Quantum groupoids

Abstract

We introduce a general notion of quantum universal enveloping algebroids (QUE algebroids), or quantum groupoids, as a unification of quantum groups and star-products. Some basic properties are studied including the twist construction and the classical limits. In particular, we show that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. Conversely, we formulate a conjecture on the existence of a quantization for any Lie bialgebroid, and prove this conjecture for the special case of regular triangular Lie bialgebroids. As an application of this theory, we study the dynamical quantum groupoid ${\cal D}\otimes_{\hbar} U_{\hbar}(\frakg)$, which gives an interpretation of the quantum dynamical Yang-Baxter equation in terms of Hopf algebroids.

48 pages, typos and minor mistakes corrected, references updadted. Comm. Math. Physics, (to appear)

References in corpus (7)

• Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups
• Finite Quantum Groupoids and Their Applications
• A Quasi-Hopf algebra interpretation of quantum 3-j and 6-j symbols and difference equations
• Courant Algebroids and Strongly Homotopy Lie Algebras
• Exchange dynamical quantum groups
• Quantum groupoids and deformation quantization
• On classification of dynamical r-matrices

Cited by in corpus (83)

• Finite Quantum Groupoids and Their Applications
• Hochschild cohomology and Atiyah classes
• Skew-monoidal categories and bialgebroids
• From Atiyah Classes to Homotopy Leibniz Algebras
• Living in Curved Momentum Space
• Tannaka-Krein duality for Hopf algebroids
• Characteristic Classes of SL(N)-Bundles and Quantum Dynamical Elliptic R-Matrices
• K-Poincare-Hopf algebra and Hopf algebroid structure of phase space from twist
• Twists, realizations and Hopf algebroid structure of kappa-deformed phase space
• Deformed Covariant Quantum Phase Spaces as Hopf Algebroids
• Caldararu's conjecture and Tsygan's formality
• Poincaré--Birkhoff--Witt isomorphisms and Kapranov dg-manifolds
• $\kappa$-Deformed Phase Space, Hopf Algebroid and Twisting
• Triangulations, orientals, and skew monoidal categories
• Formality theorems for Hochschild chains in the Lie algebroid setting
• Remarks on simple interpolation between Jordanian twists
• Lie-deformed quantum Minkowski spaces from twists: Hopf-algebraic versus Hopf-algebroid approach
• Lie algebra type noncommutative phase spaces are Hopf algebroids
• Gauge Theory on Twisted $\kappa$-Minkowski: Old Problems and Possible Solutions
• Snyder-type spaces, twisted Poincaré algebra and addition of momenta
• Elliptic Quantum Group U_{q,p}(\hat{sl}_2), Hopf Algebroid Structure and Elliptic Hypergeometric Series
• Galois-Type Extensions and Equivariant Projectivity
• Differential forms and k-Minkowski spacetime from extended twist
• Quantum gravity on polygons and $\mathbb{R}\times {\mathbb{Z}}_n$ FLRW model
• Formality and Kontsevich--Duflo type theorems for Lie pairs
• Twisted bialgebroids versus bialgebroids from a Drinfeld twist
• Quantum Twist-Deformed D=4 Phase Spaces with Spin Sector and Hopf Algebroid Structures
• Integration on algebraic quantum groupoids
• On duality of algebraic quantum groupoids
• Formality for Lie algebroids
• Extending general covariance: Moyal-type noncommutative manifolds
• Formal exponential map for graded manifolds
• Hochschild cohomology of dg manifolds associated to integrable distributions
• Generalized quantum phase spaces for the $\kappa$-deformed extended Snyder model
• Hopf Algebroids, Bimodule Connections and Noncommutative Geometry
• The relative tensor product and a minimal fiber product in the setting of $C^{\ast }$-algebras
• Symmetric ordering and Weyl realizations for quantum Minkowski spaces
• Heisenberg double H(B^*) as a braided commutative Yetter-Drinfeld module algebra over the Drinfeld double
• Weak Hopf algebras and quantum groupoids
• Hopf Algebroid Twists for Deformation Quantization of Linear Poisson Structures
• Hochschild cohomology for Lie algebroids
• Fedosov dg manifolds associated with Lie pairs
• A Hopf algebra associated to a Lie pair
• Twist star products and Morita equivalence
• Duality functors for quantum groupoids
• Morita base change in quantum groupoids
• Structures in BC_N Ruijsenaars-Schneider models
• A Lie-Rinehart algebra with no antipode
• On the classical and quantum momentum map
• Hopf algebras arising from dg manifolds
• On Hopf algebroid structure of kappa-deformed Heisenberg algebra
• Various instances of Harish-Chandra pairs
• Duality and products in algebraic (co)homology theories
• Dynamical reflection equation
• Double Groupoids, Orbifolds, and the Symplectic Category
• Polyvector fields and polydifferential operators associated with Lie pairs
• Hopf algebras in noncommutative geometry
• Regular multiplier Hopf algebroids II. Integration on and duality of algebraic quantum groupoids
• Morita invariance of equivariant and Hopf-cyclic cohomology of module algebras over Hopf algebroids
• Scalar extension Hopf algebroids
• Cyclic Cohomology of Corings
• Twist deformations leading to kappa-Poincare Hopf algebra and their application to physics
• Two ${\theta }_{\mu \nu }$ -deformed covariant relativistic quantum phase spaces as Poincare-Hopf algebroids
• Multiplier Hopf algebroids. Basic theory and examples
• Cyclic Cohomology of Hopf algebras and Hopf Algebroids
• Simple physical applications of a groupoid structure
• FRT Construction for Dynamical Yang-Baxter Maps
• A category of quantum categories
• Construction of Hopf algebroids
• Integral theory for left Hopf left bialgebroids
• Poisson geometry and deformation quantization near a pseudoconvex boundary
• Grothendieck-Verdier functors
• Formality theorem with coefficients in a module
• Hopf algebroids with balancing subalgebra
• Free Lie algebroids and the space of paths
• Dynamical Yang-Baxter maps and Hopf algebroids associated with s-sets
• Quantum groupoids and dynamical categories
• Para-Hopf algebroids and their cyclic cohomology
• Two Equivalent Realizations of Trigonometric Dynamical Affine Quantum Group $U_{q,x}(\widehat{s{l}_2})=U_{q,\lambda }(\widehat{s{l}_2})$, Drinfeld Currents and Hopf Algebroid Structures
• Dynamical quantum determinants and Pfaffians
• Duality and self-duality for dynamical quantum groups
• A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids
• Morita base change in Hopf-cyclic (co)homology