Equivariant quantization of Poisson homogeneous spaces and Kostant's problem

Karolinsky, Eugene; Stolin, Alexander; Tarasov, Vitaly

doi:10.1016/j.jalgebra.2014.03.033

Cited by 6 publications

(9 citation statements)

References 23 publications

(44 reference statements)

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“…In this respect, in a seminal paper [26], Drinfel'd proved that PHS are in one-to-one correspondence with orbits of a natural action of the Lie group G on the variety of Lagrangian subalgebras L(D(G)) of the Drinfel'd double D (G). Explicit examples of PHS as well as some classification results can be found, for instance, in [16,25,[35][36][37][38][39][52][53][54][55]. It is worth stressing the existence of a relevant relation between PHS and the dynamical classical Yang-Baxter equation, which allowed the quantization of certain PHS in terms of G-equivariant star products (see [25]) and has been used, for instance, in the context of gauge fixing in (2 + 1)-gravity [56,57].…”

Section: Poisson Homogeneous Spacesmentioning

confidence: 99%

“…It seems natural to investigate how the conditions of coisotropy, coreductivity and cosymmetry define a very specific subset within the family of all possible Lie bialgebra structures for Lorentzian Lie algebras g Λ with commutation rules of the form (36) (see [20,22,[58][59][60] for classification approaches to Lorentzian Lie bialgebras). It is well-known [58] that in (2 + 1) and (3 + 1) dimensions all (A)dS and Poincaré Lie bialgebras (g Λ , [•, •], δ) are coboundary ones, which means that all of them can be obtained through r-matrices in the form:…”

Section: Lorentzian Lie Bialgebrasmentioning

confidence: 99%

“…where t ⊂ g is the orthogonal complement of h. We remark that this condition is not invariant when a conjugate subgroup H is considered [16,[35][36][37][38][39]. In this paper we will restrict our study to the case of the so-called pointed PHS, in which the isotropy subgroup H (and hence the origin of the homogeneous space) will be fixed.…”

Section: Introductionmentioning

confidence: 99%

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Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces

Ballesteros¹,

Gutiérrez-Sagredo²,

Mercati³

2021

J. Phys. A: Math. Theor.

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Quantum homogeneous spaces are noncommutative spaces with quantum group covariance. Their semiclassical counterparts are Poisson homogeneous spaces, which are quotient manifolds of Lie groups M = G/H equipped with an additional Poisson structure π which is compatible with a Poisson–Lie structure Π on G. Since the infinitesimal version of Π defines a unique Lie bialgebra structure δ on the Lie algebra , we exploit the idea of Lie bialgebra duality in order to study the notion of complementary dual homogeneous space M ⊥ = G*/H ⊥ of a given homogeneous space M with respect to a coisotropic Lie bialgebra. Then, by considering the natural notions of reductive and symmetric homogeneous spaces, we extend these concepts to M ⊥ thus showing that an even richer duality framework between M and M ⊥ arises from them. In order to analyze physical implications of these notions, the case of M being a Minkowski or (anti-) de Sitter Poisson homogeneous spacetime is fully studied, and the corresponding complementary dual reductive and symmetric spaces M ⊥ are explicitly constructed in the case of the well-known κ-deformation, where the cosmological constant Λ is introduced as an explicit parameter in order to describe all Lorentzian spaces simultaneously. In particular, the fact that M ⊥ is a reductive space is shown to provide a natural condition for the representation theory of the quantum analogue of M that ensures the existence of physically meaningful uncertainty relations between the noncommutative spacetime coordinates. Finally, despite these dual spaces M ⊥ are not endowed in general with a G*-invariant metric, we show that their geometry can be described by making use of K-structures.

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Section: Poisson Homogeneous Spacesmentioning

confidence: 99%

Section: Lorentzian Lie Bialgebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces

Ballesteros¹,

Gutiérrez-Sagredo²,

Mercati³

2021

J. Phys. A: Math. Theor.

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“…The proof is based on the fact that A exhausts all of the locally finite part of the U q (g)module End(M). This is the answer to Kostant's problem for quantum groups, [12].…”

Section: Homogeneous Vector Bundles Over Quantum Spheresmentioning

confidence: 96%

Equivariant vector bundles over quantum spheres

Mudrov¹

2017

Preprint

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We quantize homogeneous vector bundles over an even complex sphere S 2n as onesided projective modules over its quantized coordinate ring. We realize them in two different ways: as locally finite C-homs between pseudo-parabolic Verma modules and as induced modules of the quantum orthogonal group. Based on this alternative, we study representations of a quantum symmetric pair related to S 2n q and prove their complete reducibility.

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“…In order to obtain a quantum algebra that can be viewed as the quantum counterpart of the algebra of functions on G/H one needs to consider also the so-called coisotropic subgroups since in many relevant quantum groups G q the quantisation of the subgroup H ⊂ G is not a quantum subgroup. This foundational issue has been treated in a number of works such as [17][18][19][20][21][22][23], and several examples of quantum homogeneous spaces have been explicitly constructed, see [24][25][26][27][28][29][30][31][32][33][34][35] and references therein.…”

Section: Introductionmentioning

confidence: 99%

AdS Poisson homogeneous spaces and Drinfel’d doubles

Ballesteros¹,

Meusburger²,

Naranjo³

2017

J. Phys. A: Math. Theor.

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The correspondence between Poisson homogeneous spaces over a Poisson-Lie group G and Lagrangian Lie subalgebras of the classical double D(g) is revisited and explored in detail for the case in which g = D(a) is a classical double itself. We apply these results to give an explicit description of some coisotropic 2d Poisson homogeneous spaces over the group SL(2, R) ∼ = SO(2, 1), namely 2d anti de Sitter space, 2d hyperbolic space and the lightcone in 3d Minkowski space. We show how each of these spaces is obtained as a quotient with respect to a Poisson-subgroup for one of the three inequivalent Lie bialgebra structures on sl(2, R) and as a coisotropic one for the others.We then construct families of coisotropic Poisson homogeneous structures for 3d anti de Sitter space AdS 3 and show that the ones that are quotients by a Poisson subgroup are determined by a three-parameter family of classical r-matrices for so(2, 2), while the non Poisson-subgroup cases are much more numerous. In particular, we present the two Poisson homogeneous structures on AdS 3 that arise from two Drinfel'd double structures on SO(2, 2). The first one realises AdS 3 as a quotient of SO(2, 2) by the Poisson-subgroup SL(2, R), while the second one, the non-commutative spacetime of the twisted κ-AdS deformation, realises AdS 3 as a coisotropic Poisson homogeneous space.

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Equivariant quantization of Poisson homogeneous spaces and Kostant's problem

Cited by 6 publications

References 23 publications

Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces

Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces

Equivariant vector bundles over quantum spheres

AdS Poisson homogeneous spaces and Drinfel’d doubles

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