Lie Algebras of Polynomial Growth

Homology, Homotopy and Applications, 2002

Let L be an infinite dimensional graded Lie algebra that is either the homotopy Lie algebra π * (ΩX) ⊗ Q for a finite ndimensional CW complex X, or else the homotopy Lie algebra for a local noetherian commutative ring R (U L = Ext R (Ik, Ik)) in which case put n = (embdim − depth)(R). Theorem: (i) The integers λ k = k+n−2 q=k dimL i grow faster than any polynomial in k. (ii) For some finite sequence x 1 ,. .. , x d of elements in L and some N , any y ∈ L N satisfies: some [x i , y] = 0.

Proceedings of the American Mathematical Society, 2007

Let L be a connected finite type graded Lie algebra. If dim L = ∞ and gldim L < ∞, then log index L = α > 0. If, moreover, α < ∞, then for some d, d−1 i=1 dim L k+i = e kα k , where α k → log index L as k → ∞ .

Annales Scientifiques de l’École Normale Supérieure, 2003

If A is a graded connected algebra then we define a new invariant, polydepth A, which is finite if Ext * A (M, A) = 0 for some A-module M of at most polynomial growth. THEOREM 1: If f : X → Y is a continuous map of finite category, and if the orbits of H * (ΩY) acting in the homology of the homotopy fibre grow at most polynomially, then H * (ΩY) has finite polydepth. THEOREM 5: If L is a graded Lie algebra and polydepth UL is finite then either L is solvable and UL grows at most polynomially or else for some integer d and all r, k+d i=k+1 dim Li k r , k some k(r).

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

Inventiones Mathematicae, 1996

The class of ÿnite dimensional algebras (associative, with an identity) over an algebraically closed ÿeld k may be divided into two disjoint classes. One class consists of tame algebras for which the indecomposable modules occur, in each dimension, in a ÿnite number of discrete and a ÿnite number of one-parameter families. The second class is formed by the wild algebras whose representation theory is as complicated as the study of ÿnite dimensional vector spaces together with two non-commuting endomorphisms, for which the classiÿcation up to isomorphism is a well-known unsolved problem. A convenient way to determine whether a given algebra A is tame (and to describe its representations) consists in ÿnding a simply connected cover of a suitable degeneration of A. We are interested in geometric and homological properties of tame simply connected algebras. Recently, several interesting characterizations of polynomial growth simply connected algebras and their module categories have been obtained , , . Following , an algebra A is called strongly simply connected if its ordinary quiver has no oriented cycles and the ÿrst Hochschild cohomology H 1 (C; C) of any full convex subcategory C of A vanishes. Moreover, A is of polynomial growth if there exists a positive integer m such that the indecomposable A-modules occur, in each dimension d, in a ÿnite number of discrete and at most d m one-parameter families (see ).

arXiv: Rings and Algebras, 2017

A nilpotent Lie algebra ${\mathfrak g}$ is said to be naturally graded if it is isomorphic to its associated graded Lie algebra ${\rm gr} \mathfrak{g}$ with respect to filtration by ideals of the lower central series. This concept is equivalent to the concept of the Carnot algebra arising in sub-Riemannian geometry and the geometric control theory. We classify finite-dimensional and infinite-dimensional naturally graded Lie algebras (Carnot algebras) ${\mathfrak g}=\oplus_{i=1}^{{+}\infty}{\mathfrak g}_i$ with properties $$ [{\mathfrak g}_1, {\mathfrak g}_i]={\mathfrak g}_{i{+}1}, \; \dim{{\mathfrak g}_i}+\dim{{\mathfrak g}_{i{+}1}} \le 3, \; i \ge 1. $$ For growth functions of such Lie algebras, we have the estimate $F(n) \le \frac{3}{2}n{+}1$.

arXiv: Rings and Algebras, 2013

In 1964 Golod and Shafarevich found that, provided that the number of relations of each degree satisfies some bounds, there exist infinitely dimensional algebras satisfying the relations. Such algebras have come to be referred to as Golod-Shafarevich algebras. This paper provides bounds for the growth function on images of Golod-Shafarevich algebras based upon the number of defining relations, thus extending results from [13], [14]. We also provide lower bounds of growth for constructed algebras, permitting the construction in many cases of algebras with a given growth function. Recently several open questions about algebras satisfying a prescribed number of defining relations have arisen as a consequence of the study of noncommutative singularities. This paper additionally solves one such question, posed by Donovan and Wemyss in [3].

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 Pure and Applied Algebra, 2014

The Lie algebra sl 2 = sl 2 (K) of 2 × 2 traceless matrices over a field K has only three nontrivial G-gradings when G is a group, the ones induced by G = Z 2 , Z 2 × Z 2 and Z. Here we prove that when char(K) = 0, the variety var G (sl 2) of G-graded Lie algebras generated by sl 2 , is a minimal variety of exponential growth, and in case G = Z 2 × Z 2 or Z, var G (sl 2) has almost polynomial growth.

Proceedings of the American Mathematical Society, 2000

Let A be an associative algebras over a field of characteristic zero. We prove that the codimensions of A are polynomially bounded if and only if any finite dimensional algebra B with Id(A) = Id(B) has an explicit decomposition into suitable subalgebras; we also give a decomposition of the n-th cocharacter of A into suitable Sn-characters. We give similar characterizations of finite dimensional algebras with involution whose *-codimension sequence is polynomially bounded. In this case we exploit the representation theory of the hyperoctahedral group.