Introduction to sh Lie algebras for physicists

Abstract

Closed string field theory leads to a generalization of Lie algebra which arose naturally within mathematics in the study of deformations of algebraic structures. It also appeared in work on higher spin particles \cite{BBvD}. Representation theoretic analogs arose in the mathematical analysis of the Batalin-Fradkin-Vilkovisky approach to constrained Hamiltonians. A major goal of this paper is to see the relevant formulas, especially in closed string field theory, as a generalization of those for a differential graded Lie algebra, hopefully describing the mathematical essentials in terms accessible to {\it physicists}.

14 pages AMSTEX, UNC-MATH-92/2

References in corpus (1)

• Algebraic Structures and Differential Geometry in 2D String Theory

Cited by in corpus (327)

• Double Field Theory
• Consistent couplings between fields with a gauge freedom and deformations of the master equation
• Higher derived brackets and homotopy algebras
• Reduction of Courant algebroids and generalized complex structures
• Nonassociative Star Product Deformations for D-brane Worldvolumes in Curved Backgrounds
• n-ary algebras: a review with applications
• $L_{\infty }$ Algebras and Field Theory
• Unfolding Mixed-Symmetry Fields in AdS and the BMV Conjecture: I. General Formalism
• On operad structures of moduli spaces and string theory
• Higher-Spin Interactions: four-point functions and beyond
• Parent field theory and unfolding in BRST first-quantized terms
• Čech cocycles for differential characteristic classes -- An infinity-Lie theoretic construction
• Noncommutative homotopy algebras associated with open strings
• Chiral Higher Spin Theories and Self-Duality
• Lagrange structure and quantization
• The sh Lie structure of Poisson brackets in field theory
• Chern-Simons Gauge Theory coupled with BF Theory
• L-infinity algebras from multisymplectic geometry
• On the category of Lie n-algebroids
• On Conformal, SL(4,R) and Sp(8,R) Symmetries of 4d Massless Fields
• Higher-order simple Lie algebras
• Leibniz algebroids, twistings and exceptional generalized geometry
• Double Copy from Homotopy Algebras
• Deformation theory and rational homotopy type
• Hamiltonian operators and l*-coverings
• Categorified Symplectic Geometry and the Classical String
• Homological perturbation theory for algebras over operads
• Homotopy algebras inspired by classical open-closed string field theory
• Deformation Theory and the Batalin-Vilkovisky Master Equation
• Noncommutative differential calculus, homotopy BV algebras and formality conjectures
• 3-Cocycles, Non-Associative Star-Products and the Magnetic Paradigm of R-Flux String Vacua
• NS-NS Sector of Closed Superstring Field Theory
• Higher-Spin Theory and Space-Time Metamorphoses
• Homotopy Algebra Morphism and Geometry of Classical String Field Theory
• The Schouten-Nijenhuis bracket, cohomology and generalized Poisson structures
• Lectures on deformations of complex manifolds
• From Atiyah Classes to Homotopy Leibniz Algebras
• Formal solution of the master equation via HPT and deformation theory
• General Yang-Mills type gauge theories for p-form gauge fields: From physics-based ideas to a mathematical framework OR From Bianchi identities to twisted Courant algebroids
• String Field Theory -- A Modern Introduction
• A Proof of Tsygan's Formality Conjecture for an Arbitrary Smooth Manifold
• On the generalizations of Poisson structures
• Scattering Amplitude Recursion Relations in BV Quantisable Theories
• Lectures on AKSZ Sigma Models for Physicists
• The Embedding Tensor, Leibniz-Loday Algebras, and Their Higher Gauge Theories
• L-infinity algebra connections and applications to String- and Chern-Simons n-transport
• Products of Floer cohomology of torus fibers in toric Fano manifolds
• Deformations and homotopy theory of relative Rota-Baxter Lie algebras
• Higher Derived Brackets for Arbitrary Derivations
• Gauge invariant double copy of Yang-Mills theory: the quartic theory
• Higher Symplectic Geometry
• The Gauge Structure of Double Field Theory follows from Yang-Mills Theory
• Leibniz bialgebras, relative Rota-Baxter operators and the classical Leibniz Yang-Baxter equation
• Deformations of Batalin-Vilkovisky algebras
