International Journal of Modern Physics A, ISSN 0217-751X, 06/2016, Volume 31, Issue 18, p. 1650105

In this paper, we realize the Lie algebra g l ( ∞ ) in one homotopy category, and we find that tau functions can be obtained from complexes in this homotopy category by taking Grothendieck group...

fermion | tau function | boson | Lie algebra gl(∞) | Homotopy category | Lie algebra gl(infinity) | CATEGORIFICATION | PHYSICS, NUCLEAR | PHYSICS, PARTICLES & FIELDS | Algebra

fermion | tau function | boson | Lie algebra gl(∞) | Homotopy category | Lie algebra gl(infinity) | CATEGORIFICATION | PHYSICS, NUCLEAR | PHYSICS, PARTICLES & FIELDS | Algebra

Journal Article

Communications in Contemporary Mathematics, ISSN 0219-1997, 12/2014, Volume 16, Issue 6, pp. 1450007 - 1-1450007-49

...}$ may be interpreted as vector fields on P. In particular, they possess a Lie bracket and act on characteristic cohomology $\overline{H...

characteristic cohomology | Lie-Rinehart algebras | Foliations | homotopy algebras | MATHEMATICS | LAGRANGIAN-FORMALISM | BRACKETS | MATHEMATICS, APPLIED | DEFORMATIONS | CONSERVATION-LAWS | C-SPECTRAL SEQUENCE | Algebra | Functions (mathematics) | Manifolds | Bundling | Brackets | Integrals | Mathematical analysis | Isomorphism | Fields (mathematics)

characteristic cohomology | Lie-Rinehart algebras | Foliations | homotopy algebras | MATHEMATICS | LAGRANGIAN-FORMALISM | BRACKETS | MATHEMATICS, APPLIED | DEFORMATIONS | CONSERVATION-LAWS | C-SPECTRAL SEQUENCE | Algebra | Functions (mathematics) | Manifolds | Bundling | Brackets | Integrals | Mathematical analysis | Isomorphism | Fields (mathematics)

Journal Article

Journal of Noncommutative Geometry, ISSN 1661-6952, 2012, Volume 6, Issue 3, pp. 539 - 602

This paper provides an explicit cofibrant resolution of the operad encoding Batalin-Vilkovisky algebras...

Batalin-Vilkovisky algebra | Vertex algebras | Topological conformal field theory | Homotopy algebras | Koszul duality theory | Gerstenhaber algebra | Framed little disc | Maurer-Cartan equation | Operad | OPERADS | HIGHER DERIVED BRACKETS | MATHEMATICS, APPLIED | vertex algebras | REPRESENTATIONS | FIELD-THEORIES | operad framed little disc | PHYSICS, MATHEMATICAL | DEFORMATION-THEORY | MATHEMATICS | PRODUCTS | KOSZUL DUALITY | HOCHSCHILD COHOMOLOGY | topological conformal field theory | homotopy algebras | CHIRAL DIFFERENTIAL-OPERATORS | DELIGNE CONJECTURE

Batalin-Vilkovisky algebra | Vertex algebras | Topological conformal field theory | Homotopy algebras | Koszul duality theory | Gerstenhaber algebra | Framed little disc | Maurer-Cartan equation | Operad | OPERADS | HIGHER DERIVED BRACKETS | MATHEMATICS, APPLIED | vertex algebras | REPRESENTATIONS | FIELD-THEORIES | operad framed little disc | PHYSICS, MATHEMATICAL | DEFORMATION-THEORY | MATHEMATICS | PRODUCTS | KOSZUL DUALITY | HOCHSCHILD COHOMOLOGY | topological conformal field theory | homotopy algebras | CHIRAL DIFFERENTIAL-OPERATORS | DELIGNE CONJECTURE

Journal Article

Algebras and representation theory, ISSN 1572-9079, 2015, Volume 19, Issue 1, pp. 1 - 5

In this paper, we construct a Lie 2-algebra associated to every Leibniz algebra via the skew-symmetrization.

