Advances in mathematics (New York. 1965), ISSN 0001-8708, 2016, Volume 290, pp. 163 - 207

We study VB-groupoids and VB-algebroids, which are vector bundles in the realm of Lie groupoids and Lie algebroids...

Poisson groupoids | VB-algebroids | Double Lie algebroids | VB-groupoids | LA-groupoids | BRACKETS | INTEGRABILITY | REPRESENTATIONS | 2ND-ORDER GEOMETRY | HOMOTOPY | BIALGEBROIDS | MATHEMATICS | INTEGRATION | DUALITY

Poisson groupoids | VB-algebroids | Double Lie algebroids | VB-groupoids | LA-groupoids | BRACKETS | INTEGRABILITY | REPRESENTATIONS | 2ND-ORDER GEOMETRY | HOMOTOPY | BIALGEBROIDS | MATHEMATICS | INTEGRATION | DUALITY

Journal Article

Journal of geometry and physics, ISSN 0393-0440, 2019, Volume 142, pp. 254 - 273

.... In particular, we define symplectic almost Lie 2-algebroids and show how they correspond to pre-Courant algebroids...

Q-manifolds | Supermanifolds | Courant algebroids | VB-Courant algebroids | Cochain complex | TANGENT LIFTS | HIGHER ANALOGS | PHYSICS, MATHEMATICAL | MATHEMATICS | DOUBLE LIE ALGEBROIDS | DIRAC STRUCTURES | GRADED BUNDLES | MANIFOLDS | GEOMETRY

Q-manifolds | Supermanifolds | Courant algebroids | VB-Courant algebroids | Cochain complex | TANGENT LIFTS | HIGHER ANALOGS | PHYSICS, MATHEMATICAL | MATHEMATICS | DOUBLE LIE ALGEBROIDS | DIRAC STRUCTURES | GRADED BUNDLES | MANIFOLDS | GEOMETRY

Journal Article

Advances in mathematics (New York. 1965), ISSN 0001-8708, 2010, Volume 223, Issue 4, pp. 1236 - 1275

A VB -algebroid is essentially defined as a Lie algebroid object in the category of vector bundles...

Characteristic classes | Representation | Superconnection | Double category | Lie algebroid | MATHEMATICS | COHOMOLOGY

Characteristic classes | Representation | Superconnection | Double category | Lie algebroid | MATHEMATICS | COHOMOLOGY

Journal Article

QUARTERLY JOURNAL OF MATHEMATICS, ISSN 0033-5606, 09/2019, Volume 70, Issue 3, pp. 1039 - 1089

VB-groupoids and algebroids are vector bundle objects in the categories of Lie groupoids and Lie algebroids, respectively, and they are related via the Lie functor...

MATHEMATICS | BRACKETS | DOUBLE LIE ALGEBROIDS | BUNDLES | CLASSICAL PSEUDOGROUPS | REPRESENTATIONS | QUANTUM | DEFORMATIONS | MANIFOLDS | HOMOTOPY | BIALGEBRAS

MATHEMATICS | BRACKETS | DOUBLE LIE ALGEBROIDS | BUNDLES | CLASSICAL PSEUDOGROUPS | REPRESENTATIONS | QUANTUM | DEFORMATIONS | MANIFOLDS | HOMOTOPY | BIALGEBRAS

Journal Article

Journal of Homotopy and Related Structures, ISSN 2193-8407, 6/2018, Volume 13, Issue 2, pp. 287 - 319

We show that a double Lie algebroid, together with a chosen decomposition, is equivalent to a pair of 2-term representations up to homotopy satisfying compatibility conditions which extend the notion...

Algebra | Functional Analysis | Algebraic Topology | Representations up to homotopy | Secondary 17B66 | Double Lie algebroids | Mathematics | Primary 53D17 | Number Theory | Matched pairs | 18D05 | MATHEMATICS | MORPHISMS | 2ND-ORDER GEOMETRY | POISSON GROUPOIDS | BIALGEBROIDS | Mathematics - Differential Geometry

Algebra | Functional Analysis | Algebraic Topology | Representations up to homotopy | Secondary 17B66 | Double Lie algebroids | Mathematics | Primary 53D17 | Number Theory | Matched pairs | 18D05 | MATHEMATICS | MORPHISMS | 2ND-ORDER GEOMETRY | POISSON GROUPOIDS | BIALGEBROIDS | Mathematics - Differential Geometry

Journal Article

Journal of Geometry and Physics, ISSN 0393-0440, 02/2013, Volume 64, Issue 1, pp. 155 - 173

... −1 and 0 on degree 1 NQ-manifolds. In a more conventional language this means: strict actions of Lie algebra crossed modules on Lie algebroids...