• From the Berkovits formulation to the Witten formulation in open superstring field theory
• Towards an M5-Brane Model I: A 6d Superconformal Field Theory
• Non-Commutative Batalin-Vilkovisky Algebras, Homotopy Lie Algebras and the Courant Bracket
• Representations of Homotopy Lie-Rinehart Algebras
• Leibniz 2-algebras and twisted Courant algebroids
• Open-closed homotopy algebra in mathematical physics
• Deformations and obstructions of pairs (X,D)
• Semiregularity and obstructions of complete intersections
• Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry
• An algebraic proof of Bogomolov-Tian-Todorov theorem
• Multisymplectic formulation of vielbein gravity. De Donder-Weyl formulation, Hamiltonian (n-1)-forms
• Extended Riemannian Geometry I: Local Double Field Theory
• Higher Poisson Brackets and Differential Forms
• Off-shell Gauge Fields from BRST Quantization
• Twisted Rota-Baxter operators and Reynolds operators on Lie algebras and NS-Lie algebras
• Semistrict Higher Gauge Theory
• $A_{\infty }$ structure from the Berkovits formulation of open superstring field theory
• Homotopy Lie Superalgebra in Yang-Mills Theory
• Gauge independent kinematic algebra of self-dual Yang-Mills theory
• General properties of the expansion methods of Lie algebras
• Hidden Q-structure and Lie 3-algebra for non-abelian superconformal models in six dimensions
• Simultaneous deformations and Poisson geometry
• A new formulation of higher parallel transport in higher gauge theory
• Formal Maurer-Cartan Structures: from CFT to Classical Field Equations
• Poincaré--Birkhoff--Witt isomorphisms and Kapranov dg-manifolds
• Leibniz Gauge Theories and Infinity Structures
• Introduction to graded geometry
• Braided $L_{\infty }$-Algebras, Braided Field Theory and Noncommutative Gravity
• Non-Abelian Antibrackets
• Effective Batalin--Vilkovisky theories, equivariant configuration spaces and cyclic chains
• On the Strong Homotopy Lie-Rinehart Algebra of a Foliation
• Non-commutative deformation of Chern-Simons theory
• $L_{\infty }$-Algebras, the BV Formalism, and Classical Fields
• 4-d semistrict higher Chern-Simons theory I
• Tensor hierarchy algebras and extended geometry II: Gauge structure and dynamics
• $L_{\infty }$ algebras for extended geometry from Borcherds superalgebras
• Gauge PDE and AKSZ-type Sigma Models
• Lie 2-bialgebras
• BRST analysis of general mechanical systems
• SFT-inspired Algebraic Structures in Gauge Theories
• $Sp(8)$ invariant higher spin theory, twistors and geometric BRST formulation of unfolded field equations
• Chiral Higher Spin Gravity and Convex Geometry
• Braided Symmetries in Noncommutative Field Theory
• Lie n-algebras of BPS charges
• The character map in (twisted differential) non-abelian cohomology
• Open superstring field theory based on the supermoduli space
• Towards an M5-Brane Model II: Metric String Structures
• Higher order BRST and anti-BRST operators and cohomology for compact Lie algebras
• Non-abelian Extensions of Lie 2-algebras
• From $L_{\infty }$-algebroids to higher Schouten/Poisson structures
• A Riccati type PDE for light-front higher helicity vertices
• Cutkosky Rules and Unitarity (Violation) in D-instanton Amplitudes
• Structure of Higher Spin Gauge Interactions
• The Pontryagin Class for Pre-Courant Algebroids
• The controlling $L_{\infty }$-algebra, cohomology and homotopy of embedding tensors and Lie-Leibniz triples
• On the Strong Homotopy Associative Algebra of a Foliation
• Slightly Broken Higher Spin Symmetry: General Structure of Correlators
• From Loop Space Mechanics to Nonabelian Strings
• Characteristic Cohomology and Observables in Higher Spin Gravity
• Weak Lie 2-bialgebra
• Nonabelian higher derived brackets
• Lie 2-algebra models
• Topological Field Theories and Geometry of Batalin-Vilkovisky Algebras
• A cohomological framework for homotopy moment maps
• Adjusted Parallel Transport for Higher Gauge Theories
• Graded Geometry, $Q$-Manifolds, and Microformal Geometry
• Higher Gauge Structures in Double and Exceptional Field Theory
• Homotopy Algebras via Resolutions of Operads
• $L_{\infty }$ algebras and Tensor Hierarchies in Exceptional Field Theory and Gauged Supergravity
• Non-abelian Gerbes and Enhanced Leibniz Algebras
• M-theory from the Superpoint
