5. Deformations of algebras

It is proven that Rankin-Cohen brackets form an associativedeformation of the algebra of polynomials whose coeffcients are holomorphicfunctions on the upper half-plane.

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Speciality and deformations of algebras

Algebra ◽

10.1515/9783110805697.345 ◽

2012 ◽

Author(s):

I. P. Shestakov

Keyword(s):

Deformations Of Algebras

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Deformations of algebras defined by tilting bundles

Journal of Algebra ◽

10.1016/j.jalgebra.2018.07.031 ◽

2018 ◽

Vol 513 ◽

pp. 388-434

Author(s):

Joseph Karmazyn

Keyword(s):

Deformations Of Algebras

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Quantum group actions, twisting elements, and deformations of algebras

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2006.01.011 ◽

2007 ◽

Vol 208 (1) ◽

pp. 371-389

Author(s):

Georgia Benkart ◽

Sarah Witherspoon

Keyword(s):

Quantum Group ◽

Group Actions ◽

Quantum Group Actions ◽

Deformations Of Algebras

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DEFORMATIONS OF ALGEBRAS OF OBSERVABLES AND THE CLASSICAL LIMIT OF QUANTUM MECHANICS

Reviews in Mathematical Physics ◽

10.1142/s0129055x93000243 ◽

1993 ◽

Vol 05 (04) ◽

pp. 775-806 ◽

Cited By ~ 7

Author(s):

N. P. LANDSMAN

Keyword(s):

Classical Limit ◽

Wigner Distribution ◽

Distribution Functions ◽

Poisson Manifolds ◽

Ideal Space ◽

Convergence Criterion ◽

Yang Mills ◽

C Algebra ◽

Classical Counterpart ◽

Deformations Of Algebras

The quantum algebra of observables of a particle moving on a homogeneous configuration space Q = G/H, the transformation group C*-algebra C* (G, G/H), is deformed into its classical counterpart C0 ((T*G)/H). The Poisson structure of the latter is obtained as the classical limit of the quantum commutator. The superselection sectors of both algebras describe the particle moving in an external Yang–Mills field. Analytical aspects of deformation theory, such as the nature of the limit ħ → 0, are studied in detail. A physically motivated convergence criterion in ħ is introduced. The Weyl–Moyal quantization formalism, and the associated use of Wigner distribution functions, is generalized from flat phase spaces T*ℝn to Poisson manifolds of the form (T*G)/H. The classical limit of quantum states as well as of superselection sectors is investigated. The former is handled by introducing the notion of a classical germ, generalizing coherent states. The latter is analyzed by studying the Jacobson topology on the primitive ideal space of a certain continuous field of C*-algebras, constructed from the classical and the quantum algebras of observables. The symplectic leaves of (T*G)/H are confirmed to be the correct classical analogue of the quantum superselection sectors of C*(G, G/H).

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