Double Lie algebroid | Higher Lie group action | NQ-manifold | Higher Lie algebra action | Lie algebroid | BRACKETS | MATHEMATICS, APPLIED | GROUPOIDS | PHYSICS, MATHEMATICAL | L-INFINITY-ALGEBRAS | Algebra

Double Lie algebroid | Higher Lie group action | NQ-manifold | Higher Lie algebra action | Lie algebroid | BRACKETS | MATHEMATICS, APPLIED | GROUPOIDS | PHYSICS, MATHEMATICAL | L-INFINITY-ALGEBRAS | Algebra

Journal Article

International Journal of Geometric Methods in Modern Physics, ISSN 0219-8878, 05/2006, Volume 3, Issue 3, pp. 559 - 575

.... One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids...

Double vector bundle | Lagrangian function | Euler-Lagrange equations | Lie algebroid | LAGRANGIAN SUBMANIFOLDS | BUNDLES | POISSON | LIE ALGEBROIDS | DYNAMICS | double vector bundle | PHYSICS, MATHEMATICAL

Double vector bundle | Lagrangian function | Euler-Lagrange equations | Lie algebroid | LAGRANGIAN SUBMANIFOLDS | BUNDLES | POISSON | LIE ALGEBROIDS | DYNAMICS | double vector bundle | PHYSICS, MATHEMATICAL

Journal Article

Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), ISSN 1815-0659, 2013, Volume 9, p. 014

.... The search for the right notion of a "double" for Lie bialgebroids led to the definition of Courant algebroids...

Derived bracket | Lie bialgebroid | Courant algebroid | Generalized geometry | Dirac structure | Dorfman bracket | Loday algebra | Lie algebroid | Leibniz algebra | DRINFELD DOUBLES | generalized geometry | derived bracket | GERSTENHABER ALGEBRAS | 2ND-ORDER GEOMETRY | POISSON GROUPOIDS | FORMULATION | PHYSICS, MATHEMATICAL | BIALGEBROIDS | EHRESMANN DOUBLES | DOUBLE LIE ALGEBROIDS | DIRAC STRUCTURES | MANIFOLDS | History and Overview | Mathematics | Symplectic Geometry | Differential Geometry

Derived bracket | Lie bialgebroid | Courant algebroid | Generalized geometry | Dirac structure | Dorfman bracket | Loday algebra | Lie algebroid | Leibniz algebra | DRINFELD DOUBLES | generalized geometry | derived bracket | GERSTENHABER ALGEBRAS | 2ND-ORDER GEOMETRY | POISSON GROUPOIDS | FORMULATION | PHYSICS, MATHEMATICAL | BIALGEBROIDS | EHRESMANN DOUBLES | DOUBLE LIE ALGEBROIDS | DIRAC STRUCTURES | MANIFOLDS | History and Overview | Mathematics | Symplectic Geometry | Differential Geometry

Journal Article

Reports on Mathematical Physics, ISSN 0034-4877, 2010, Volume 66, Issue 2, pp. 251 - 276

We show how to extend the construction of Tulczyjew triples to Lie algebroids via graded manifolds...

graded manifolds | Lie algebroids | supermanifolds | higher Poisson brackets | higher Schouten brackets | double vector bundles | Supermanifolds | Double vector bundles | Graded manifolds | Higher Poisson brackets | Higher Schouten brackets | BRACKET | GENERALIZED POISSON STRUCTURES | QUANTIZATION | ANTIBRACKETS | PHYSICS, MATHEMATICAL | Manifolds | Mathematical analysis | Construction

graded manifolds | Lie algebroids | supermanifolds | higher Poisson brackets | higher Schouten brackets | double vector bundles | Supermanifolds | Double vector bundles | Graded manifolds | Higher Poisson brackets | Higher Schouten brackets | BRACKET | GENERALIZED POISSON STRUCTURES | QUANTIZATION | ANTIBRACKETS | PHYSICS, MATHEMATICAL | Manifolds | Mathematical analysis | Construction

Journal Article

Advances in Mathematics, ISSN 0001-8708, 08/2019, Volume 352, pp. 406 - 482

.... In this paper, we extend Kontsevich's formality theorem to Lie pairs, a framework which includes a range of diverse geometric contexts such as complex manifolds, foliations, and g-manifolds...

