Homotopy Lie Algebra Research Articles

Non-Commutative Batalin-Vilkovisky Algebras, Homotopy Lie Algebras and the Courant Bracket

We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent Δ operator, we define a non-commutative generalization of the higher Koszul brackets, which are used in a generalized Batalin-Vilkovisky algebra, and we show that they form a homotopy Lie algebra. Secondly, we investigate higher, so-called derived brackets built from symmetrized, nested Lie brackets with a fixed nilpotent Lie algebra element Q. We find the most general Jacobi-like identity that such a hierarchy satisfies. The numerical coefficients in front of each term in these generalized Jacobi identities are related to the Bernoulli numbers. We suggest that the definition of a homotopy Lie algebra should be enlarged to accommodate this important case. Finally, we consider the Courant bracket as an example of a derived bracket. We extend it to the “big bracket” of exterior forms and multi-vectors, and give closed formulas for the higher Courant brackets.

On the Golod property of Stanley–Reisner rings

Recently in [M. Jöllenbeck, On the multigraded Hilbert and Poincaré series of monomial rings, J. Pure Appl. Algebra 207 (2) (2006) 261–298] the second author made a conjecture about the structure of Ext A ∗ ( k , k ) as an N × N n -graded vector space, where A is a monomial ring over a field k, that is, the quotient of a polynomial ring P = k [ x 1 , … , x n ] by a monomial ideal, and he verified this conjecture for several classes of such rings. Using the results of [A. Berglund, Poincaré series and homotopy Lie algebras of monomial rings, Licentiate thesis, Stockholm University, http://www.math.su.se/reports/2005/6/, 2005] by the first author, we are able to prove this conjecture in general. In particular we get a new explicit formula for the multigraded Hilbert series of Ext A ∗ ( k , k ) . A surprising consequence of our results is that a monomial ring A is Golod if and only if the product on Tor ∗ P ( A , k ) is trivial. For Stanley–Reisner rings of flag complexes we get a complete combinatorial characterization of Golodness. We introduce the concept of ‘minimally non-Golod complexes,’ and show that boundary complexes of stacked polytopes are minimally non-Golod. Finally we discuss the relation between minimal non-Golodness and the Gorenstein ∗ property for simplicial complexes.

Associative homotopy Lie algebras and Wronskians

We analyze representations of Schlessinger-Stasheff associative homotopy Lie algebras by higher-order differential operators. W-transformations of chiral embeddings of a complex curve related with the Toda equations into Kahler manifolds are shown to be endowed with the homotopy Lie-algebra structures. Extensions of the Wronskian determinants preserving Schlessinger-Stasheff algebras are constructed for the case of n ≥ 1 independent variables.

On the homotopy Lie algebra of an arrangement

Let A be a graded-commutative, connected k-algebra generated in degree 1. The homotopy Lie algebra g_A is defined to be the Lie algebra of primitives of the Yoneda algebra, Ext_A(k,k). Under certain homological assumptions on A and its quadratic closure, we express g_A as a semi-direct product of the well-understood holonomy Lie algebra h_A with a certain h_A-module. This allows us to compute the homotopy Lie algebra associated to the cohomology ring of the complement of a complex hyperplane arrangement, provided some combinatorial assumptions are satisfied. As an application, we give examples of hyperplane arrangements whose complements have the same Poincar\'e polynomial, the same fundamental group, and the same holonomy Lie algebra, yet different homotopy Lie algebras.

Formula omitted] algebra structures from Lie algebra deformations

In this paper, we will show how to kill the obstructions to Lie algebra deformations via a method which essentially embeds a Lie algebra into a strong homotopy Lie algebra or L ∞ algebra such that the sh-Lie algebra is the same homotopy type as the original Lie algebra.

A class of strongly homotopy Lie algebras with simplified sh-Lie structures

Given a complex that is a differential graded vector space, it is known that a single mapping defined on a space of it where the homology is non-trivial extends to a strongly homotopy Lie algebra (on the graded space) when that mapping satisfies two conditions. This strongly homotopy Lie algebra is non-trivial (it is not a Lie algebra); however we show that one can obtain an sh-Lie algebra where the only non-zero mappings defining it are the lower order mappings. This structure applies to a significant class of examples. Moreover in this case the graded space can be replaced by another graded space, with only three non-zero terms, on which the same sh-Lie structure exists.

