Haar states and Lévy processes on the unitary dual group

Cébron, Guillaume; Ulrich, Michael R.

doi:10.1016/j.jfa.2015.12.004

Cited by 10 publications

(23 citation statements)

References 29 publications

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“…Indeed, the multiplication µ : G × G → G of a group translates into ∆ : C(G) → C(G × G) on the algebraic level, but we have C(G × G) ∼ = C(G) ⊗ C(G) rather than C(G × G) ∼ = C(G) * C(G). However, dual groups have been studied as reasonable noncommutative analogues of groups in various settings, see for instance [CU15] for an overview on the literature. The main example of a dual group is Brown's algebra [Bro81] denoted by U n or U nc n .…”

Section: Unitary Easy Groupsmentioning

confidence: 99%

Unitary Easy Quantum Groups: The Free Case and the Group Case

Tarrago

Weber

2016

Int Math Res Notices

163

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Abstract. Easy quantum groups have been studied intensively since the time they were introduced by Banica and Speicher in 2009. They arise as a subclass of (C * -algebraic) compact matrix quantum groups in the sense of Woronowicz. Due to some Tannaka-Krein type result, they are completely determined by the combinatorics of categories of (set theoretical) partitions. So far, only orthogonal easy quantum groups have been considered in order to understand quantum subgroups of the free orthogonal quantum group O + n . We now give a definition of unitary easy quantum groups using colored partitions to tackle the problem of finding quantum subgroups of U + n . In the free case (i.e. restricting to noncrossing partitions), the corresponding categories of partitions have recently been classified by the authors by purely combinatorial means. There are ten series showing up each indexed by one or two discrete parameters, plus two additional quantum groups. We now present the quantum group picture of it and investigate them in detail. We show how they can be constructed from other known examples using generalizations of Banica's free complexification. For doing so, we introduce new kinds of products between quantum groups.We also study the notion of easy groups. IntroductionIn order to provide a new notion of symmetries adapted to the situation in operator algebras, Woronowicz introduced compact matrix quantum groups in 1987 in [Wor87]. The idea is roughly to take a compact Lie group G ⊆ M n (C) and to pass to the algebra C(G) of continuous functions over it. It turns out that this algebra fulfills all axioms of a C * -algebra with the special feature that the multiplication is commutative. If we now also dualize main properties of the group law µ : G×G → G to properties of a map ∆ : C(G) → C(G×G) ∼ = C(G)⊗C(G) (the comultiplication), and consider noncommutative C * -algebras equiped with such a map ∆, we are right in the heart of the definition of a compact matrix quantum group (see Section 3.1 for details). It generalizes the notion of a compact matrix group. The reader not familiar with quantum groups might also first jump to Section 5 and read it as a motivation.Date: December 2, 2015. 2010 Mathematics Subject Classification. 20G42 (Primary); 05A18, 46L54 (Secondary).

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Section: Unitary Easy Groupsmentioning

confidence: 99%

Unitary Easy Quantum Groups: The Free Case and the Group Case

Tarrago

Weber

2016

Int Math Res Notices

163

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“…Note that free compressions of unitary variables correspond to quantum random variables on the unitary dual group, as studied in [CU16], and Haar unitaries correspond to the particular case of the Haar trace on the unitary dual group. Another particular case of interest is the unitary free Brownian motion U(t), which has a bounded density whenever t > 0 (see [Voi99, Proposition 1.6]): the free compressions (U i,j (t)) n i,j=1 of the unitary free Brownian motion have maximal ∆ whenever t > 0.…”

Section: Theorem 34 [Shl06 Theorem 1] Let δ Be a Non-commutative Deri...mentioning

confidence: 99%

Universality of free random variables: atoms for non-commutative rational functions

