2).…”

Section: The Path Spaces Are Related By T ( γ) = T (γ) × T (P) Moreomentioning

confidence: 96%

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Description of the characters and factor representations of the infinite symmetric inverse semigroup

Вершик

Nikitin

2011

Funct Anal Its Appl

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Section: The Path Spaces Are Related By T ( γ) = T (γ) × T (P) Moreomentioning

confidence: 96%

“…In [2] it was shown that the class of finite inverse semigroups generates exactly the class of involutive semisimple bialgebras.…”

Section: Involutive Bialgebras and Semigroup Algebras Of Inverse Semimentioning

confidence: 99%

Description of the characters and factor representations of the infinite symmetric inverse semigroup

Вершик

Nikitin

2011

Funct Anal Its Appl

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“…Vershik proved a result which shows the connection between finite generalized group bialgebras and finite-dimensional bialgebras, it can be found in [10].…”

Section: Generalized Groups and Weak Hopf Algebrasmentioning

confidence: 97%

Topology-preserving quantum deformation with non-numerical parameter

Aukhadiev

Grigoryan

Lipacheva

2013

J. Phys.: Conf. Ser.

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Abstract. We introduce a class of compact quantum semigroups, that we call semigroup deformations of compact Abelian qroups. These objects arise from reduced semigroup C * -algebras, the generalization of the Toeplitz algebra. We study quantum subgroups, quantum projective spaces and quantum quotient groups for such objects, and show that the group is contained as a compact quantum subgroup in the deformation of itself. The connection with the weak Hopf algebra notion is described. We give a grading on the C * -algebra of the compact quantum semigroups constructed. PreliminariesThis work is devoted to constructing compact quantum semigroups using non-commutative C * -algebras, generated by isometries. The procedure starts with considering a compact Abelian group G and its Pontryagin dual group Γ. The method described in section 2 represents a sort of deformation of the group G, which preserves its topology and multiplication.Unlike the standard quantization procedure, the deformation parameter here is not a number, but a set of elements. More precisely, any semigroup S generating group Γ can play a role of deformation parameter. It is shown that this deformation depends essentially on the choice of S, giving different results for different semigroups generating the same group G.Another special feature of this method is that the object QS r (S) obtained as deformation is not a quantum group, but a compact quantum semigroup with some unique properties. Notion of the compact quantum semigroup is given in section 1.1. We show that QS r (S) can be regarded as an extension of G, since G ⊂ QS r (S) as a compact quantum subgroup (see section 3). Similarly to the classical case this extension generates an action of G on QS r (S). This action and quantum projective space are described in section 3.

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Krein duality, positive 2-algebras, and dilation of comultiplications

Vershik

2007

Funct Anal Its Appl

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The Krein-Tannaka duality for compact groups was a generalization of the Pontryaginvan Kampen duality for locally compact Abelian groups and a remote predecessor of the theory of tensor categories. It is less known that it found applications in algebraic combinatorics ("Krein algebras"). Later, this duality was substantially extended: in [29], the notion of involutive algebras in positive vector duality was introduced. In this paper, we reformulate the notions of this theory using the language of bialgebras (and Hopf algebras) and introduce the class of involutive bialgebras and positive 2-algebras. The main goal of the paper is to give a precise statement of a new problem, which we consider as one of the main problems in this field, concerning the existence of dilations (embeddings) of positive 2-algebras in involutive bialgebras, or, in other words, the problem of describing subobjects of involutive bialgebras; we define two types of subobjects of bialgebras, strict and nonstrict ones. The dilation problem is illustrated by the example of the Hecke algebra, which is viewed as a positive involutive 2-algebra. We consider in detail only the simplest situation and classify two-dimensional Hecke algebras for various values of the parameter q, demonstrating the difference between the two types of dilations. We also prove that the class of finite-dimensional involutive semisimple bialgebras coincides with the class of semigroup algebras of finite inverse semigroups.

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Krein duality, positive 2-algebras, and dilation of comultiplications

Cited by 9 publications

References 20 publications

Description of the characters and factor representations of the infinite symmetric inverse semigroup

Description of the characters and factor representations of the infinite symmetric inverse semigroup

Topology-preserving quantum deformation with non-numerical parameter

Krein duality, positive 2-algebras, and dilation of comultiplications

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