An abstract interface to higher spin gauge field theory

A comprehensive approach to the theory of higher spin gauge fields is proposed. By explicitly separating out details of implementation from general principles, it becomes possible to focus on the bare minimum of requirements that such a theory must satisfy. The abstraction is based on a survey of the progress that has been achieved since relativistic wave equations for higher spin fields were first considered in the 1930s. As a byproduct, a formalism is obtained that is abstract enough to describe a wide class of classical field theories. The formalism, viewed as syntax, can then be semantically mapped to a category of strongly homotopy Lie algebras, thus showing that the theory in some sense exists, at least as an abstract mathematical structure. Still, a concrete physicslike, implementation remains to be constructed. Lacking deep physical insight into the problem, an implementation in terms of generalized vertex operators is set up within which a brute force iterative determination of the first few orders in the interaction can be attempted.

VERSAL DEFORMATIONS OF THREE-DIMENSIONAL LIE ALGEBRAS AS L∞ALGEBRAS

We consider versal deformations of 0|3-dimensional L∞algebras, also called strongly homotopy Lie algebras, which correspond precisely to ordinary (non-graded) three-dimensional Lie algebras. The classification of such algebras is well-known, although we shall give a derivation of this classification using an approach of treating them as L∞algebras. Because the symmetric algebra of a three-dimensional odd vector space contains terms only of exterior degree less than or equal to three, the construction of versal deformations can be carried out completely. We give a characterization of the moduli space of Lie algebras using deformation theory as a guide to understanding the picture.

Higher derived brackets and homotopy algebras

We give a construction of homotopy algebras based on “higher derived brackets”. More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element Δ . Given this, we introduce an infinite sequence of higher brackets on the image of the projector, and explicitly calculate their Jacobiators in terms of Δ 2 . This allows to control higher Jacobi identities in terms of the “order” of Δ 2 . Examples include Stasheff's strongly homotopy Lie algebras and variants of homotopy Batalin–Vilkovisky algebras. There is a generalization with Δ replaced by an arbitrary odd derivation. We discuss applications and links with other constructions.

ON THE DENOMINATOR OF THE POINCARÉ SERIES FOR MONOMIAL QUOTIENT RINGS#

ABSTRACT Let S = 𝕂[x 1, …, x n ] be a polynomial ring over a field 𝕂 and I a monomial ideal of S. It is well known that the Poincaré series of 𝕂 over S/I is rational. We describe the coefficients of the denominator of the series and study the multigraded homotopy Lie algebra of S/I.

Strongly homotopy Lie algebras of one even and two odd dimensions

We classify strongly homotopy Lie algebras—also called L ∞ algebras—of one even and two odd dimensions, which are related to 2 | 1 -dimensional Z 2 -graded Lie algebras. What makes this case interesting is that there are many nonequivalent L ∞ examples, in addition to the Z 2 -graded Lie algebra (or superalgebra) structures, yet the moduli space is simple enough that we can give a complete classification up to equivalence.

Homotopy Lie algebras, lower central series and the Koszul property

Let X and Y be finite-type CW-complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k-rescaling of the rational cohomology ring of X. Assume H^*(X,Q) is a Koszul algebra. Then, the homotopy Lie algebra pi_*(Omega Y) tensor Q equals, up to k-rescaling, the graded rational Lie algebra associated to the lower central series of pi_1(X). If Y is a formal space, this equality is actually equivalent to the Koszulness of H^*(X,Q). If X is formal (and only then), the equality lifts to a filtered isomorphism between the Malcev completion of pi_1(X) and the completion of [Omega S^{2k+1}, Omega Y]. Among spaces that admit naturally defined homological rescalings are complements of complex hyperplane arrangements, and complements of classical links. The Rescaling Formula holds for supersolvable arrangements, as well as for links with connected linking graph.

BV Quantization of Topological Open Membranes

We study bulk-boundary correlators in topological open membranes. The basic example is the open membrane with a WZ coupling to a 3-form. We view the bulk interaction as a deformation of the boundary string theory. This boundary string has the structure of a homotopy Lie algebra, which can be viewed as a closed string field theory. We calculate the leading order perturbative expansion of this structure. For the 3-form field we find that the C-field induces a trilinear bracket, deforming the Lie algebra structure. This paper is the first step towards a formal universal quantization of general quasi-Lie bialgebroids.