• The $L_{\infty }$-deformations of associative Rota-Baxter algebras and homotopy Rota-Baxter operators
• An Abstract Interface to Higher Spin Gauge Field Theory
• L-infinity Algebras From Multicontact Geometry
• "Nonlinear pullbacks" of functions and $L_{\infty }$-morphisms for homotopy Poisson structures
• Multi-operator brackets acting thrice
• Supersymmetric Deformations of Maximally Supersymmetric Gauge Theories
• Local BRST cohomology in (non-)Lagrangian field theory
• $L_{\infty }$-Algebra Models and Higher Chern-Simons Theories
• Presymplectic AKSZ formulation of Einstein gravity
• Extended Riemannian Geometry II: Local Heterotic Double Field Theory
• Deformations of coisotropic submanifolds for fibrewise entire Poisson structures
• The Non-Abelian Self-Dual String and the (2,0)-Theory
• A Lie based 4-dimensional higher Chern-Simons theory
• Deformation Theory (lecture notes)
• Formality and Kontsevich--Duflo type theorems for Lie pairs
• Extended deformation functors
• $L_{\infty }$-Algebras of Einstein-Cartan-Palatini Gravity
• Poisson-Jacobi reduction of homogeneous tensors
• Minimal models of field theories: SDYM and SDGR
• Cohomology of Filippov algebras and an analogue of Whitehead's lemma
• Towards Unifying Structures in Higher Spin Gauge Symmetry
• Twisted Courant algebroids and coisotropic Cartan geometries
• Cohomological constraint to deformations of compact Kaehler manifolds
• Courant sigma model and $L_{\infty }$-algebras
• Existence and unicity of co-moments in multisymplectic geometry
• Infinity-enhancing of Leibniz algebras
• L-infinity structures on mapping cones
• Homotopy algebra of open-closed strings
• Infinity Algebras, Cohomology and Cyclic Cohomology, and Infinitesimal Deformations
• Strong Homotopy Lie Algebras, Generalized Nahm Equations and Multiple M2-branes
• BRST Formalism and Zero Locus Reduction
• $A_{\infty }$ Algebras from Slightly Broken Higher Spin Symmetries
• $A_{\infty }$ Algebras and the Cohomology of Moduli Spaces
• Leibniz-Yang-Mills Gauge Theories and the 2-Higgs Mechanism
• Variational tricomplex of a local gauge system, Lagrange structure and weak Poisson bracket
• Double field theory algebroid and curved $L_{\infty }$-algebras
• Microformal geometry and homotopy algebras
• On-shell equivalence of two formulations for superstring field theory
• $\mathcal{W}$ algebras are L${}_{\infty }$ algebras
• Gauge Invariant Perturbation Theory via Homotopy Transfer
• Formality conjecture for minimal surfaces of Kodaira dimension 0
• Homotopy Algebras in Higher Spin Theory
• Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories
• Generalized Higher Gauge Theory
• Categorification of Pre-Lie Algebras and Solutions of 2-graded Classical Yang-Baxter Equations
• Formality theorems for Hochschild complexes and their applications
• Strongly homotopy Lie bialgebras and Lie quasi-bialgebras
• Semiheaps and Ternary Algebras in Quantum Mechanics Revisited
• Graded manifolds and Drinfeld doubles for Lie bialgebroids
• Canonical Transformations of Local Functionals and sh-Lie Structures
• Higher Poincare Lemma and Integrability
• Variational Lie algebroids and homological evolutionary vector fields
• Algebraic formulation of higher gauge theory
• $A_{\infty }/L_{\infty }$ structure and alternative action for WZW-like superstring field theory
• Higher Derived Brackets and Deformation Theory I
• Classical and quantum aspects of the extended antifield formalism
• Supersymmetric Yang-Mills Theory as Higher Chern-Simons Theory
• Tulczyjew triples and higher Poisson/Schouten structures on Lie algebroids
• Higher Structures in M-Theory
• Spacetime Diffeomorphisms and Topological W-Infinity Symmetry in Two Dimensional Topological String Theory
• Bootstrapping Non-commutative Gauge Theories from L${}_{\infty }$ algebras
• Courant algebroids from categorified symplectic geometry
• Intrinsic brackets and the $L_{\infty }$-deformation theory of bialgebras
• L-infinity algebras and higher analogues of Dirac structures and Courant algebroids
• Automorphisms of Strong Homotopy Lie Algebras of Local Observables
• Cohomological characterizations of non-abelian extensions of strict Lie 2-algebras
• L-infinity algebras, Cartan homotopies and period maps
