scholarly journals On the Lie-formality of Poisson manifolds

2008 ◽  
Vol 2 (2) ◽  
pp. 361-384 ◽  
Author(s):  
G. Sharygin ◽  
D. Talalaev
Keyword(s):  
Lie Algebra ◽  
Present Note ◽  
Poisson Manifold ◽  
Poisson Manifolds ◽  
De Rham ◽  
Graded Lie Algebra ◽  
Differential Graded Lie Algebra

AbstractIn the present note we prove formality of the differential graded Lie algebra of de Rham forms on a smooth Poisson manifold.

scholarly journals NON-ABELIAN POISSON MANIFOLDS FROM D-BRANES

2004 ◽  
Vol 19 (34) ◽  
pp. 2541-2548
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Lie Algebra ◽  
Gauge Group ◽  
Matrix Multiplication ◽  
Poisson Manifold ◽  
Poisson Manifolds ◽  
Classical Dynamics ◽  
Planck’S Constant ◽  
C Algebra ◽  
Simple Lie Algebra ◽  
Finite Dimensional

Superimposed D-branes have matrix-valued functions as their transverse coordinates, since the latter take values in the Lie algebra of the gauge group inside the stack of coincident branes. This leads to considering a classical dynamics where the multiplication law for coordinates and/or momenta, being given by matrix multiplication, is non-Abelian. Quantization further introduces noncommutativity as a deformation in powers of Planck's constant ℏ. Given an arbitrary simple Lie algebra [Formula: see text] and an arbitrary Poisson manifold ℳ, both finite-dimensional, we define a corresponding C⋆-algebra that can be regarded as a non-Abelian Poisson manifold. The latter provides a natural framework for a matrix-valued classical dynamics.

scholarly journals Batalin-Vilkovisky formality for Chern-Simons theory

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Ezra Getzler
Keyword(s):  
Field Theory ◽  
Lie Algebra ◽  
Simons Theory ◽  
Semisimple Lie Algebra ◽  
General Field ◽  
Graded Lie Algebra ◽  
Variational Complex ◽  
Differential Graded Lie Algebra ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract We prove that the differential graded Lie algebra of functionals associated to the Chern-Simons theory of a semisimple Lie algebra is homotopy abelian. For a general field theory, we show that the variational complex in the Batalin-Vilkovisky formalism is a differential graded Lie algebra.

scholarly journals Explicit symmetric DGLA models of 3-cells

2020 ◽  
pp. 1-15
Author(s):  
Itay Griniasty ◽  
Ruth Lawrence
Keyword(s):  
Differential Geometry ◽  
Lie Algebra ◽  
Graded Lie Algebra ◽  
Explicit Formulae ◽  
Discrete Analogues ◽  
Differential Graded Lie Algebra

Abstract We give explicit formulae for differential graded Lie algebra (DGLA) models of 3-cells. In particular, for a cube and an n-faceted banana-shaped 3-cell with two vertices, n edges each joining those two vertices, and n bi-gon 2-cells, we construct a model symmetric under the geometric symmetries of the cell fixing two antipodal vertices. The cube model is to be used in forthcoming work for discrete analogues of differential geometry on cubulated manifolds.

scholarly journals On Deformations of Pairs (Manifold, Coherent Sheaf)

2019 ◽  
Vol 71 (5) ◽  
pp. 1209-1241 ◽  
Author(s):  
Donatella Iacono ◽  
Marco Manetti
Keyword(s):  
Lie Algebra ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Coherent Sheaf ◽  
Closed Field ◽  
Trace Map ◽  
Deformation Problem ◽  
Graded Lie Algebra ◽  
Infinitesimal Deformations ◽  
Differential Graded Lie Algebra

AbstractWe analyse infinitesimal deformations of pairs $(X,{\mathcal{F}})$ with ${\mathcal{F}}$ a coherent sheaf on a smooth projective variety $X$ over an algebraically closed field of characteristic 0. We describe a differential graded Lie algebra controlling the deformation problem, and we prove an analog of a Mukai–Artamkin theorem about the trace map.

The gauge action, DG Lie algebras and identities for Bernoulli numbers

2017 ◽  
Vol 29 (2) ◽  
pp. 277-286
Author(s):  
Urtzi Buijs ◽  
José G. Carrasquel-Vera ◽  
Aniceto Murillo
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bernoulli Numbers ◽  
Symmetry Relation ◽  
Linear Combinations ◽  
Gauge Action ◽  
Graded Lie Algebra ◽  
Differential Graded Lie Algebra

AbstractIn this paper we prove a family of identities for Bernoulli numbers parameterized by triples of integers ${(a,b,c)}$ with ${a+b+c=n-1}$, ${n\geq 4}$. These identities are deduced by translating into homotopical terms the gauge action on the Maurer–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 of our family and they satisfy a particular symmetry relation.

scholarly journals Quantized Nambu-Poisson manifolds in a 3-Lie algebra reduced model

2011 ◽  
Vol 2011 (4) ◽  
Author(s):  
Joshua DeBellis ◽  
Christian Sämann ◽  
Richard J. Szabo
Keyword(s):  
Lie Algebra ◽  
Poisson Manifolds ◽  
Reduced Model
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