Homotopy Lie algebra of classifying spaces for hyperbolic coformal 2-cones

In this paper, we show that the rational homotopy Lie algebra of classifying spaces for certain types of hyperbolic coformal 2-cones is not nilpotent.

Hyperbolic spaces at large primes and a conjecture of Moore

A simply connected finite complex X is called elliptic if its rational homotopy Lie algebra is of finite dimension and hyperbolic otherwise. According to a conjecture of Moore, there exists an exponent for the p-torsion part of π ∗(X) if and only if X is elliptic. In this note, it is shown that, provided the prime p is sufficiently large, a hyperbolic space with p-torsion free loop space homology has no exponent in the p-torsion of the homotopy groups. For a class of formal spaces, this result is obtained for every odd prime.

Elliptic rational spaces whose cohomologies and homotopies are isomorphic

We give two simply connected elliptic 79-dimensional closed smooth manifolds whose rational homotopy types are different. But both their rational cohomology rings and rational homotopy Lie algebras are isomorphic.

Deformation theory of infinity algebras

This work explores the deformation theory of algebraic structures in a very general setting. These structures include associative algebras, Lie algebras, and the infinity versions of these structures, the strongly homotopy associative and Lie algebras. In all these cases the algebraic structure is determined by an element of a certain graded Lie algebra which determines a differential on the Lie algebra. We work out the deformation theory in terms of the Lie algebra of coderivations of an appropriate coalgebra structure and construct a universal infinitesimal deformation as well as a miniversal formal deformation. By working at this level of generality, the main ideas involved in deformation theory stand out more clearly.

Cyclic operads and algebra of chord diagrams

We prove that the algebra $ \cal A $ of chord diagrams, the dual to the associated graded algebra of Vassiliev knot invariants, is isomorphic to the universal enveloping algebra of a Casimir Lie algebra in a certain tensor category (the PROP for Casimir Lie algebras). This puts on a firm ground a known statement that the algebra $ \cal A $ looks and behaves like a universal enveloping algebra. An immediate corollary of our result is the conjecture of [BGRT] on the Kirillov-Duflo isomorphism for algebras of chord diagrams.¶ Our main tool is a general construction of a functor from the category $ \tt CycOp $ of cyclic operads to the category $ \tt ModOp $ of modular operads which is left adjoint to the tree part functor $ {\tt ModOp} \to {\tt CycOp} $ . The algebra of chord diagrams arises when this construction is applied to the operad $ {\tt LIE} $ . Another example of this construction is Kontsevich's graph complex which corresponds to the operad $ {\tt LIE}_\infty $ for homotopy Lie algebras.

Homotopy Lie algebras and Poincaré series of algebras with monomial relations

To every homogeneous ideal of a polynomial ring S over a Þeld K, Macaulay assigned an ideal generated by monomials in the indeterminates and with the same Hilbert function. Thus, from the point of view of Hilbert series residue rings modulo monomial ideals display the most general behavior. The homo- logical perspective reveals a very dierent picture. Two aspects are particularly relevant to this paper: If I is generated by monomials, then the Poincare series of the residue Þeld k of S=I is rational by Backelin (7), and the homotopy Lie algebra of S=I is Þnitely generated by Backe- lin and Roos (8). Constructions of Anick (1) and Roos (15), respectively, show that these properties may fail for general homogeneous ideals. Recenly, Gasharov, Peeva, and Welker (12) showed that some homological properties of S=I, such as being Golod, de- pend only on combinatorial data gathered from a minimal set of monomial generators. Here we prove that these data determine the Poincare series of k over S=I, along with most of its homotopy Lie algebra. As a consequence, we obtain the surprising result that if the number of generators of the ideal I is Þxed, then the number of such Poincare series is Þnite, even when K ranges over all Þelds. To Jan-Erik Roos on his sixty-Þfth birthday

Growth and Lie brackets in the homotopy Lie algebra

Let L be an inÞnite dimensional graded Lie algebra that is either the homotopy Lie algebra oE(SX) aQ for a Þnite ndimensional CW complex X, or else the homotopy Lie algebra for a local noetherian commutative ring R (UL = ExtR(Ik;Ik)) in which case put n = (embdim A depth)(R).