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QUANTIZATION OF SOME POISSONLIE DYNAMICAL rMATRICES AND POISSON HOMOGENEOUS SPACES
, 2004
"... Abstract. PoissonLie (PL) dynamical rmatrices are generalizations of dynamical rmatrices, where the base is a PoissonLie group. We prove analogues of basic results for these rmatrices, namely constructions of (quasi)Poisson groupoids and of Poisson homogeneous spaces. We introduce a class of PL ..."
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Abstract. PoissonLie (PL) dynamical rmatrices are generalizations of dynamical rmatrices, where the base is a PoissonLie group. We prove analogues of basic results for these rmatrices, namely constructions of (quasi)Poisson groupoids and of Poisson homogeneous spaces. We introduce a class of PL dynamical rmatrices, associated to nondegenerate Lie bialgebras with a splitting; this is a generalization of trigonometric rmatrices with an abelian base. We prove a composition theorem for PL dynamical rmatrices, and construct quantizations of the polarized PL dynamical rmatrices. This way, we obtain quantizations of Poisson homogeneous structures on G/L (G a semisimple Lie group, L a Levi subgroup), thereby generalizing earlier constructions.
Irreducible highestweight modules and equivariant quantization
, 2005
"... The notion of deformation quantization, motivated by ideas coming from both physics and mathematics, was introduced in classical papers [2, 7, 8]. Roughly speaking, a deformation quantization of a Poisson manifold (P, { ,}) is a formal ..."
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The notion of deformation quantization, motivated by ideas coming from both physics and mathematics, was introduced in classical papers [2, 7, 8]. Roughly speaking, a deformation quantization of a Poisson manifold (P, { ,}) is a formal
Quantum groupoids and . . .
, 2003
"... In this paper we realize dynamical categories introduced in our previous paper as categories of modules over bialgebroids. We study bialgebroids arising in this way. We define quasitriangular structure on bialgebroids and present examples of quasitriangular bialgebroids related to dynamical categori ..."
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In this paper we realize dynamical categories introduced in our previous paper as categories of modules over bialgebroids. We study bialgebroids arising in this way. We define quasitriangular structure on bialgebroids and present examples of quasitriangular bialgebroids related to dynamical categories. We show that dynamical twists over an arbitrary base give rise to bialgebroid twists. We prove that the classical dynamical rmatrices over an arbitrary base manifold
Equivariant quantization of Poisson homogeneous spaces and Kostant’s problem
, 908
"... Let g be a finite dimensional split semisimple Lie algebra and λ a weight of g. Let F be the algebra of quantized regular functions on the connected simply connected group G corresponding to g. In the present paper we introduce a certain subspace F ′ of F (which is not necessary a subalgebra of F) a ..."
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Let g be a finite dimensional split semisimple Lie algebra and λ a weight of g. Let F be the algebra of quantized regular functions on the connected simply connected group G corresponding to g. In the present paper we introduce a certain subspace F ′ of F (which is not necessary a subalgebra of F) and endow it with an associative ⋆product using the socalled reduced fusion element. We prove that the algebra (F ′ (),⋆) is isomorphic to EndL(λ) fin, where L(λ) is the irreducible highest weight Ǔqgmodule and “fin ” stands for the subalgebra of the locally finite elements with respect to the adjoint action of Ǔqg. The introduced ⋆product has some limiting properties what enables us to prove Kostant’s problem for Ǔqg in certain cases. We remind the reader that this means that ( EndL(λ)) fin coincides with the image of Ǔqg in EndL(λ). We also note that if λ is such that 〈λ,α ∨ i 〉 = 0 for some simple roots αi and generic otherwise, then (F,⋆) is a Ǔqginvariant quantization of the Poisson homogeneous space G/K, where K is the stabilizer of λ.
On dynamical smash product
, 708
"... In the theory of dynamical YangBaxter equation, with any Hopf algebra H and a certain Hmodule and Hcomodule algebra L (base algebra) one associates a monoidal ..."
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In the theory of dynamical YangBaxter equation, with any Hopf algebra H and a certain Hmodule and Hcomodule algebra L (base algebra) one associates a monoidal