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Mixed Hodge structures and representations of fundamental groups of algebraic varieties

Advances in Mathematics, ISSN 0001-8708, 06/2019, Volume 349, pp. 869 - 910

Given a complex variety X, a linear algebraic group G and a representation ρ of the fundamental group π1(X,x) into G, we develop a framework for constructing a...

Hodge theory | [formula omitted] algebras | Representation varieties | Complex algebraic geometry | Formal deformation theory | Fundamental groups | MATHEMATICS | L-infinity algebras | Algebraic Geometry | Mathematics

Hodge theory | [formula omitted] algebras | Representation varieties | Complex algebraic geometry | Formal deformation theory | Fundamental groups | MATHEMATICS | L-infinity algebras | Algebraic Geometry | Mathematics

Journal Article

Advances in Mathematics, ISSN 0001-8708, 01/2016, Volume 288, pp. 527 - 575

Given a Lie group G, one constructs a principal G-bundle on a manifold X by taking a cover U→X, specifying a transition cocycle on the cover, and descending...

Simplicial manifolds | String 2-group | Bundle gerbes | MATHEMATICS | CATEGORIES | L-INFINITY-ALGEBRAS

Simplicial manifolds | String 2-group | Bundle gerbes | MATHEMATICS | CATEGORIES | L-INFINITY-ALGEBRAS

Journal Article

Letters in mathematical physics, ISSN 1573-0530, 2018, Volume 108, Issue 9, pp. 2055 - 2097

We describe some $$L_{\infty }$$ L∞ model for the local period map of a compact Kähler manifold. Applications include the study of deformations with associated...

Geometry | 32G20 | 14D07 | Theoretical, Mathematical and Computational Physics | Complex Systems | Group Theory and Generalizations | Yukawa algebra | L-infinity algebras | 13D10 | Physics | Formal period maps | LIE-ALGEBRAS | SPACE | COHOMOLOGY | DEFORMATIONS | CONSTRUCTION | MANIFOLDS | PHYSICS, MATHEMATICAL | HOMOTOPY-THEORY | GEOMETRY | Analysis | Algebra

Geometry | 32G20 | 14D07 | Theoretical, Mathematical and Computational Physics | Complex Systems | Group Theory and Generalizations | Yukawa algebra | L-infinity algebras | 13D10 | Physics | Formal period maps | LIE-ALGEBRAS | SPACE | COHOMOLOGY | DEFORMATIONS | CONSTRUCTION | MANIFOLDS | PHYSICS, MATHEMATICAL | HOMOTOPY-THEORY | GEOMETRY | Analysis | Algebra

Journal Article

Georgian mathematical journal, ISSN 1572-9176, 2018, Volume 25, Issue 4, pp. 623 - 627

A decade ago, I wrote a tribute to Tornike Kadeishvili in honor of his 60th birthday [J. Stasheff, A twisted tale of cochains and connections, Georgian Math....

55R | 57T | Hochschild and Harrison cohomology | 55P45 | Hirsch algebras | Maurer–Cartan | (co)bar construction | twisted differential | 18G55 | Twisting cochain | and L | algebra | Maurer-Cartan | MATHEMATICS | A(infinity)- and L-infinity-algebra | COHOMOLOGY | SH-LIE-ALGEBRAS | CONSTRUCTION

55R | 57T | Hochschild and Harrison cohomology | 55P45 | Hirsch algebras | Maurer–Cartan | (co)bar construction | twisted differential | 18G55 | Twisting cochain | and L | algebra | Maurer-Cartan | MATHEMATICS | A(infinity)- and L-infinity-algebra | COHOMOLOGY | SH-LIE-ALGEBRAS | CONSTRUCTION

Journal Article

Applied Categorical Structures, ISSN 0927-2852, 8/2017, Volume 25, Issue 4, pp. 505 - 538

This paper establishes a procedure that splits the operations in any algebraic operad, generalizing previous notions of splitting algebraic structures, from...