Homotopy Lie algebra | Hochschild-Kostant-Rosenberg theorem | Todd class | Kontsevich's formality theorem | Lie algebroid | Duflo's theorem | DRINFELD DOUBLES | ALGEBROIDS | INDEX THEOREM | HOMOTOPY | EHRESMANN DOUBLES | MATHEMATICS | COHOMOLOGY | DEFORMATION QUANTIZATION | ROZANSKY-WITTEN INVARIANTS | Algebra

Homotopy Lie algebra | Hochschild-Kostant-Rosenberg theorem | Todd class | Kontsevich's formality theorem | Lie algebroid | Duflo's theorem | DRINFELD DOUBLES | ALGEBROIDS | INDEX THEOREM | HOMOTOPY | EHRESMANN DOUBLES | MATHEMATICS | COHOMOLOGY | DEFORMATION QUANTIZATION | ROZANSKY-WITTEN INVARIANTS | Algebra

Journal Article

Journal of Geometry and Physics, ISSN 0393-0440, 06/2016, Volume 104, pp. 89 - 111

.... According to a result of Liu, Weinstein and Xu, Poisson homogeneous spaces of a Poisson groupoid are in correspondence with suitable Dirac structures in the Courant algebroid defined by the Lie...

Poisson groupoids | Dirac manifolds | Lie groupoids | Homogeneous spaces | Courant algebroids | MATHEMATICS | DOUBLE LIE ALGEBROIDS | INTEGRATION | PHYSICS, MATHEMATICAL

Poisson groupoids | Dirac manifolds | Lie groupoids | Homogeneous spaces | Courant algebroids | MATHEMATICS | DOUBLE LIE ALGEBROIDS | INTEGRATION | PHYSICS, MATHEMATICAL

Journal Article

Journal of Sciences, Islamic Republic of Iran, ISSN 1016-1104, 06/2016, Volume 27, Issue 3, pp. 279 - 285

Journal Article

Mathematische Zeitschrift, ISSN 1432-1823, 2019, Volume 294, Issue 3-4, pp. 1181 - 1225

This paper belongs to a series of works aiming at exploring generalized (complex) geometry in odd dimensions. Holomorphic Jacobi manifolds were introduced and...

MATHEMATICS | Lie groupoid | DOUBLE LIE ALGEBROIDS | BUNDLES | Homogeneous Poisson structure | REDUCTION | Multiplicative tensor | DEFORMATIONS | Spencer operator | Holomorphic Jacobi structure | Holomorphic Poisson structure

MATHEMATICS | Lie groupoid | DOUBLE LIE ALGEBROIDS | BUNDLES | Homogeneous Poisson structure | REDUCTION | Multiplicative tensor | DEFORMATIONS | Spencer operator | Holomorphic Jacobi structure | Holomorphic Poisson structure

Journal Article

Pacific journal of mathematics, ISSN 0030-8730, 2013, Volume 266, Issue 2, pp. 329 - 365

...) and multiplicative closed 2-forms such as symplectic groupoids. We prove that for every source simply connected Lie groupoid G with Lie algebroid AG, there exists a one-to-one...

Lie algebroids | Lie groupoids | Multiplicative Dirac structures | MATHEMATICS | SYMPLECTIC GROUPOIDS | DOUBLE LIE ALGEBROIDS | POISSON | INTEGRATION | TANGENT | LIFTS | multiplicative Dirac structures | GEOMETRY

Lie algebroids | Lie groupoids | Multiplicative Dirac structures | MATHEMATICS | SYMPLECTIC GROUPOIDS | DOUBLE LIE ALGEBROIDS | POISSON | INTEGRATION | TANGENT | LIFTS | multiplicative Dirac structures | GEOMETRY

Journal Article

Journal of geometry and physics, ISSN 0393-0440, 2009, Volume 59, Issue 9, pp. 1285 - 1305

A natural explicit condition is given ensuring that an action of the multiplicative monoid of non-negative reals on a manifold F comes from homotheties of a...