17B99 | Associative Rings and Algebras | Omni-Lie algebras | Courant algebroids | Non-associative Rings and Algebras | Commutative Rings and Algebras | Mathematics | Lie 2-algebras | Leibniz algebras | 55U15 | MATHEMATICS | HOMOTOPY | Algebra

17B99 | Associative Rings and Algebras | Omni-Lie algebras | Courant algebroids | Non-associative Rings and Algebras | Commutative Rings and Algebras | Mathematics | Lie 2-algebras | Leibniz algebras | 55U15 | MATHEMATICS | HOMOTOPY | Algebra

Journal Article

Letters in Mathematical Physics, ISSN 0377-9017, 5/2017, Volume 107, Issue 5, pp. 861 - 885

... $$L_\infty $$ L ∞ -algebra, which is proposed to be the geometric structure on the dual of a Lie 2-algebra...

Theoretical, Mathematical and Computational Physics | Complex Systems | Lie 2-algebras | Physics | Maurer–Cartan elements | 53D17 | Geometry | Symplectic NQ-manifolds | 17B99 | L_\infty $$ L ∞ -algebras | Courant algebroids | Group Theory and Generalizations | Homotopy Poisson manifolds | algebras | BRACKETS | REDUCTION | Maurer-Cartan elements | L-infinity-algebras | PHYSICS, MATHEMATICAL | BIALGEBROIDS | QUASI | GEOMETRY | Algebra

Theoretical, Mathematical and Computational Physics | Complex Systems | Lie 2-algebras | Physics | Maurer–Cartan elements | 53D17 | Geometry | Symplectic NQ-manifolds | 17B99 | L_\infty $$ L ∞ -algebras | Courant algebroids | Group Theory and Generalizations | Homotopy Poisson manifolds | algebras | BRACKETS | REDUCTION | Maurer-Cartan elements | L-infinity-algebras | PHYSICS, MATHEMATICAL | BIALGEBROIDS | QUASI | GEOMETRY | Algebra

Journal Article

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

... (that is, manifolds endowed with an action of a Lie algebra g). The spaces tot(Γ(Λ•A∨)⊗RTpoly•) and tot(Γ(Λ•A∨)⊗RDpoly•) associated with a Lie pair...

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

2011, Mathematical surveys and monographs, ISBN 9780821805015, Volume 179, xii, 364

Book

8.
Super-Lie n-algebra extensions, higher WZW models and super-p-branes with tensor multiplet fields

International Journal of Geometric Methods in Modern Physics, ISSN 0219-8878, 02/2015, Volume 12, Issue 2, pp. 1550018 - 1-1550018-35

...-) Lie n-algebra homotopy theory (the true home of the "FDA"-language used in the supergravity literature...

Lie n-algebras | Lie superalgebras | p-branes | WZW model | DIVISION-ALGEBRAS | PHYSICS, MATHEMATICAL | Tensors | Algebra | Homotopy theory | M theory | Condensates | Mathematical analysis | Mathematical models | Branes | Symmetry

Lie n-algebras | Lie superalgebras | p-branes | WZW model | DIVISION-ALGEBRAS | PHYSICS, MATHEMATICAL | Tensors | Algebra | Homotopy theory | M theory | Condensates | Mathematical analysis | Mathematical models | Branes | Symmetry

Journal Article

manuscripta mathematica, ISSN 0025-2611, 5/2019, Volume 159, Issue 1, pp. 161 - 170

Let (L, d) be a differential graded Lie algebra, where $$L={\mathbb {L}}(V)$$ L = L ( V ) is free as graded Lie algebra...

Geometry | 16W60 | Topological Groups, Lie Groups | Calculus of Variations and Optimal Control; Optimization | Mathematics, general | Algebraic Geometry | Mathematics | Number Theory | Primary 55P62 | Secondary 55P60 | MATHEMATICS | Analysis | Algebra | Mathematics - Algebraic Topology

Geometry | 16W60 | Topological Groups, Lie Groups | Calculus of Variations and Optimal Control; Optimization | Mathematics, general | Algebraic Geometry | Mathematics | Number Theory | Primary 55P62 | Secondary 55P60 | MATHEMATICS | Analysis | Algebra | Mathematics - Algebraic Topology

Journal Article

Journal of Geometry and Physics, ISSN 0393-0440, 12/2012, Volume 62, Issue 12, pp. 2389 - 2400

Let g be a simplicial Lie algebra with Moore complex Ng of length k. Let G be the simplicial Lie group integrating g, such that each Gn is simply connected...