• Constructions of L${}_{\infty }$ algebras and their field theory realizations
• Infinity Algebras, Massey Products, and Deformations
• On the L${}_{\infty }$ structure of Poisson gauge theory
• Higher-Dimensional Algebra VI: Lie 2-Algebras
• Descent of Deligne-Getzler $\infty$-groupoids
• Cohomology and deformations of dendriform algebras, and ${\mathrm{Dend}}_{\infty }$-algebras
• BFV-complex and higher homotopy structures
• A survey on deformations, cohomologies and homotopies of relative Rota-Baxter Lie algebras
• Magnetic Domains
• Review of deformation theory I: Concrete formulas for deformations of algebraic structures
• Systematics of Higher-spin Light-front Interactions
• Twilled 3-Lie algebras, generalized matched pairs of 3-Lie algebras and O-operators
• Thick morphisms, higher Koszul brackets, and $L_{\infty }$-algebroids
• Formalite $G_{\infty }$ adaptee et star-representations sur des sous-varietes coisotropes
• T-duality in rational homotopy theory via $L_{\infty }$-algebras
• Field Theory Equivalences as Spans of $L_{\infty }$-algebras
• Multibraces on the Hochschild complex
• On the infinity category of homotopy Leibniz algebras
• HS in flat spacetime. The effective action method
• Deformations and homotopy theory of Rota-Baxter algebras of any weight
• Derived Differentiable Manifolds
• The (secret?) homological algebra of the Batalin-Vilkovisky approach
• $\mathcal{O}$-operators on Lie $\infty$-algebras with respect to Lie $\infty$-actions
• Higher categorified algebras versus bounded homotopy algebras
• Odd Jacobi structures and classical BV-gauge systems
• Sh-Lie algebras Induced by Gauge Transformations
• Homotopy Lie Algebras of Gravity and their Braided Deformations
• A short note on infinity-groupoids and the period map for projective manifolds
• A Class of Strongly Homotopy Lie Algebras with Simplified sh-Lie Structures
• $L_{\infty }$ and $A_{\infty }$ structures: then and now
• Deformations of path algebras of quivers with relations
• Covariant phase space and $L_{\infty }$ algebras
• Symplectic structure in open string field theory II: Sliding lump
• Higher Koszul brackets on the cotangent complex
• Basic curvature and the Atiyah cocycle in gauge theory
• Split Courant algebroids as $L_{\infty }$-structures
• Odd Jacobi manifolds: general theory and applications to generalised Lie algebroids
• Versal Deformations of Three Dimensional Lie Algebras as L-infinity Algebras
• Lagrangian field theories: ind/pro-approach and L-infinity algebra of local observables
• Worldline quantization of field theory, effective actions and $L_{\infty }$ structure
• $L_{\infty }$ algebras for extended geometry
• The modular class of a singular foliation
• Deformations and cohomologies of embedding tensors on 3-Lie algebras
• Atiyah classes of strongly homotopy Lie pairs
• Higher curvature Bianchi identities, generalised geometry and $L_{\infty }$ algebras
• Homotopy Rota-Baxter operators, homotopy $\mathcal{O}$-operators and homotopy post-Lie algebras
• Universal Lie Formulas for Higher Antibrackets
• Geometry of differential operators, odd Laplacians, and homotopy algebras
• The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective
• Kirillov structures up to homotopy
• Deformation Theory of Infinity Algebras
• Homotopy relative Rota-Baxter Lie algebras, triangular $L_{\infty }$-bialgebras and higher derived brackets
• Symmetry breaking mechanisms of the 3BF action for the Standard Model coupled to gravity
• Non-commutative gauge symmetry from strong homotopy algebras
• Calculus of multilinear differential operators, operator $L_{\infty }$-algebras and ${IBL}_{\infty }$-algebras
• Gauge equivalences for foliations and pre-symplectic structures
• Compatibility, multi-brackets and integrability of systems of PDEs
• On the Structure of Quantum L${}_{\infty }$ algebras
• From topological field theory to deformation quantization and reduction
• L-Infinity Structures on Spaces with 3 One-Dimensional Components
• On higher Poisson and Koszul--Schouten brackets
• Relation between the Reducibility Structures and between the Master Actions in the Witten Formulation and the Berkovits Formulation of Open Superstring Field Theory
• Nambu-like odd brackets on supermanifolds
• On the $L_{\infty }$ formulation of Chern-Simons theories
