Higher categorified algebras versus bounded homotopy algebras (original) (raw)
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The structure of homotopy Lie algebras
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
Homotopy Commutative Algebra and 2-Nilpotent Lie Algebra
Springer Proceedings in Mathematics & Statistics, 2014
The homotopy transfer theorem due to Tornike Kadeishvili induces the structure of a homotopy commutative algebra, or C ∞-algebra, on the cohomology of the free 2-nilpotent Lie algebra. The latter C ∞-algebra is shown to be generated in degree one by the binary and the ternary operations.
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Building on the seminal works of Quillen [12] and Sullivan [16], Bousfield and Guggenheim [3] developed a "homotopy theory" for commutative differential graded algebras (cdgas) in order to study the rational homotopy theory of topological spaces. This "homotopy theory" is a certain categorical framework, invented by Quillen, that provides a useful model for the non-abelian analogs of the derived categories used in classical homological algebra. In this masters thesis, we use K. Brown’s generalization [5] of Quillen’s formalism to present a homotopy theory for the category of semi-free, finite-type cdgas over a field k of characteristic 0. In this homotopy theory, the "weak homotopy equivalences" are a refinement of those used by Bousfield and Guggenheim. As an application, we show that the category of finite-dimensional Lie algebras over k faithfully embeds into our homotopy category of cdgas via the Chevalley-Eilenberg construction. Moreover, we prove ...
European Journal of Pure and Applied Mathematics, 2021
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On Defining Relations of Certain Infinite-Dimensional Lie ALGEBRAS1
AMERICAN MATHEMATICAL SOCIETY, 1981
ABSTRACT. In this note we prove a conjecture stated in [2] about defining relations of the so-called Kac-Moody Lie algebras. In the finite-dimensional case this is Serre's theorem [5]. The basic idea is to map the ideal of relations into a Verma module and then to use the ( ...