Hypercrossed complex | Dg Lie algebra | Simplicial Lie algebra | 1-jet | MATHEMATICS, APPLIED | PHYSICS, MATHEMATICAL | HOMOTOPY THEORY | GEOMETRY | Algebra

Hypercrossed complex | Dg Lie algebra | Simplicial Lie algebra | 1-jet | MATHEMATICS, APPLIED | PHYSICS, MATHEMATICAL | HOMOTOPY THEORY | GEOMETRY | Algebra

Journal Article

Geometriae Dedicata, ISSN 0046-5755, 8/2013, Volume 165, Issue 1, pp. 111 - 133

.... Equivalently, we classify 2-step nilpotent Lie algebras in dimension 7. This classification also recovers the real homotopy type of 7-dimensional 2-step nilmanifolds.

Geometry | 11E04 | Minimal model | 17B30 | Rational homotopy | Nilmanifolds | Mathematics | Nilpotent Lie algebras | Primary: 55P62 | Secondary: 22E25 | MATHEMATICS | CLASSIFICATION | Analysis | Algebra

Geometry | 11E04 | Minimal model | 17B30 | Rational homotopy | Nilmanifolds | Mathematics | Nilpotent Lie algebras | Primary: 55P62 | Secondary: 22E25 | MATHEMATICS | CLASSIFICATION | Analysis | Algebra

Journal Article

Journal of Algebra, ISSN 0021-8693, 02/2017, Volume 472, pp. 437 - 479

We extend the classical characterization of a finite-dimensional Lie algebra g in terms of its Maurer–Cartan algebra...

[formula omitted]-algebra | Sh Lie–Rinehart algebra | Higher homotopies | Foliation | Quasi Lie–Rinehart algebra | Maurer–Cartan algebra | algebra | MATHEMATICS | Maurer-Cartan algebra | Sh Lie-Rinehart algebra | L-infinity-algebra | Quasi Lie-Rinehart algebra | Algebra

[formula omitted]-algebra | Sh Lie–Rinehart algebra | Higher homotopies | Foliation | Quasi Lie–Rinehart algebra | Maurer–Cartan algebra | algebra | MATHEMATICS | Maurer-Cartan algebra | Sh Lie-Rinehart algebra | L-infinity-algebra | Quasi Lie-Rinehart algebra | Algebra

Journal Article

Mathematische Zeitschrift, ISSN 1432-1823, 2013, Volume 275, Issue 1-2, pp. 245 - 292

.... In this paper, we compute the rational homotopy Lie algebra of symplectomorphism groups of the 3-point blow-up of the projective plane...

Almost complex structures | 57R17 | 57S05 | 57T20 | Homotopy type | Symplectomorphism group | Mathematics, general | Mathematics | 53D35 | TOPOLOGY | SPACE | MATHEMATICS | SYMPLECTIC BALLS | COMPLEX STRUCTURES | Analysis | Algebra

Almost complex structures | 57R17 | 57S05 | 57T20 | Homotopy type | Symplectomorphism group | Mathematics, general | Mathematics | 53D35 | TOPOLOGY | SPACE | MATHEMATICS | SYMPLECTIC BALLS | COMPLEX STRUCTURES | Analysis | Algebra

Journal Article

Communications in Contemporary Mathematics, ISSN 0219-1997, 06/2017, Volume 19, Issue 3, p. 1650034

We study the extension of a Lie algebroid by a representation up to homotopy, including semidirect products of a Lie algebroid with such representations...

Representation up to homotopy | integration | Lie 2-groupoids | Courant algebroids | Lie 2-algebroids | BRACKETS | MATHEMATICS, APPLIED | REPRESENTATIONS | HOMOTOPY | L-INFINITY-ALGEBRAS | MATHEMATICS | GROUPOIDS | GEOMETRY

Representation up to homotopy | integration | Lie 2-groupoids | Courant algebroids | Lie 2-algebroids | BRACKETS | MATHEMATICS, APPLIED | REPRESENTATIONS | HOMOTOPY | L-INFINITY-ALGEBRAS | MATHEMATICS | GROUPOIDS | GEOMETRY

Journal Article

Forum Mathematicum, ISSN 0933-7741, 03/2017, Volume 29, Issue 2, pp. 277 - 286

...–Cartan set of a differential graded Lie algebra. We show that Euler and Miki’s identities, well-known and apparently non-related formulas, are linear combinations...

