An Invitation to Quantum Groups and Duality

Timmermann, Thomas

doi:10.4171/043

Cited by 205 publications

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“…Since we will be dealing with non-commutative L p -spaces, we stick to the von Neumann algebra setting. For an introduction to the theory of locally compact quantum groups we refer to [12] or [18], where the results below are summarized. See also [19] were a simple von Neumann algebraic approach to quantum groups is presented.…”

Section: Locally Compact Quantum Groupsmentioning

confidence: 99%

“…Let A ⊆ M be the Hopf algebra of the underlying algebraic quantum group. We mention that A is the Hopf algebra of matrix coefficients of irreducible, unitary corepresentations of M and refer to [18] for more explanation. LetÂ be its dual, which is the space of linear functionals on A of the form ϕ( · x), where x ∈ A, [20, Theorem 1.2].…”

Section: Fourier Theorymentioning

confidence: 99%

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The $L^p$-Fourier transform on locally compact quantum groups

Caspers¹

2013

J. Operator Theory

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Abstract. Using interpolation properties of non-commutative L p -spaces associated with an arbitrary von Neumann algebra, we define a L p -Fourier transform 1 ≤ p ≤ 2 on locally compact quantum groups. We show that the Fourier transform determines a distinguished choice for the interpolation parameter as introduced by Izumi. We define a convolution product in the L p -setting and show that the Fourier transform turns the convolution product into a product. IntroductionThe Fourier transform is one of the most powerful tools coming from abstract harmonic analysis. Many classical applications, in particular in the direction of L p -spaces, can be found in for example [6]. Here we extend this tool by giving a definition of a Fourier transform on the non-commutative L p -spaces associated with a locally compact quantum group. This gives a link between quantum groups and non-commutative measure theory.Recall that the Fourier transform on locally compact abelian groups can be defined in an L p -setting for p any real number between 1 and 2. This is done in the following way. Let G be a locally compact group and letĜ be its Pontrjagin dual. For a L 1 -function f on G, we define its Fourier transformf to be the function onĜ, which is defined byThenf is a continuous function onĜ vanishing at infinity. So we can consider this transform as a bounded mapthenf is a L 2 -function onĜ and this map extends to a unitary map. It is known that the Fourier transform can be generalized in a L p -setting by means of the Riesz-Thorin theorem, see [1]. The statement of this theorem directly implies the following. For any p, with 1 ≤ p ≤ 2, the linear mapf →f extends uniquely to a bounded mapThis map F p is known as the L p -Fourier transform.Date: May 10th, 2011.

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Section: Locally Compact Quantum Groupsmentioning

confidence: 99%

Section: Fourier Theorymentioning

confidence: 99%

The $L^p$-Fourier transform on locally compact quantum groups

Caspers¹

2013

J. Operator Theory

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“…By passing to finite sums, we can drop the condition that the elements x and y are positive in the C*-algebra, since the usual C*-algebraic decomposition of a element into four positive elements can be made to work in any unital *-closed algebra, and the algebraic elements are a unital *-closed algebra (see, for example, [13] (Theorem 5.4.1)). We thus see that…”

Section: Lemmamentioning

confidence: 99%

“…Each compact-type C*-algebraic quantum group carries with it an algebraic quantum group as a dense subset, and an enveloping Hopf-von Neumann algebra. See [13] for a discussion. The algebraic elements of a compact-type C*-algebraic quantum group A will be denoted A 0 and the enveloping Hopf-von Neumann algebra by A * * .…”

Section: Introductionmentioning

confidence: 99%

Cuntz Semigroups of Compact-Type Hopf C*-Algebras

Kucerovsky

2017

Axioms

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Abstract:The classical Cuntz semigroup has an important role in the study of C*-algebras, being one of the main invariants used to classify recalcitrant C*-algebras up to isomorphism. We consider C*-algebras that have Hopf algebra structure, and find additional structure in their Cuntz semigroups. We show that in many cases, isomorphisms of Cuntz semigroups that respect this additional structure can be lifted to Hopf algebra (bi)isomorphisms, up to a possible flip of the co-product. This shows that the Cuntz semigroup provides an interesting invariant of C*-algebraic quantum groups.