Arizmendi¹,

Cébron²,

Speicher³

et al. 2021

Preprint

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We address the following question: what can one say, for a tuple (Y 1 , . . . , Y d ) of normal operators in a tracial operator algebra setting with prescribed sizes of the eigenspaces for each Y i , about the sizes of the eigenspaces for any non-commutative polynomial P (Y 1 , . . . , Y d ) in those operators? We show that for each polynomial P there are unavoidable eigenspaces, which occur in P (Y 1 , . . . , Y d ) for any (Y 1 , . . . , Y d ) with the prescribed eigenspaces for the marginals. We will describe this minimal situation both in algebraic terms -where it is given by realizations via matrices over the free skew field and via rank calculations -and in analytic terms -where it is given by freely independent random variables with prescribed atoms in their distributions. The fact that the latter situation corresponds to this minimal situation allows to draw many new conclusions about atoms in polynomials of free variables. In particular, we give a complete description of atoms in the free commutator and the free anti-commutator. Furthermore, our results do not only apply to polynomials, but much more general also to non-commutative rational functions. O.A. was supported by Conacyt grant A1-S-9764. G.C. and S.Y. were partly supported by the Project MESA (ANR-18-CE40-006) of the French National ResearchAgency (ANR). O.A. and R.S. gratefully acknowledge financial support by SFB TRR 195.in other words, for each such rational expression R in d non-commuting variables, ...,Y d ) . Note that this means that for determining the atoms in R(X 1 , . . . , X d ), for X 1 , . . . , X d being * -free, we can model the non-discrete parts in µ X i by any non-atomic distribution; choosing semicircular distributions for this seems to be a good option.Another consequence of Theorem 1.2 is that it allows us to define, for each noncommutative rational function R, a corresponding rational convolution R on discrete sub-probability measures on C, according to. The proof of our general results will proceed via modeling the situation of given marginal distributions, in the case of rational weights for the atoms, by a purely algebraic object, namely by matrices over the free field. The free field is the universal field of fractions of non-commutative polynomials. Those matrices over the free field contain thus all algebraic information common to all realizations by operator tuples (Y 1 , . . . , Y d ) with the given atoms in their marginals. An unavoidable atom at 0 in the distribution of P (Y 1 , . . . , Y d ) corresponds in this model to the non-invertibility of the matrices. By results of [MSY19], the special operator tuples (X 1 , . . . , X d ), where X 1 , . . . , X d are free, yield an analytic model for this algebraic situation. The inner rank on the algebraic side is then the same as the von Neumann operator algebraic rank on the analytic side, which in turn corresponds to the size of the atom in P (X 1 , . . . , X d ) at 0. (By shifting P by a constant, we can of course move the location of any atom to 0.) This shows the...

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“…Furthermore, inspired by computations in (Kirillov, 2004) page 59-61 about the Plancherel measure in the unitary of the 3-dimensional Heisenberg Lie group, we shall generalize this result to the Plancherel measure in the unitary dual of -dimensional Heisenberg Lie group corresponding to its irreducible unitary representation presented in (Corwin & Greenleaf, 1990) page 48. In the previous work we can see the notions and some applications of unitary dual groups (Bagarello & Russo, 2018), (Baraquin, 2017), and (Cebron & Ulrich, 2016).…”

Section: Keywordmentioning

confidence: 99%

On Properties of the (2n+1)-Dimensional Heisenberg Lie Algebra

Kurniadi¹

2020

JTAM

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In the present paper, we study some properties of the Heisenberg Lie algebra of dimension . The main purpose of this research is to construct a real Frobenius Lie algebra from the Heisenberg Lie algebra of dimension . To achieve this, we exhibit how to compute the derivation of the Heisenberg Lie algebra by following Oom’s result. In this research, we use a literature review method to some related papers corresponding to a derivation of a Lie algebra, Frobenius Lie algebras, and Plancherel measure. Determining a conjecture of a real Frobenius Lie algebra is obtained. As the main result, we prove that conjecture. Namely, for the given the Heisenberg Lie algebra, there exists a commutative subalgebra of dimension one such that its semi direct sum is a real Frobenius Lie algebra of dimension . Futhermore, in the notion of the Lie group of the Heisenberg Lie algebra which is called the Heisenberg Lie group, we compute the generalized character of its group and we determine the Plancherel measure of the unitary dual of the Heisenberg Lie group. As our contributions, we complete some examples of Frobenius Lie algebras obtained from a nilpotent Lie algebra and we also give alternative computations to find the Plancherel measure of the Heisenberg Lie group.

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Haar states and Lévy processes on the unitary dual group

Cited by 10 publications

References 29 publications

Unitary Easy Quantum Groups: The Free Case and the Group Case

Unitary Easy Quantum Groups: The Free Case and the Group Case

Universality of free random variables: atoms for non-commutative rational functions

On Properties of the (2n+1)-Dimensional Heisenberg Lie Algebra

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