17B01 | 55U35 | Bernoulli numbers | homotopy theory of Lie algebras | 11B68 | Gauge action | MATHEMATICS | MATHEMATICS, APPLIED | MODELS

17B01 | 55U35 | Bernoulli numbers | homotopy theory of Lie algebras | 11B68 | Gauge action | MATHEMATICS | MATHEMATICS, APPLIED | MODELS

Journal Article

DOCUMENTA MATHEMATICA, ISSN 1431-0643, 2018, Volume 23, pp. 189 - 240

In order to solve two problems in deformation theory, we establish natural structures of homotopy Lie algebras and of homotopy associative algebras on tensor products of algebras of different types...

MATHEMATICS | infinity-morphisms | homotopy Lie algebras | Maurer-Cartan elements | Operads | homotopy associative algebras

MATHEMATICS | infinity-morphisms | homotopy Lie algebras | Maurer-Cartan elements | Operads | homotopy associative algebras

Journal Article

Communications in Contemporary Mathematics, ISSN 0219-1997, 04/2005, Volume 7, Issue 2, pp. 145 - 165

We consider versal deformations of 0|3-dimensional L∞ algebras, also called strongly homotopy Lie algebras, which correspond precisely to ordinary (non-graded...

algebras | Versal deformations | Strongly homotopy Lie algebras | Lie algebras | Moduli space

algebras | Versal deformations | Strongly homotopy Lie algebras | Lie algebras | Moduli space

Journal Article

Letters in Mathematical Physics, ISSN 0377-9017, 4/2012, Volume 100, Issue 1, pp. 29 - 50

.... In particular, just as a symplectic manifold gives a Poisson algebra of functions, any 2-plectic manifold gives a Lie 2-algebra of 1-forms and functions...

Geometry | 17B55 | 70S05 | strongly homotopy Lie algebras | Theoretical, Mathematical and Computational Physics | 53D05 | Group Theory and Generalizations | multisymplectic geometry | Statistical Physics, Dynamical Systems and Complexity | classical field theory | Physics

Geometry | 17B55 | 70S05 | strongly homotopy Lie algebras | Theoretical, Mathematical and Computational Physics | 53D05 | Group Theory and Generalizations | multisymplectic geometry | Statistical Physics, Dynamical Systems and Complexity | classical field theory | Physics

Journal Article

Letters in mathematical physics, ISSN 1573-0530, 2012, Volume 102, Issue 2, pp. 223 - 244

Given a strict Lie 2-algebra, we can integrate it to a strict Lie 2-group by integrating the corresponding Lie algebra crossed module...

Lie 2-groups | Theoretical, Mathematical and Computational Physics | Lie 2-algebras | Statistical Physics, Dynamical Systems and Complexity | L ∞ -algebras | Physics | Geometry | 18D10 | Secondary 18B40 | Primary 17B55 | L ∞ -morphisms | integration | Group Theory and Generalizations | crossed modules | algebras | morphisms | BRACKETS | REPRESENTATIONS | EXTENSIONS | L-infinity-morphisms | HOMOTOPY | L-infinity-algebras | PHYSICS, MATHEMATICAL

Lie 2-groups | Theoretical, Mathematical and Computational Physics | Lie 2-algebras | Statistical Physics, Dynamical Systems and Complexity | L ∞ -algebras | Physics | Geometry | 18D10 | Secondary 18B40 | Primary 17B55 | L ∞ -morphisms | integration | Group Theory and Generalizations | crossed modules | algebras | morphisms | BRACKETS | REPRESENTATIONS | EXTENSIONS | L-infinity-morphisms | HOMOTOPY | L-infinity-algebras | PHYSICS, MATHEMATICAL

Journal Article

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