Dendriform algebra | Rota-Baxter operator | Mathematics | Theory of Computation | L ∞ $L_{\infty }$ algebra | Pre-Lie algebra | Geometry | Successor | A ∞ $A_{\infty }$ algebra | Splitting | Convex and Discrete Geometry | Operad | n -algebra | Mathematical Logic and Foundations | n-algebra | algebra | LIE-ALGEBRAS | MATHEMATICS | PRODUCTS | A(infinity) algebra | L-infinity algebra | Computer science | Algebra

Dendriform algebra | Rota-Baxter operator | Mathematics | Theory of Computation | L ∞ $L_{\infty }$ algebra | Pre-Lie algebra | Geometry | Successor | A ∞ $A_{\infty }$ algebra | Splitting | Convex and Discrete Geometry | Operad | n -algebra | Mathematical Logic and Foundations | n-algebra | algebra | LIE-ALGEBRAS | MATHEMATICS | PRODUCTS | A(infinity) algebra | L-infinity algebra | Computer science | Algebra

Journal Article

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

We study Maurer–Cartan elements on homotopy Poisson manifolds of degree n. They unify many twisted or homotopy structures in Poisson geometry and mathematical...

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

International Mathematics Research Notices, ISSN 1073-7928, 2013, Volume 2013, Issue 16, pp. 3678 - 3721

We construct a model for the string group as an infinite-dimensional Lie group. In a second step, we extend this model by a contractible Lie group to a Lie...

DIMENSIONAL LIE-GROUPS | MATHEMATICS | PRINCIPAL BUNDLES | UNIVERSAL COMPLEXIFICATIONS | STACKS | 2-GROUPS | L-INFINITY-ALGEBRAS | NON-ABELIAN EXTENSIONS

DIMENSIONAL LIE-GROUPS | MATHEMATICS | PRINCIPAL BUNDLES | UNIVERSAL COMPLEXIFICATIONS | STACKS | 2-GROUPS | L-INFINITY-ALGEBRAS | NON-ABELIAN EXTENSIONS

Journal Article

Applied Categorical Structures, ISSN 0927-2852, 12/2016, Volume 24, Issue 6, pp. 845 - 873

For a Koszul operad 𝒫 $\mathcal {P}$ , there are several existing approaches to the notion of a homotopy between homotopy morphisms of homotopy 𝒫 $\mathcal...

Geometry | Homotopy morphisms | 18D50 | Convex and Discrete Geometry | Homotopy algebras | Models for operads | Mathematics | Theory of Computation | 18G55 | Mathematical Logic and Foundations | OPERADS | MATHEMATICS | MODELS | CATEGORY | DEFORMATIONS | L-INFINITY-ALGEBRAS | Analysis | Algebra | Computer Science

Geometry | Homotopy morphisms | 18D50 | Convex and Discrete Geometry | Homotopy algebras | Models for operads | Mathematics | Theory of Computation | 18G55 | Mathematical Logic and Foundations | OPERADS | MATHEMATICS | MODELS | CATEGORY | DEFORMATIONS | L-INFINITY-ALGEBRAS | Analysis | Algebra | Computer Science

Journal Article

Mathematische Zeitschrift, ISSN 0025-5874, 4/2013, Volume 273, Issue 3, pp. 981 - 997

We describe the rational homotopy type of any component of the based mapping space map*(X,Y) as an explicit L ∞ algebra defined on the (desuspended and...

Mapping space | Secondary 54C35 | Rational homotopy | Mathematics, general | Mathematics | Primary 55P62 | L ∞ -algebras | algebras | MATHEMATICS | LIE-ALGEBRA | MODELS | L infinity-algebras | Analysis | Algebra

Mapping space | Secondary 54C35 | Rational homotopy | Mathematics, general | Mathematics | Primary 55P62 | L ∞ -algebras | algebras | MATHEMATICS | LIE-ALGEBRA | MODELS | L infinity-algebras | Analysis | Algebra

Journal Article

International Journal of Geometric Methods in Modern Physics, ISSN 0219-8878, 03/2016, Volume 13, Issue 3

First we show that, associated to any Poisson vector field E on a Poisson manifold (M, pi), there is a canonical Lie algebroid structure on the first jet...

