Quantum groups and deformation quantization: Explicit approaches and implicit aspects

DOI10.1063/1.1786681zbMath1071.53052OpenAlexW2082749417MaRDI QIDQ4833509

Murray Gerstenhaber, Philippe Bonneau, Anthony Giaquinto, Daniel Sternheimer

Publication date: 15 December 2004

Published in: Journal of Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1063/1.1786681

Mathematics Subject Classification ID

Related Items (24)

Least uncertainty principle in deformation quantization ⋮ On quasi-Hopf smash products and twisted tensor products of quasialgebras. ⋮ Some bialgebroids constructed by Kadison and Connes-Moscovici are isomorphic. ⋮ Minimal parabolic quantum groups in twist deformations ⋮ L-R-smash product for (quasi-)Hopf algebras. ⋮ General twisting of algebras. ⋮ The global dimension of L-R twisted smash products. ⋮ L-R-smash biproducts, double biproducts and a braided category of Yetter-Drinfeld-Long bimodules. ⋮ Quantization: Deformation and/or functor? ⋮ Quantization of semi-classical twists and noncommutative geometry ⋮ Gorenstein global dimensions and representation dimensions for L-R smash products. ⋮ Parabolic twists for the linear algebras \(A_{n-1}\) ⋮ L-R smash products for multiplier Hopf algebras. ⋮ L-R-Smash Products and L-R-Twisted Tensor Products of Algebras ⋮ Hom-L-R-smash products, Hom-diagonal crossed products and the Drinfeld double of a Hom-Hopf algebra. ⋮ On L-R Smash Products of Hopf Algebras ⋮ Duality theorem for L-R crossed coproducts ⋮ Orderings and non-formal deformation quantization ⋮ Cyclic homology of Brzeziński's crossed products and of braided Hopf crossed products. ⋮ A duality theorem for weak L-R smash products. ⋮ Phase space quantum mechanics ⋮ Quantum duality in quantum deformations ⋮ Reconstruction of universal Drinfeld twists from representations ⋮ Twisted Smash Products and L-R Smash Products for Biquasimodule Hopf Quasigroups

Cites Work

• The quantum group structure associated with non-linearly extended Virasoro algebras
• Représentations \({}^*\) des groupes exponentiels. \((^*\)- representations of exponential groups)
• Dynamical Yang-Baxter equations, quasi-Poisson homogeneous spaces, and quantization
• Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction
• Quantum deformation of Lorentz group
• Quantum R-matrices and factorization problems
• Universal deformation formulae, symplectic Lie groups and symmetric spaces
• An explicit *-product on the cotangent bundle of a Lie group
• Deformation theory applied to quantization and statistical mechanics
• A q-difference analogue of \(U({\mathfrak g})\) and the Yang-Baxter equation
• *-products in the method of orbits for nilpotent groups
• Some ideas about quantization
• Six generator \(q\)-deformed Lorentz algebra
• A natural and rigid model of quantum groups
• Quantization of tensor representations and deformation of matrix bialgebras
• Semi-direct products of Hopf algebras
• Deformation theory and quantization. I: Deformations of symplectic structures
• Bialgebra actions, twists, and universal deformation formulas
• Deformation quantization on cotangent bundles
• Operads and motives in deformation quantization
• Lessons from quantum field theory: Hopf algebras and spacetime geometries
• Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group
• The hidden group structure of quantum groups: Strong duality, rigidity and preferred deformations
• Boundary solutions of the classical Yang-Baxter equation
• Noncommutative deformation theory
• Generalization and exact deformations of quantum groups
• Four-dimensional simply connected symplectic symmetric spaces
• Nonstandard solutions of the Yang-Baxter equation
• Boundary solutions of the quantum Yang-Baxter equation and solutions in three dimensions
• Algebraic deformations arising from orbifolds with discrete torsion.
• Moduli space and structure of noncommutative 3-spheres
• Generalized Jordanian \(R\)-matrices of Cremmer-Gervais type.
• Cohomology and associated deformations for not necessarily co-associative bialgebras
• On the twisted convolution product and the Weyl transformation of tempered distributions
• Nilpotent Fourier transform and applications
• Deformation quantization of Heisenberg manifolds
• Invariant star products and representations of compact semisimple Lie groups
• Strict quantization of solvable symmetric spaces
• Deformation quantization of Poisson manifolds
• Quantum coadjoint orbits of \(\text{GL}(n)\) and generalized Verma modules
• A star-product approach to noncompact quantum groups
• Quantization of Kähler manifolds. IV
• Quantum Weyl algebras
• A candidate for a noncompact quantum group
• Moyal product and representations of solvable Lie groups
• On Poisson homogeneous spaces of Poisson-Lie groups
• Quantization of Lie bialgebras. I
• Quantization of Poisson-Lie groups and applications
• Deformation quantization and Nambu mechanics
• On deformations of complex analytic structures. III: Stability theorems for complex structures
• The cohomology structure of an associative ring
• On the deformation of rings and algebras. III
• A cohomological construction of quantization functors of Lie bialgebras
• Invariant *-products on coadjoint orbits and the Shapovalov pairing
• On the deformation of rings and algebras
• Multiparameter quantum groups and twisted quasitriangular Hopf algebras
• A class of operator algebras which are determined by groups
• Geometrical theory of contractions of groups and representations
• Deformations of Lie group and Lie algebra representations
• A THEOREM ON STABILITY OF COMPLEX STRUCTURES
• On rational solutions of Yang-Baxter equation for $\mathfrak{sl}(n)$.
• Bialgebra cohomology, deformations, and quantum groups.
• Generalised Deformations, Koszul Resolutions, Moyal Products
• Actions of multiplier hopf algebras
• Extended Jordanian twists for Lie algebras
• TOPOLOGICAL QUANTUM DOUBLE
• A q-deformed Lorentz algebra
• Berezin’s quantization on flag manifolds and spherical modules
• ELEMENTARY PARABOLIC TWIST
• On the Quantum Correction For Thermodynamic Equilibrium
• Symbolic calculus on the time-frequency half-plane
• Generalized Hamiltonian Dynamics
• Tamarkins proof of Kontsevich formality theorem
• Star exponentials for any ordering of the elements of the inhomogeneous symplectic Lie algebra
• Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras
• A noncommutative Hopf structure on 𝒞 ∞[SL(2,C) as a quantum Lorentz group]
• Deformation and Contraction of Lie Algebras
• On the Contraction of Groups and Their Representations
• On the principles of elementary quantum mechanics
• Noncommutative field theory
• Proof of the GGS conjecture
• The Donald-Flanigan problem for finite reflection groups
• Noncommutative manifolds, the instanton algebra and isospectral deformations
• Deformation quantization of algebraic varieties.
• Unnamed Item

This page was built for publication: Quantum groups and deformation quantization: Explicit approaches and implicit aspects