Young Tableaux and Homotopy Commutative Algebras (original) (raw)

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.

A∞-Algebras and the Cyclic Bar Complex

Illinois J. Math, 1990

This paper arose from our use of Chen's theory of iterated integrals as a tool in the study of the complex of S 1-equivariant differential forms on the free loop space LX of a manifold X (see [2]). In trying to understand the behaviour of the iterated integral map with respect to products, we were led to a natural product on the space of S 1-equivariant differential forms Ω(Y)[u] of a manifold Y with circle action, where u is a variable of degree 2. This product is not associative but is homotopy associative in a precise way; indeed there is whole infinite family of "higher homotopies". It turns out that this product structure is an example of Stasheff's A ∞-algebras, which are a generalization of differential graded algebras (dgas). Using the iterated integral map, it is a straightforward matter to translate this product structure on the space of S 1-equivariant differential forms on LX into formulas on the cyclic bar complex of Ω(X). Our main goal in this paper is to show that in general, the cyclic bar complex of a commutative dga A has a natural A ∞-structure and we give explicit formulas for this structure. In particular, this shows that the cyclic homology of A has a natural associative product, but it is a much stronger result, since it holds at the chain level. Thus, it considerably strengthens the results of Hood and Jones [3]. We also show how to construct the cyclic bar complex of an A ∞-algebra, and in particular define its cyclic homology. As hinted at in [2], this construction may have applications to the problem of giving models for the S 1 × S 1-equivariant cohomology of double loop spaces LL(X) of a manifold and, since the space of equivariant differential forms on a smooth S 1-manifold Y is an A ∞-algebra, to the problem of finding models for the space of S 1 × S 1-equivariant differential forms on LY. Although the methods that we use were developed independently, they bear a strong resemblance with those of Quillen [6]. Finally, we discuss in our general context the Chen normalization of the cyclic bar complex of an A ∞-algebra. This is a quotient of the cyclic bar complex by a complex of degenerate chains which is acyclic if A is connected, and which was shown by Chen to coincide with the kernel of the iterated integral map in the case A = Ω(X). This normalization is an important tool, since it allows us to remove a large contractible sub-complex of the cyclic bar complex. The first two sections of this paper are devoted to generalities concerning coalgebras and A ∞algebras; a good reference for further background on coalgebras is the book of McCleary [5]. The cyclic bar complex of an A ∞-algebra is constructed in Section 3, the A ∞-structure on the cyclic bar complex of a commutative dga in Section 4, and we discuss Chen normalization in Section 5. All our algebra will be carried out over a fixed coefficient ring K; in fact nothing will be lost by thinking of the case where K is the integers Z. In particular, all tensor products are taken over K unless explicitly stated otherwise. We will make use of the sign-convention in the category of Z 2-graded K-modules, which may be phrased as follows: the canonical map S 21 from V 1 ⊗ V 2 to V 2 ⊗ V 1 is defined by S 21 (v 1 ⊗ v 2) = (−1) |v1||v2| v 2 ⊗ v 1. *In the preprint of [2], the maps m andm are exchanged, for which we beg the reader's forgiveness.

Commutative combinatorial Hopf algebras

Journal of Algebraic Combinatorics, 2008

We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general construction based on graphs, and its non-commutative dual is realized in three different ways, in particular as the Grossman-Larson algebra of heap ordered trees. Extensions to endofunctions, parking functions, set compositions, set partitions, planar binary trees and rooted forests are discussed. Finally, we introduce one-parameter families interpolating between different structures constructed on the same combinatorial objects.

Infinitesimal and B∞-algebras, finite spaces, and quasi-symmetric functions

Journal of Pure and Applied Algebra, 2015

Finite topological spaces are in bijective correspondence with preorders on finite sets. We undertake their study using combinatorial tools that have been developed to investigate general discrete structures. A particular emphasis will be put on recent topological and combinatorial Hopf algebra techniques. We will show that the linear span of finite spaces carries generalized Hopf algebraic structures that are closely connected with familiar constructions and structures in topology (such as the one of cogroups in the category of associative algebras that has appeared e.g. in the study of loop spaces of suspensions). The most striking results that we obtain are certainly that the linear span of finite spaces carries the structure of the enveloping algebra of a B∞-algebra, and that there are natural (Hopf algebraic) morphisms between finite spaces and quasi-symmetric functions. In the process, we introduce the notion of Schur-Weyl categories in order to describe rigidity theorems for cogroups in the category of associative algebras and related structures, as well as to account for the existence of natural operations (graded permutations) on them.

4 on the Homology and Homotopy of Commutative Shuffle Algebras

2016

For commutative algebras there are three important homology theories, Harrison homology, André-Quillen homology and Gamma-homology. In general these differ, unless one works with respect to a ground field of characteristic zero. We show that the analogues of these homology theories agree in the category of pointed commutative monoids in symmetric sequences and that Hochschild homology always possesses a Hodge decomposition in this setting. In addition we prove that the category of pointed differential graded commutative monoids in symmetric sequences has a model structure and that it is Quillen equivalent to the model category of pointed simplicial commutative monoids in symmetric sequences.

On the homotopy Lie algebra of an arrangement

Michigan Mathematical Journal, 2005

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.

Infinitesimal and BinftyB\_\inftyBinfty-algebras, finite spaces, and quasi-symmetric functions

arXiv: Algebraic Topology, 2014

Finite topological spaces are in bijective correspondence with preorders on finite sets. We undertake their study using combinatorial tools that have been developed to investigate general discrete structures. A particular emphasis will be put on recent topological and combinatorial Hopf algebra techniques. We will show that the linear span of finite spaces carries generalized Hopf algebraic structures that are closely connected with familiar constructions and structures in topology (such as the one of cogroups in the category of associative algebras that has appeared e.g. in the study of loop spaces of suspensions). The most striking results that we obtain are certainly that the linear span of finite spaces carries the structure of the enveloping algebra of a BinftyB\_\inftyBinfty--algebra, and that there are natural (Hopf algebraic) morphisms between finite spaces and quasi-symmetric functions. In the process, we introduce the notion of Schur-Weyl categories in order to describe rigidity the...

A Cartan–Eilenberg approach to homotopical algebra

Journal of Pure and Applied Algebra, 2010

In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence between its localization with respect to weak equivalences and the localised category of cofibrant objets with respect to strong equivalences. This equivalence allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and categories of functors defined on a category endowed with a cotriple (comonad) and taking values on a category of complexes of an abelian category. In the latter case there are examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications of our theory, we establish a very general acyclic models theorem.