L ∞-Algebra | Lie groupoid | cosymplectic structure | matched pair of Lie algebroids | Poisson structure | Atiyah class | Lichnerowicz-Poisson cohomology | TOPOLOGY | PHYSICS, MATHEMATICAL | JACOBI | LIE ALGEBROIDS | GROUPOIDS | L-infinity-algebra | MATCHED PAIRS

L ∞-Algebra | Lie groupoid | cosymplectic structure | matched pair of Lie algebroids | Poisson structure | Atiyah class | Lichnerowicz-Poisson cohomology | TOPOLOGY | PHYSICS, MATHEMATICAL | JACOBI | LIE ALGEBROIDS | GROUPOIDS | L-infinity-algebra | MATCHED PAIRS

Journal Article

International Journal of Geometric Methods in Modern Physics, ISSN 0219-8878, 08/2012, Volume 9, Issue 5, pp. 1250043 - 1250031

The semidirect product of a Lie algebra and a 2-term representation up to homotopy is a Lie 2-algebra. Such Lie 2-algebras include many examples arising from...

butterfly | Representations up to homotopy | integration | algebras | omni-Lie algebras | Lie 2-algebras | crossed modules | COURANT ALGEBROIDS | HOMOTOPY | L-infinity-algebras | PHYSICS, MATHEMATICAL | Lie groups | Representations

butterfly | Representations up to homotopy | integration | algebras | omni-Lie algebras | Lie 2-algebras | crossed modules | COURANT ALGEBROIDS | HOMOTOPY | L-infinity-algebras | PHYSICS, MATHEMATICAL | Lie groups | Representations

Journal Article

Journal of Homotopy and Related Structures, ISSN 2193-8407, 6/2016, Volume 11, Issue 2, pp. 309 - 332

By using homotopy transfer techniques in the context of rational homotopy theory, we show that if $$C$$ C is a coalgebra model of a space $$X$$ X , then the...

Algebra | L_\infty $$ L ∞ -algebra | Functional Analysis | Mapping space | Secondary 54C35 | Algebraic Topology | Rational homotopy | Mathematics | Number Theory | Primary 55P62 | algebra | MATHEMATICS | L-infinity-algebra | LIE-ALGEBRA | Mathematics - Algebraic Topology

Algebra | L_\infty $$ L ∞ -algebra | Functional Analysis | Mapping space | Secondary 54C35 | Algebraic Topology | Rational homotopy | Mathematics | Number Theory | Primary 55P62 | algebra | MATHEMATICS | L-infinity-algebra | LIE-ALGEBRA | Mathematics - Algebraic Topology

Journal Article

Compositio mathematica, ISSN 0010-437X, 02/2013, Volume 149, Issue 2, pp. 264 - 294

Given two Lie 2-groups, we study the problem of integrating a weak morphism between the corresponding Lie 2-algebras to a weak morphism between the Lie...

Lie 2-group | L ∞-algebra | Lie crossed module | connected cover | Lie butterfly | Lie 2-algebra | MATHEMATICS | L-infinity-algebra | Algebraic group theory | Construction | Butterflies | Lie groups

Lie 2-group | L ∞-algebra | Lie crossed module | connected cover | Lie butterfly | Lie 2-algebra | MATHEMATICS | L-infinity-algebra | Algebraic group theory | Construction | Butterflies | Lie groups

Journal Article

Letters in Mathematical Physics, ISSN 0377-9017, 1/2013, Volume 103, Issue 1, pp. 79 - 112

We give a general treatment of the Maurer–Cartan equation in homotopy algebras and describe the operads and formal differential geometric objects governing the...