• Infinity Algebras and the Homology of Graph Complexes
• Nijenhuis forms on $L_{\infty }$-algebras and Poisson geometry
• Secondary products in supersymmetric field theory
• Off-Shell Covariantization of Algebroid Gauge Theories
• All Stable Characteristic Classes of Homological Vector Fields
• Pre-NQ Manifolds and Correspondence Spaces: the Nilmanifold Example
• Noncommutative calculus and the Gauss-Manin connection
• Higher structures in rational homotopy theory
• Twisted Weil Algebras for the String Lie 2-Algebra
• Gerbes in Geometry, Field Theory, and Quantisation
• Equivalences of coisotropic submanifolds
• On the Existence of an L${}_{\infty }$ structure for the Super-Virasoro Algebra
• Nonlinear realisations of Lie superalgebras
• On the associative homotopy Lie algebras and the Wronskians
• Brackets in the jet-bundle approach to field theory
• Strongly Homotopy Lie Algebras from Multisymplectic Geometry
• Strongly Homotopy Lie Algebras of One Even and Two Odd Dimensions
• Multisymplectic Structures and Higher Momentum Maps
• Stability of fixed points of Dirac structures
• Derived brackets and sh Leibniz algebras
• $L_{\infty }$-structure on Barzdell's complex for monomial algebras
• Cohomology theory of averaging algebras, $L_{\infty }$-structures and homotopy averaging algebras
• The $L_{\infty }$-algebra of the S-matrix
• Twisting on pre-Lie algebras and quasi-pre-Lie bialgebras
• Homotopy Poisson algebras, Maurer-Cartan elements and Dirac structures of CLWX 2-algebroids
• Generalized symmetries as homotopy Lie algebras
• $L_{\infty }$-algebra of an unobstructed deformation functor
• Deformations of Courant Algebroids and Dirac Structures via Blended Structures
• Cohomology and deformations of Courant pairs
• Parametrized L-infinity Spaces
• $L_{\infty }$ algebra structures of Lie algebra deformations
• Commutative $n$-ary superalgebras with an invariant skew-symmetric form
• Categorification of VB-Lie algebroids and VB-Courant algebroids
• Strong homotopy Lie algebras, homotopy Poisson manifolds and Courant algebroids
• The $L_{\infty }$-algebra of a symplectic manifold
• Gauged Double Field Theory as an $L_{\infty }$ algebra
• Omni-Lie Color Algebras and Lie Color 2-Algebras
• Lie-Rinehart algebra $\simeq$ acyclic Lie $\infty$-algebroid
• Dg Loday-Pirashvili modules over Lie algebras
• Compatible $L_{\infty }$-algebras
• Cyclic Homology of Fukaya Categories and the Linearized Contact Homology
• Multisymplectic observable reduction using constraint triples
• A supergeometric approach to $L_{\infty }$-bialgebras
• Extended Bargmann FDA and non-relativistic gravity
• On the L-infinity description of the Hitchin Map
• Extended Field Theories as higher Kaluza-Klein theories
• Cumulants, Koszul brackets and homological perturbation theory for commutative $B{V}_{\infty }$ and $IB{L}_{\infty }$ algebras
• Lectures on Symplectic Geometry, Poisson Geometry, Deformation Quantization and Quantum Field Theory
• BRST Extension of the Non-Linear Unfolded Formalism
• The higher-algebraic skeleton of the superstring -- a case study
• $L_{\infty }$-algebras and membrane sigma models
• Generalized Higher Gauge Theory and M5-brane dynamics. Proceedings: Higher structures in String and M-theory, Tohoku Forum for Creativity, March 7-11 2016
• Strongly Homotopy Chiral Algebroids
• The Maurer-Cartan structure of BRST differential
• Higher Lie and Leibniz algebras
• On exact Courant algebras
• Extensions of L_infinity algebras of two even and one odd dimension
• Classification and versal deformations of L_\infinity algebras on a 2|1-dimensional space
• Omni-Lie Superalgebras and Lie 2-superalgebras
• Lift of $C\_\infty$ and $L\_\infty$ morphisms to $G\_\infty$ morphisms
• Formality theorem with coefficients in a module
• Comparison of 3-dimensional Z-graded and Z_2-graded L_Infinity algebras
• Examples of infinity and Lie algebras and their versal deformations
• Hypersurfaces and generalized deformations
• Examples of Homotopy Lie Algebras
• On the tensor structure of BRST differential and it's application
• Chicken or egg? A hierarchy of homotopy algebras
• Homology of L_{\infty}-Algebras and Cyclic Homology
• Chain extensions of D-algebras and their applications
• A Formality Theorem for Hochschild Chains
• Free Homotopy Algebras