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“…There are many fine references for this material, one of which is [25]. We define SU q (2) as the universal * -algebra over C on the generators a and c satisfying these relations:…”

Section: Definition 75 the Dequantized Algebra Associated To A Is Dementioning

confidence: 99%

Toeplitz Quantization for Non-commutating Symbol Spaces such as SU_q(2)

Sontz

2016

Communications in Mathematics

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Toeplitz quantization is defined in a general setting in which the symbols are the elements of a possibly non-commutative algebra with a conjugation and a possibly degenerate inner product. We show that the quantum group SU q (2) is such an algebra. Unlike many quantization schemes, this Toeplitz quantization does not require a measure. The theory is based on the mathematical structures defined and studied in several recent papers of the author; those papers dealt with some specific examples of this new Toeplitz quantization. Annihilation and creation operators are defined as densely defined Toeplitz operators acting in a quantum Hilbert space, and their commutation relations are discussed. At this point Planck's constant is introduced into the theory. Due to the possibility of non-commuting symbols, there are now two definitions for anti-Wick quantization; these two definitions are equivalent in the commutative case. The Toeplitz quantization introduced here satisfies one of these definitions, but not necessarily the other. This theory should be considered as a second quantization, since it quantizes non-commutative (that is, already quantum) objects. The quantization theory presented here has two essential features of a physically useful quantization: Planck's constant and a Hilbert space where natural, densely defined operators act.Keywords: Toeplitz quantization, non-commutating symbols, creation and annihilation operators, canonical commutation relations, anti-Wick quantization, second quantization of a quantum group MSC 2010: 47B35, 81S99 1 IntroductionThe history of Toeplitz operators covers a bit over one hundred years and includes many major works, far too numerous to mention here. For a recent reference that will give the reader some first links to that extensive literature, see Section 3.5 in [18]. Speaking for myself, the papers [6], [7] and [11] have been rather influential. But the study of Toeplitz operators with symbols coming from a non-commutative algebra seems to be limited mostly to cases where the algebra is a matrix algebra or is some other quite specific noncommutative algebra such as in two recent works, [22] and [23], of the author. The papers [22] and [23] can be considered as two rather elaborated examples of the theory presented here. That study is continued in this paper, but in a much more general setting intended to clarify the mathematical structures at play in those two examples. A new example, the quantum group SU q (2) as symbol space, will also be presented here.The paper is organized as follows. After presenting the foundations of this theory in the next section, we define and analyze in Section 3 the Toeplitz quantization. In particular, Toeplitz operators are defined as densely defined operators acting in a quantum Hilbert space. The symbols of these Toeplitz operators come from a possibly non-commutative algebra A, which in physics terminology serves as the phase space for the theory. The common domain of these Toeplitz operators is P, a pre-Hilbert space and sub-a...

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An Invitation to Quantum Groups and Duality

Cited by 205 publications

References 0 publications

The $L^p$-Fourier transform on locally compact quantum groups

The $L^p$-Fourier transform on locally compact quantum groups

Cuntz Semigroups of Compact-Type Hopf C*-Algebras

Toeplitz Quantization for Non-commutating Symbol Spaces such as SU_q(2)

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An Invitation to Quantum Groups and Duality

Cited by 205 publications

References 0 publications

The $L^p$-Fourier transform on locally compact quantum groups

The $L^p$-Fourier transform on locally compact quantum groups

Cuntz Semigroups of Compact-Type Hopf C*-Algebras

Toeplitz Quantization for Non-commutating Symbol Spaces such as SUq(2)

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Product

Resources

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Toeplitz Quantization for Non-commutating Symbol Spaces such as SU_q(2)