Lie bialgebroids | Courant algebroids | Drinfeld doubles | BRST formalism | Double vector bundles | Graded super-manifolds | LEGENDRE TRANSFORMATION | 2ND-ORDER GEOMETRY | TANGENT | POISSON GROUPOIDS | PHYSICS, MATHEMATICAL | BIALGEBROIDS | MATHEMATICS | DOUBLE LIE ALGEBROIDS | DUALITY | LIFTS | Mathematics - Differential Geometry

Lie bialgebroids | Courant algebroids | Drinfeld doubles | BRST formalism | Double vector bundles | Graded super-manifolds | LEGENDRE TRANSFORMATION | 2ND-ORDER GEOMETRY | TANGENT | POISSON GROUPOIDS | PHYSICS, MATHEMATICAL | BIALGEBROIDS | MATHEMATICS | DOUBLE LIE ALGEBROIDS | DUALITY | LIFTS | Mathematics - Differential Geometry

Journal Article

Communications in Mathematical Physics, ISSN 0010-3616, 11/2015, Volume 339, Issue 3, pp. 1003 - 1020

... (Commun Math Phys 315(2): 279–310, 2012): Double Lie algebroids arose in the works on double Lie groupoids...

Quantum Physics | Statistical Physics, Dynamical Systems and Complexity | Mathematical Physics | Classical and Quantum Gravitation, Relativity Theory | Theoretical, Mathematical and Computational Physics | Physics | DRINFELD DOUBLES | BRACKETS | DOUBLE LIE ALGEBROIDS | GERSTENHABER ALGEBRAS | 2ND-ORDER GEOMETRY | MANIFOLDS | PHYSICS, MATHEMATICAL | BIALGEBROIDS | EHRESMANN DOUBLES | Algebra

Quantum Physics | Statistical Physics, Dynamical Systems and Complexity | Mathematical Physics | Classical and Quantum Gravitation, Relativity Theory | Theoretical, Mathematical and Computational Physics | Physics | DRINFELD DOUBLES | BRACKETS | DOUBLE LIE ALGEBROIDS | GERSTENHABER ALGEBRAS | 2ND-ORDER GEOMETRY | MANIFOLDS | PHYSICS, MATHEMATICAL | BIALGEBROIDS | EHRESMANN DOUBLES | Algebra

Journal Article

Journal of Symplectic Geometry, ISSN 1527-5256, 2017, Volume 15, Issue 3, pp. 741 - 783

A VB-groupoid is a Lie groupoid equipped with a compatible linear structure. In this paper, we describe a correspondence, up to isomorphism, between VB-groupoids and 2-term representations up to homotopy of Lie groupoids...

MATHEMATICS | DRINFELD DOUBLES | ALGEBROIDS | BIALGEBROIDS | 2ND-ORDER GEOMETRY | EHRESMANN DOUBLES

MATHEMATICS | DRINFELD DOUBLES | ALGEBROIDS | BIALGEBROIDS | 2ND-ORDER GEOMETRY | EHRESMANN DOUBLES

Journal Article

Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), ISSN 1815-0659, 2009, Volume 5, p. 051

.... This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non-Abelian higher dimensional structures...

Compact quantum groupoids | Quantum algebraic topology (QAT) | Weak C-Hopf algebroids | Higher dimensional quantum symmetries | Quantum algebras: von Neumann algebra factors | Lie algebras | Fluctuating quantum spacetimes | Quantum groupoid C-algebras | Weak C-Hopf and graded Lie algebras | Intense gravitational fields | Symmetry breaking | Paracrystals | Poisson-Lie manifolds and quantum gravity theories | Quantization procedures | Algebraic topology of quantum systems | Paragroups and Kac algebras | Convolution algebras and quantum algebroids | Groupoid and functor representations in relation to extended quantum symmetries in QAT | Groupoids and algebroids | Quantum fundamental groupoids | Lie algebroids | Relativistic quantum gravity (RQG) | Extended quantum symmetries | Spin networks and spin glasses | Applications of generalized van Kampen theorem (GvKT) to quantum spacetime invariants | Hamiltonian algebroids in quantum gravity | Grassmann-Hopf | Quantum double groupoids and algebroids | Supergravity and supersymmetry theories | Quantum groups and ring structures | Tensor products of algebroids and categories | Nuclear Fréchet spaces and GNS representations of quantum state spaces (QSS) | Superfluids | HOPF-ALGEBRAS | quantization procedures | REPRESENTATIONS | INVARIANTS | fluctuating quantum spacetimes | relativistic quantum gravity (RQG) | PHYSICS, MATHEMATICAL | convolution algebras and quantum algebroids | quantum algebras: von Neumann algebra factors | quantum fundamental groupoids | intense gravitational fields | quantum groups and ring structures | quantum groupoid C-algebras | groupoid and functor representations in relation to extended quantum symmetries in QAT | quantum algebraic topology (QAT) | paragroups and Kac algebras | algebraic topology of quantum systems | extended quantum symmetries | symmetry breaking, paracrystals, superfluids, spin networks and spin glasses | GALOIS THEORY | applications of generalized van Kampen theorem (GvKT) to quantum spacetime invariants | KOCHEN-SPECKER THEOREM | higher dimensional quantum symmetries | Grassmann-Hopf, weak C-Hopf and graded Lie algebras | CATEGORIES | tensor products of algebroids and categories | groupoids and algebroids | compact quantum groupoids | nuclear Frechet spaces and GNS representations of quantum state spaces (QSS) | GROUPOIDS | supergravity and supersymmetry theories | weak C-Hopf algebroids | QUANTIZATION | quantum double groupoids and algebroids | TOPOS PERSPECTIVE | HAAR MEASURE | weak C-Hopf and graded Lie algebras | symmetry breaking | spin networks and spin glasses | superfluids | nuclear Fréchet spaces and GNS representations of quantum state spaces (QSS) | paracrystals