Theoretical, Mathematical and Computational Physics | Statistical Physics, Dynamical Systems and Complexity | 16E45 | Physics | Geometry | 17B55 | 17B66 | operad | 18D50 | A-infinity algebra | differential graded Lie algebra | Maurer–Cartan element | Group Theory and Generalizations | L-infinity algebra | twisting | Maurer-Cartan element | OPERADS | DUALITY | PHYSICS, MATHEMATICAL | Algebra

Theoretical, Mathematical and Computational Physics | Statistical Physics, Dynamical Systems and Complexity | 16E45 | Physics | Geometry | 17B55 | 17B66 | operad | 18D50 | A-infinity algebra | differential graded Lie algebra | Maurer–Cartan element | Group Theory and Generalizations | L-infinity algebra | twisting | Maurer-Cartan element | OPERADS | DUALITY | PHYSICS, MATHEMATICAL | Algebra

Journal Article

International Journal of Mathematics, ISSN 0129-167X, 07/2012, Volume 23, Issue 7, pp. 1250053 - 1250030

We identify dglas that control infinitesimal deformations of the pairs (manifold, Higgs bundle) and of Hitchin pairs. As a consequence, we recover known...

algebras | Hitchin pairs | differential graded Lie algebras | Deformation theory | MATHEMATICS | COMPACT KAHLER-MANIFOLDS | L-infinity-algebras | MODULI | Bundling | Manifolds | Descriptions | Obstructions | Deformation | Mathematical analysis | Standards

algebras | Hitchin pairs | differential graded Lie algebras | Deformation theory | MATHEMATICS | COMPACT KAHLER-MANIFOLDS | L-infinity-algebras | MODULI | Bundling | Manifolds | Descriptions | Obstructions | Deformation | Mathematical analysis | Standards

Journal Article

Journal of Noncommutative Geometry, ISSN 1661-6952, 2015, Volume 9, Issue 1, pp. 161 - 184

A well-known theorem of Kapranov states that the Atiyah class of the tangent bundle TX of a complex manifold X makes the shifted tangent bundle TX [-1] into a...

Jet bundle | Differential graded algebra | L ∞ -algebra | Atiyah class | Formal geometry | Formal neighborhood | MATHEMATICS | MATHEMATICS, APPLIED | LIE ALGEBROIDS | differential graded algebra | L-infinity-algebra | DEFORMATION QUANTIZATION | PHYSICS, MATHEMATICAL | jet bundle | formal geometry | GEOMETRY

Jet bundle | Differential graded algebra | L ∞ -algebra | Atiyah class | Formal geometry | Formal neighborhood | MATHEMATICS | MATHEMATICS, APPLIED | LIE ALGEBROIDS | differential graded algebra | L-infinity-algebra | DEFORMATION QUANTIZATION | PHYSICS, MATHEMATICAL | jet bundle | formal geometry | GEOMETRY

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. On the other hand, the...

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

GEORGIAN MATHEMATICAL JOURNAL, ISSN 1072-947X, 2009, Volume 16, Issue 2, pp. 305 - 310

We consider a particular finite dimensional example of an L-infinity algebra in which a 2-dimensional Lie algebra acts on a 1-dimensional vector space in a...

MATHEMATICS | L(infinity)algebra | open-closed homotopy algebra

MATHEMATICS | L(infinity)algebra | open-closed homotopy algebra

Journal Article

Differential Geometry and its Applications, ISSN 0926-2245, 06/2020, Volume 70, p. 101631

Consider a closed non-degenerate 3-form ω with an infinitesimal action of a Lie algebra g. Motivated by the fact that the observables associated to ω form a...

Moment map | Lie 2-algebra | Multisymplectic geometry | MATHEMATICS | MATHEMATICS, APPLIED | L-INFINITY-ALGEBRAS

Moment map | Lie 2-algebra | Multisymplectic geometry | MATHEMATICS | MATHEMATICS, APPLIED | L-INFINITY-ALGEBRAS

Journal Article

LINEAR ALGEBRA AND ITS APPLICATIONS, ISSN 0024-3795, 05/2017, Volume 520, pp. 16 - 31

We detect higher order Whitehead products on the homology H of a differential graded Lie algebra L in terms of higher brackets in the transferred L. structure...

Rational homotopy theory | MATHEMATICS | MATHEMATICS, APPLIED | Higher Whitehead product | MODELS | L-infinity-algebra | DEFORMATION-THEORY

Rational homotopy theory | MATHEMATICS | MATHEMATICS, APPLIED | Higher Whitehead product | MODELS | L-infinity-algebra | DEFORMATION-THEORY

Journal Article

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