Engel elements in the homotopy Lie algebra

Commentarii Mathematici Helvetici, 2009

In this paper we consider a graded Lie algebra, L, of finite depth m, and study the interplay between the depth of L and the growth of the integers dim L i. A subspace W in a graded vector space V is called full if for some integers d , N , q, dim V k Ä d P kCq iDk dim W i , i N. We define an equivalence relation on the subspaces of V by U W if U and W are full in U C W. Two subspaces V , W in L are then called L-equivalent (V L W) if for all ideals K L, V \ K W \ K. Then our main result asserts that the set L of L-equivalence classes of ideals in L is a distributive lattice with at most 2 m elements. To establish this we show that for each ideal I there is a Lie subalgebra E L such that E \ I D 0, E˚I

Publications mathématiques de l'IHÉS, 1982

L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ THE HOMOTOPY LIE ALGEBRA FOR FINITE COMPLEXES by YVES FfiLIX (1), STEPHEN HALPERIN (2) and JEAN-CLAUDE THOMAS (3) i. Introduction.-A generic question in topology asks how geometric restrictions on a topological space S are reflected in restrictions on^(S). A classical example is this: which discrete groups G admit a finite GW complex as classifying space? In this paper we shall deal with an analogous question for Lie algebras and simply connected spaces. Henceforth, and throughout this paper, we shall consider only those spaces which are simply connected and have the homotopy type ofGW complexes whose rational homology is finite dimensional in each degree. Such spaces will be called i-connected GW spaces of finite Qj-tyRe. For such spaces 7Tp(S)®Q^ is finite dimensional (each p), and the Whitehead product in ^(S), transferred to T^(OS) by the canonical isomorphism, makes T^(QS) ® Q, into a connected graded Lie algebra of finite type {i.e. finite dimensional in each degree): the rational homotopy Lie algebra of S. A striking result of Quillen [QJ asserts that every connected graded Lie algebra (over QJ of finite type arises in this way. The situation for finite complexes is very different, and the question referred to above, which forms the starting point of this paper, can be stated as the Problem.-What restrictions are imposed on the rational homotopy Lie algebra of a space S, if S is a finite, i-connected, GW complex? We shall establish serious restrictions, both on the integers dimTTp(S) ®Q,, and on the Lie structure. These restrictions, moreover, turn out to hold for the much larger class of those i-connected GW spaces of finite Q^-type whose rational Lusternik-Schnirelmann category is finite. Recall that the Lusternik Schnirelmann category of a space S, as normalized by Ganea [Ga], is the least integer m such that S can be covered by m + i open sets, each (1) Chercheur qualifie au F.N.R.S. (2) During this research the second named author enjoyed the hospitality of the Sonderforschungsbereich (40) Mathematik at the University of Bonn,

Journal of the London Mathematical Society, 1991

Kac has introduced the notion of (polynomial) growth for a graded Lie algebra. Here we consider Lie algebras L that occur as ideals either in the rational homotopy Lie algebra of a simply connected CW complex of finite type and finite category or as ideals in the homotopy Lie algebra of a local noetherian ring. Theorem. If these ideals have {finite) polynomial growth, then they are finite dimensional.

2006

use of general theory when it is convenient, for example in placing model category structures on dga and dge (Theorem 4.16) and in our brief discussion of minimal models (Section 4.6). We expect this framework to be useful in a range of applications. In one sequel to this work, we will show that the functor E applied to the cochains of a space encodes generalized Hopf invariants. The Lie coalgebra point of view thus leads to a way to pass between cochain and homotopy data where the formalism, the combinatorics, and the geometry are unified. Indeed, it was an investigation of generalized Hopf invariants which led us to the framework of this paper. Another promising application is to use the functor E to understand classical Harrison homology. Our work throughout is over a field of characteristic zero. We emphasize that we are adding a finiteness hypothesis, namely that our algebras and coalgebras are finite-dimensional in each positive degree, for the sake of linear duality theorems. Under this hypothesis the category of chain complexes is canonically isomorphic to that of cochain complexes, and we will use this isomorphism without further comment, by abuse denoting both categories by dg. To clarify when possible, we have endeavored to use V to denote a chain complex and W to denote a cochain complex. Many of the facts we prove are true without the finiteness hypothesis, as we may indicate. We plan to remove this hypothesis, as well as the (simple-) connectivity hypotheses in the third paper in this series. Contents 1. Introduction 1 2. The Eil cooperad and its pairing with the Lie operad 2 3. The perfect pairing between free Lie algebras and cofree Eil coalgebras 4 3.1. Basic manipulations of cofree Eil coalgebras 4 3.2. Duality of free algebras and cofree coalgebras 5 3.3. Cofree Eil coalgebras as quotients of cotensor coalgebras 8 4. The functors E and A, a Quillen pair 9 4.1. The Quillen functors L and C 10 4.2. The functors E and A 10 4.3. Adjointness of E and A 12 4.4. The main diagram 13 4.5. Model structures and rational homotopy theory. 15 4.6. Minimal models 17 References 18 2. The Eil cooperad and its pairing with the Lie operad The combinatorial heart of our work is a pairing between rooted and unrooted trees, developed in [14]. Definition 2.1. (1) An n-tree is an isotopy class of acyclic graph whose vertices are either trivalent or univalent, with a distinguished univalent vertex called the root, embedded in the upper half-plane with the root at the origin. Univalent vertices, other than the root, are called leaves, and they are labeled by n = {1,. .. , n}. Trivalent vertices are also called internal vertices. (2) The height of a vertex in a n-tree is the number of edges between that vertex and the root. (3) Define the nadir of a path in a n-tree to be the vertex of lowest height which it traverses. (4) A n-graph is a connected oriented acyclic graph with vertices labeled by n.

Expositiones Mathematicae, 2007

Let X be a simply connected CW complex with finitely many cells in each degree. The first part of the paper is a report on the different conjectures for the behavior of the sequence rk i (X). In the second part, we give conditions on the Lusternik-Schnirelmann category of X and on the depth of the Lie algebra * (X) ⊗ Q that imply the exponential growth in k of the sequence d i=1 rk k+i (X), for some d.