Compact quantum groupoids | Quantum algebraic topology (QAT) | Weak C-Hopf algebroids | Higher dimensional quantum symmetries | Quantum algebras: von Neumann algebra factors | Lie algebras | Fluctuating quantum spacetimes | Quantum groupoid C-algebras | Weak C-Hopf and graded Lie algebras | Intense gravitational fields | Symmetry breaking | Paracrystals | Poisson-Lie manifolds and quantum gravity theories | Quantization procedures | Algebraic topology of quantum systems | Paragroups and Kac algebras | Convolution algebras and quantum algebroids | Groupoid and functor representations in relation to extended quantum symmetries in QAT | Groupoids and algebroids | Quantum fundamental groupoids | Lie algebroids | Relativistic quantum gravity (RQG) | Extended quantum symmetries | Spin networks and spin glasses | Applications of generalized van Kampen theorem (GvKT) to quantum spacetime invariants | Hamiltonian algebroids in quantum gravity | Grassmann-Hopf | Quantum double groupoids and algebroids | Supergravity and supersymmetry theories | Quantum groups and ring structures | Tensor products of algebroids and categories | Nuclear Fréchet spaces and GNS representations of quantum state spaces (QSS) | Superfluids | HOPF-ALGEBRAS | quantization procedures | REPRESENTATIONS | INVARIANTS | fluctuating quantum spacetimes | relativistic quantum gravity (RQG) | PHYSICS, MATHEMATICAL | convolution algebras and quantum algebroids | quantum algebras: von Neumann algebra factors | quantum fundamental groupoids | intense gravitational fields | quantum groups and ring structures | quantum groupoid C-algebras | groupoid and functor representations in relation to extended quantum symmetries in QAT | quantum algebraic topology (QAT) | paragroups and Kac algebras | algebraic topology of quantum systems | extended quantum symmetries | symmetry breaking, paracrystals, superfluids, spin networks and spin glasses | GALOIS THEORY | applications of generalized van Kampen theorem (GvKT) to quantum spacetime invariants | KOCHEN-SPECKER THEOREM | higher dimensional quantum symmetries | Grassmann-Hopf, weak C-Hopf and graded Lie algebras | CATEGORIES | tensor products of algebroids and categories | groupoids and algebroids | compact quantum groupoids | nuclear Frechet spaces and GNS representations of quantum state spaces (QSS) | GROUPOIDS | supergravity and supersymmetry theories | weak C-Hopf algebroids | QUANTIZATION | quantum double groupoids and algebroids | TOPOS PERSPECTIVE | HAAR MEASURE | weak C-Hopf and graded Lie algebras | symmetry breaking | spin networks and spin glasses | superfluids | nuclear Fréchet spaces and GNS representations of quantum state spaces (QSS) | paracrystals

Journal Article

Journal fur die Reine und Angewandte Mathematik, ISSN 0075-4102, 09/2011, Volume 658, Issue 658, pp. 193 - 245

...'. Double categories, double groupoids and double vector bundles are instances, but the notion of Lie algebroid cannot readily be doubled in the Ehresmann sense, since a Lie algebroid bracket cannot...

MATHEMATICS | INTEGRABILITY | INTEGRATION | DIRAC STRUCTURES | DUALITY | SYMPLECTIC DOUBLE GROUPOIDS | MATCHED PAIRS

MATHEMATICS | INTEGRABILITY | INTEGRATION | DIRAC STRUCTURES | DUALITY | SYMPLECTIC DOUBLE GROUPOIDS | MATCHED PAIRS

Journal Article

Letters in Mathematical Physics, ISSN 0377-9017, 06/2018, pp. 1 - 51

Journal Article

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