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Differential graded Lie algebra

About: Differential graded Lie algebra is a research topic. Over the lifetime, 199 publications have been published within this topic receiving 8993 citations. The topic is also known as: dg-Lie algebra.


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TL;DR: In this paper, it was shown that every finite-dimensional Poisson manifold X admits a canonical deformation quantization, and that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the class of Poisson structures on X modulo diffeomorphisms.
Abstract: I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the ‘Formality conjecture’), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.

2,672 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that every finite-dimensional Poisson manifold X admits a canonical deformation quantization, which can be interpreted as correlators in topological open string theory.
Abstract: I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven ("Formality conjecture"), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not use explicitly the language of functional integrals. One of corollaries is a justification of the orbit method in the representation theory.

2,223 citations

Journal ArticleDOI
TL;DR: In this paper, Stasheff et al. introduced the Lie algebra of closed string theory and proved that the full Fock complex of the theory is a Lie algebra, with the BRST difierential Q.
Abstract: UNC-MATH-92/2originally April 27, 1990, revised September 24, 1992INTRODUCTION TO SH LIE ALGEBRAS FOR PHYSICISTSTom LadaJim StasheffMuch of point particle physics can be described in terms of Lie algebras andtheir representations. Closed string field theory, on the other hand, leads to ageneralization of Lie algebra which arose naturally within mathematics in the studyof deformations of algebraic structures [SS]. It also appeared in work on higherspin particles [BBvD]. Representation theoretic analogs arose in the mathematicalanalysis of the Batalin-Fradkin-Vilkovisky approach to constrained Hamiltonians[S6].The sh Lie algebra of closed string field theory [SZ], [KKS], [K], [Wies], [WZ],[Z] is defined on the full Fock complex of the theory, with the BRST differential Q.Following Zwiebach [Z], we stipulate that the string fields B

784 citations

Journal ArticleDOI
TL;DR: In this paper, Strongly homotopy lie algebras have been studied in the context of algebraic graph theory, and they are shown to be strongly homotopomorphic.
Abstract: (1995). Strongly homotopy lie algebras. Communications in Algebra: Vol. 23, No. 6, pp. 2147-2161.

576 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that there exists a neighborhood of ρ in ℜ(Γ, G) which is analytically equivalent to a cone defined by homogeneous quadratic equations.
Abstract: Let Γ be the fundamental group of a compact Kahler manifold M and let G be a real algebraic Lie group. Let ℜ(Γ, G) denote the variety of representations Γ → G. Under various conditions on ρ ∈ ℜ(Γ, G) it is shown that there exists a neighborhood of ρ in ℜ(Γ, G) which is analytically equivalent to a cone defined by homogeneous quadratic equations. Furthermore this cone may be identified with the quadratic cone in the space\(Z^1 (\Gamma ,g_{Ad\rho } )\) of Lie algebra-valued l-cocycles on Γ comprising cocyclesu such that the cohomology class of the cup/Lie product square [u, u] is zero in\(H^2 (\Gamma ,g_{Ad\rho } )\). We prove that ℜ(Γ, G) is quadratic at ρ if either (i) G is compact, (ii) ρ is the monodromy of a variation of Hodge structure over M, or (iii) G is the group of automorphisms of a Hermitian symmetric space X and the associated flat X-bundle over M possesses a holomorphic section. Examples are given where singularities of ℜ(Γ, G) are not quadratic, and are quadratic but not reduced. These results can be applied to construct deformations of discrete subgroups of Lie groups.

436 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202110
202014
201925
201815
201713
20168