Quantum isometry group of the $n$-tori

Bhowmick, Jyotishman

doi:10.1090/s0002-9939-09-09908-0

Cited by 13 publications

(18 citation statements)

References 9 publications

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“…As an example here, consider the torus T ⊂ C. A straightforward complex extension of the trick in Proposition 3.2 (2), explained in [1], shows that we have G + (T) = G(T) = U 1 . We should mention that it is true as well that we have G + (T r ) = G(T r ) = O 2 , therefore confirming the formula G + (T) = G + (T r ) ∩ U + 1 , but this result holds due to much deeper reasons, explained by Bhowmick in [8]. For more on these issues, see also [16].…”

Section: Complexification Issuessupporting

confidence: 77%

Quantum Isometries and Noncommutative Spheres

Banica

Goswami²

2010

Commun. Math. Phys.

182

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The real sphere S N −1 R appears as increasing union, over d ∈ {1, . . . , N }, of its "polygonal" versions S N −1,d−1Motivated by general classification questions for the undeformed noncommutative spheres, smooth or not, we study here the quantum isometries of S N −1,d−1 R , and of its various noncommutative analogues, obtained via liberation and twisting. We discuss as well a complex version of these results, with S N −1 R replaced by the complex sphere S N −1 C

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Section: Complexification Issuessupporting

confidence: 77%

Quantum Isometries and Noncommutative Spheres

Banica

Goswami²

2010

Commun. Math. Phys.

182

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“…As already mentioned in the introduction, the results in [10], [13], along with the recent ones in [29], and also with those in the previous section, give some substantial evidence for the conjectural statement "M classical and connected implies G(M) = G + (M)".…”

Section: Group Dual Actionssupporting

confidence: 69%

“…Observe that this kind of statement, and the above algebraic technology in general, is far below from what would be needed for attacking the conjecture in the introduction. For instance our results don't cover the torus T k , obtained as a "mixing" product of k circles, and which is known from [10] not to have genuine quantum isometries.…”

Section: Some Basic Resultsmentioning

confidence: 68%

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Quantum isometries and group dual subgroups

Banica¹,

Bhowmick²,

Commer³

2012

Ann. math. Blaise Pascal

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We study the discrete groups Λ whose duals embed into a given compact quantum group, Λ ⊂ G. In the matrix case G ⊂ U + n the embedding condition is equivalent to having a quotient map Γ U → Λ, where F = {Γ U |U ∈ U n } is a certain family of groups associated to G. We develop here a number of techniques for computing F , partly inspired from Bichon's classification of group dual subgroups Λ ⊂ S + n . These results are motivated by Goswami's notion of quantum isometry group, because a compact connected Riemannian manifold cannot have non-abelian group dual isometries.2000 Mathematics Subject Classification. 58J42 (46L87).

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“…The result can be deduced from calculations in [12], but we can also offer an explicit isomorphism. Denote the images of elements of {U iα,jβ : iα, jβ ∈ J p } in the quotient space with respect to the commutator ideal of A h (p, 0) by {Û iα,jβ : iα, jβ ∈ J p } and let the quotient C * -algebra be denoted by A com h (p, 0).…”

Section: Classical Versions and Interpretations In Terms Of Classicalmentioning

confidence: 89%

Two-parameter families of quantum symmetry groups

Banica

Skalski

2011

Journal of Functional Analysis

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We introduce and study natural two-parameter families of quantum groups motivated on one hand by the liberations of classical orthogonal groups and on the other by quantum isometry groups of the duals of the free groups. Specifically, for each pair (p, q) of non-negative integers we define and investigate quantum groups O + (p, q), B + (p, q), S + (p, q) and H + (p, q) corresponding to, respectively, orthogonal groups, bistochastic groups, symmetric groups and hyperoctahedral groups. In the first three cases the new quantum groups turn out to be related to the (dual free products of ) free quantum groups studied earlier.For H + (p, q) the situation is different and we show that H + (p, 0) ≈ QISO( F p ), where the latter can be viewed as a liberation of the classical isometry group of the p-dimensional torus.

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Quantum isometry group of the $n$-tori

Cited by 13 publications

References 9 publications

Quantum Isometries and Noncommutative Spheres

Quantum Isometries and Noncommutative Spheres

Quantum isometries and group dual subgroups

Two-parameter families of quantum symmetry groups

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