Comparison and continuity of Wick-type star products on certain coadjoint orbits

Esposito, Chiara; Schmitt, Philipp; Waldmann, Stefan

doi:10.1515/forum-2018-0302

Cited by 7 publications

(11 citation statements)

References 38 publications

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“…For the hyperbolic disc, the quantum algebra .A.D n /; " / agrees with the algebra obtained in [28] while, for the 2-sphere, .A.S 2 /; " / is the algebra considered in [15].…”

Section: Main Theorem I the Shapovalov Pairing Hsupporting

confidence: 65%

Strict quantization of coadjoint orbits

Schmitt

2021

J. Noncommut. Geom.

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For every semisimple coadjoint orbit y O of a complex connected semisimple Lie group y G, we obtain a family of y G-invariant products O " on the space of holomorphic functions on y O. For every semisimple coadjoint orbit O of a real connected semisimple Lie group G, we obtain a family of G-invariant products " on a space A.O/ of certain analytic functions on O by restriction. A.O/, endowed with one of the products " , is a G-Fréchet algebra, and the formal expansion of the products around " D 0 determines a formal deformation quantization of O, which is of Wick type if G is compact. Our construction relies on an explicit computation of the canonical element of the Shapovalov pairing between generalized Verma modules and complex analytic results on the extension of holomorphic functions.

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“…For the hyperbolic disc, the quantum algebra .A.D n /; " / agrees with the algebra obtained in [28] while, for the 2-sphere, .A.S 2 /; " / is the algebra considered in [15].…”

Section: Main Theorem I the Shapovalov Pairing Hsupporting

confidence: 65%

Strict quantization of coadjoint orbits

Schmitt

2021

J. Noncommut. Geom.

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“…We would like to remark that, similar as in [12], one can also use the description of the star product using bidifferential operators to prove its continuity. Theorem 5.26 For every h ∈ Ω, the product ⋆ red,h on P(M red ) extends to a continuous associative product on A (M red ).…”

Section: The Analytic Casementioning

confidence: 96%

“…This approach has been worked out e.g. for star products of exponential type on possibly infinitedimensional vector spaces [24,26], for the Gutt star product on the dual of a Lie algebra [13], for the 2-sphere [12], for the hyperbolic disc n [3,17], and for semisimple coadjoint orbits of semisimple connected Lie groups [23]. In the case of the hyperbolic disc the completed algebra A has a nice geometric interpretation as functions that allow an extension to holomorphic functions on some fixed, larger space.…”

Section: Introductionmentioning

confidence: 99%

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Wick Rotations in Deformation Quantization

Schmitt,

Schötz

2019

Preprint

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We study formal and non-formal deformation quantizations of a family of manifolds that can be obtained by phase space reduction from 1+n with the Wick star product in arbitrary signature. Two special cases of such manifolds are the complex projective space È n and the complex hyperbolic disc n . We generalize several older results to this setting: The construction of formal star products and their explicit description by bidifferential operators, the existence of a convergent subalgebra of "polynomial" functions, and its completion to an algebra of certain analytic functions that allow an easy characterization via their holomorphic extensions. Moreover, we find an isomorphism between the non-formal deformation quantizations for different signatures,

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“…This is the classical limit of some well-known non-formal star products, which can also be obtained by symmetry reduction of the Wick star product on 1+n , see e.g. [2,5,6,8,10,12,27].…”

Section: Introductionmentioning

confidence: 99%

Symmetry Reduction of States I

Schmitt¹,

Schötz²

2021

Preprint

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We develop a general theory of symmetry reduction of states on (possibly non-commutative) * -algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra g. The key idea advocated for in this article is that the "correct" notion of positivity on a * -algebra A is not necessarily the algebraic one whose positive elements are the sums of Hermitian squares a * a with a ∈ A, but can be a more general one that depends on the example at hand, like pointwise positivity on * -algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on A thus depends on this choice of positivity on A, and the notion of positivity on the reduced algebra A µ-red should be such that states on A µ-red are obtained as reductions of certain states on A. In the special case of the * -algebra of smooth functions on a Poisson manifold M , this reduction scheme reproduces the coisotropic reduction of M , where the reduced manifold M µ-red is just a geometric manifestation of the reduction of the evaluation functionals associated to certain points of M . However, we are mainly interested in applications to non-formal deformation quantization and therefore also discuss the reduction of the Weyl algebra, and of the polynomial algebra whose non-commutative analogs we plan to examine in future projects.

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Comparison and continuity of Wick-type star products on certain coadjoint orbits

Cited by 7 publications

References 38 publications

Strict quantization of coadjoint orbits

Strict quantization of coadjoint orbits

Wick Rotations in Deformation Quantization

Symmetry Reduction of States I

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