Irreducible highest weight modules and equivariant quantization

Karolinsky, Eugene; Stolin, Alexander; Tarasov, Vitaly

doi:10.1016/j.aim.2006.08.004

Cited by 11 publications

(16 citation statements)

References 13 publications

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“…Remark 4. Theorem 14 together with results of [15] implies that the algebras Hom U (L(λ), L(λ)⊗F ), (End L(λ)) fin , and F [0] K λ + K λ , ⋆ λ are isomorphic as right Hopf module algebras over U.…”

Section: Combining These Together We Getmentioning

confidence: 94%

“…We have ϕ * ψ ∈ Hom U (M, M ⊗ F ), and this definition equips Hom U (M, M ⊗ F ) with a unital associative algebra structure. Consider the map Φ : [15,Proposition 6]). Now we apply this to M = M(λ) and M = L(λ).…”

Section: Star Products and Fusion Elements 31 Algebra Of Intertwininmentioning

confidence: 99%

“…This idea was realized in [13,14,15] and first relations between quantization of Poisson homogeneous spaces of triangular type with the so-called Kostant's problem for the highest weight irreducible U(g)-modules were considered in [15]. It is worth to mention that, in particular, this approach enabled the authors to obtain explicit formulas for quantization of the Kirillov-Kostant-Lie Poisson bracket on reductive co-adjoint orbits of g.…”

Section: Introductionmentioning

confidence: 99%

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Equivariant quantization of Poisson homogeneous spaces and Kostant's problem

2014

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We find a partial solution to the longstanding problem of Kostant concerning description of the so-called locally finite endomorphisms of highest weight irreducible modules. The solution is obtained by means of its reduction to a far-reaching extension of the quantization problem. While the classical quantization problem consists in finding ⋆-product deformations of the commutative algebras of functions, we consider the case when the initial object is already a noncommutative algebra, the algebra of functions within q-calculus. (2000). 17B37, 17B10, 53D17, 53D55. Mathematics Subject Classifications

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Section: Combining These Together We Getmentioning

confidence: 94%

Section: Star Products and Fusion Elements 31 Algebra Of Intertwininmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Equivariant quantization of Poisson homogeneous spaces and Kostant's problem

2014

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“…Its inverse is closely related to intertwining operators [2], the dynamical Yang-Baxter equation [3], and invariant star product on homogeneous spaces [4,5].…”

Section: Introductionmentioning

confidence: 99%

Orthogonal basis for the Shapovalov form on $U_q(\mathfrak{sl}(n + 1))$

Mudrov

2015

Rev. Math. Phys.

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“…The DAF is also present there, but with no explicit connection with the Shapovalov form. Another paper of interest, [11], directly generalizes the ideas of [7]. Remarkably, the approach of [11] can be suitable for certain conjugacy classes with non-Levi isotropy subgroups, which drop from the framework of the DYBE in its present version, but still can be quantized in a similar way, [12].As an illustration, we give the star product on the homogeneous space GL(n+1)/GL(n)× GL(1) that is equviariant under the action of either classical or quantum group GL(n + 1).In this simple case the Shapovalov form can be calculated explicitly.…”

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confidence: 99%

On Dynamical Adjoint Functor

Mudrov

2011

Appl Categor Struct

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We give an explicit formula relating the dynamical adjoint functor and dynamical twist over nonalbelian base to the invariant pairing on parabolic Verma modules.As an illustration, we give explicit U sl(n) -and U sl(n) -invariant star product on projective spaces.A recipe of constructing dynamical twist was suggested in [6], through dynamical adjoint functor (DAF) between certain module categories associated with a Levi subalgebra H in a reductive (classical or quantum) universal enveloping algebra U. One of them is a certain subcategory of finite dimensional H-modules, while the other is the parabolic O-category, both equipped with tensor multiplication by finite dimensional U-modules.Remark that quantization of function algebras on homogeneous space involve only scalar PVM. General PVM appear in quantization of associated vector bundles as projective modules over function algebras, as discussed in [6]. This is where the nonabelian base really plays a role, along with the corresponding dynamical twist and DAF.An alternative approach to quantization was employed in [7], where the star product on semisimple coadjoint orbits of simple complex Lie groups was constructed directly from the Shapovalov form on scalar PVM. It was clear that the methods of [6] and [7] were close and based on similar underlying ideas. Relation of the Shapovalov form on Verma modules with the dynamical twist was already indicated in earlier works on DYBE in the special case of Cartan base, [5]. This construction had motivated the generalization for the nonabelian base, which was given in [6], however, without straight use of the Shapovalov form. A sort of "nonabelian paring" associated with the triangular factorization of (quantized) universal enveloping algebras, which is equivalent to Shapovalov form in representations, was employed in [8] for construction of the dynamical twist. It was done directly, bypassing DAF. Thus, the explicit relation of the Shapovalov form to DAF over general Levi subalgebra, which is a more fundamental object than dynamical twist, has not been given much attention in the literature. In the present work we do it in a most elementary way.We would like to mention the following two papers in connection with the present work.In [9], the dynamical twist is constructed with the use of the ABRR equation, [10]. The DAF is also present there, but with no explicit connection with the Shapovalov form. Another paper of interest, [11], directly generalizes the ideas of [7]. Remarkably, the approach of [11] can be suitable for certain conjugacy classes with non-Levi isotropy subgroups, which drop from the framework of the DYBE in its present version, but still can be quantized in a similar way, [12].As an illustration, we give the star product on the homogeneous space GL(n+1)/GL(n)× GL(1) that is equviariant under the action of either classical or quantum group GL(n + 1).In this simple case the Shapovalov form can be calculated explicitly. We show that its U(g)invariant limit coincides with the star product on the projective ...

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Irreducible highest weight modules and equivariant quantization

Cited by 11 publications

References 13 publications

Equivariant quantization of Poisson homogeneous spaces and Kostant's problem

Equivariant quantization of Poisson homogeneous spaces and Kostant's problem

Orthogonal basis for the Shapovalov form on $U_q(\mathfrak{sl}(n + 1))$

On Dynamical Adjoint Functor

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