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.
Parafermions and homotopy algebras
2013
We explore the Fock spaces of the parafermionic algebra closed by the creation and annihilation operators introduced by H.S. Green. Each parafermionic Fock space allows for a free minimal resolution by graded modules of the graded 2-step nilpotent subalgebra of the parafermionic creation operators. Such a free resolution is constructed with the help of a classical Kostant’s theorem computing suitable Lie algebra cohomologies (of the creation nilpotent subalgebra with values in the parafermionic Fock space). We give a cohomological interpretation of the Schur functions identities which have been recently discovered by Stoilova and Van der Jeugt. The endomorphisms of the parafermionic minimal free resolution close a differential graded algebra which is naturally endowed with the structure of a Stasheff homotopy algebra. 1. Parastatistics Algebras H.S. Green introduced a scheme of quantization based on algebras of non-canonical commutation relations between the creation and annihilatio...
Combinatorial Hopf algebras from PROs
Journal of Algebraic Combinatorics, 2016
We introduce a general construction that takes as input a so-called stiff PRO and that outputs a Hopf algebra. Stiff PROs are particular PROs that can be described by generators and relations with precise conditions. Our construction generalizes the classical construction from operads to Hopf algebras of van der Laan. We study some of its properties and review some examples of application. We get in particular Hopf algebras on heaps of pieces and retrieve some deformed versions of the noncommutative Faà di Bruno algebra introduced by Foissy. Contents 1.2. The natural Hopf algebra of an operad 1.3. PROs and free PROs 2. From PROs to combinatorial Hopf algebras 2.1. The Hopf algebra of a free PRO 2.2. Properties of the construction 2.3. The Hopf algebra of a stiff PRO 2.4. Related constructions 3. Examples of application of the construction 3.1. Hopf algebras of forests 3.2. The Faà di Bruno algebra and its deformations 3.3. Hopf algebra of forests of bitrees 3.4. Hopf algebra of heaps of pieces 3.5. Hopf algebra of heaps of friable pieces Concluding remarks and perspectives References
Tridendriform Algebras and Combinatorial Hopf Algebras
2009
We extend the definition of tridendriform bialgebra by introducing a weight q. The subspace of primitive elements of a q-tridendriform bialgebra is equipped with an associative product and a natural structure of brace algebra, related by a distributive law. This data is called qGerstenhaber-Voronov algebras. We prove the equivalence between the categories of connected q-tridendriform bialgebras and of q-GerstenhaberVoronov algebras. The space spanned by surjective maps, as well as the space spanned by parking functions, have natural structures of qtridendriform bialgebras, denoted ST(q) and PQSym(q)∗, in such a way that ST(q) is a sub-tridendriform bialgebra of PQSym(q)∗. Finally we show that the bialgebra ofM-permutations defined by T. Lam and P. Pylyavskyy may be endowed with a natural structure of q-tridendriform algebra which is a quotient of ST(q).
C_∞-structure on the cohomology of free 2-nilpotent Lie algebra
2019
We consider the free 2-step nilpotent graded Lie algebra and its cohomology ring. The homotopy transfer induces a homotopy commutative algebra on its cohomology ring which we describe. 1. Homotopy algebras The homotopy associative algebras, or A∞-algebras were introduced by Jim Stasheff in the 1960’s as a tool in algebraic topology for studying ‘group-like’ spaces. Homotpy algebras received a new attention and further development in the 1990’s after the discovery of their relevence into a multitude of topics in algebraic geometry, symplectic and contact geometry, knot theory, moduli spaces, deformation theory... Definition 1.1. (A∞-algebra) A homotopy associative algebra, orA∞-algebra, over K is a Z-graded vector space A = ⊕ i∈Z A i endowed with a family of graded mappings (operations) mn : A ⊗n → A, deg(mn) = 2− n n > 1 satisfying the Stasheff identities SI(n) for n > 1 SI(n) : ∑ r+s+t=n (−1)mr+1+t(Id ⊗r ⊗ms ⊗ Id ) = 0 r > 0, t > 0, s > 1 where the sum runs over all ...
On realizing diagrams of Π–algebras
Algebraic & Geometric Topology, 2006
Given a diagram of …-algebras (graded groups equipped with an action of the primary homotopy operations), we ask whether it can be realized as the homotopy groups of a diagram of spaces. The answer given here is in the form of an obstruction theory, of somewhat wider application, formulated in terms of generalized …-algebras. This extends a program begun by Dwyer, Kan, Stover, Blanc and Goerss [21; 10] to study the realization of a single …-algebra. In particular, we explicitly analyze the simple case of a single map, and provide a detailed example, illustrating the connections to higher homotopy operations.
Hopf Structures on Standard Young Tableaux
2010
We review the Poirier-Reutenauer Hopf structure on Standard Young Tableaux and show that it is a distinguished member of a family of Hopf structures. The family in question is related to deformed parastatistics. In this paper K is a field of characteristic zero and all vector spaces are over K. A K[S]-module is a collection of K[S r ]-modules of the symmetric groups S r. A H(q)-module is a collection of H r (q)-modules of the Hecke algebras H r (q).
2003
In this paper we generalize the plus-construction given by M. Livernet for algebras over rational differential graded Koszul operads to the framework of admissible operads (the category of algebras over such operads admits a closed model category structure). We follow the modern approach of J. Berrick and C. Casacuberta defining topological plus-construction as a nullification with respect to a universal acyclic space. Similarly, we construct a universal H ∗ -acyclic algebra U and we define A −→ A+ as the U -nullification of the algebra A. This map induces an isomorphism on Quillen homology and quotients out the maximal perfect ideal of π0(A). As an application, we consider for any associative algebra R the plusconstructions of gl(R) in the categories of Lie and Leibniz algebras up to homotopy. This gives rise to two new homology theories for associative algebras, namely cyclic and Hochschild homologies up to homotopy. In particular, these theories coincide with the classical cyclic...
Trialgebras and families of polytopes
Contemporary mathematics, 2004
We show that the family of standard simplices and the family of Stasheff polytopes are dual to each other in the following sense. The chain modules of the standard simplices, resp. the Stasheff polytopes, assemble to give an operad. We show that these operads are dual of each other in the operadic sense. The main result of this paper is to show that they are both Koszul operads. As a consequence the generating series of the standard simplices and the generating series of the Stasheff polytopes are inverse to each other. The two operads give rise to new types of algebras with 3 generating operations, 11 relations, respectively 7 relations, that we call associative trialgebras and dendriform trialgebras respectively. The free dendriform trialgebra, which is based on planar trees, has an interesting Hopf algebra structure, which will be dealt with in another paper. Similarly the family of cubes gives rise to an operad which happens to be self-dual for Koszul duality. Introduction. We introduce a new type of associative algebras characterized by the fact that the associative product * is the sum of three binary operations : x * y := x ≺ y + x ≻ y + x • y ,
Associahedra, cellular W-construction and products of AinftyA_\inftyAinfty-algebras
2003
Our aim is to construct a functorial tensor product of AinftyA_\inftyAinfty-algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff associahedron. These construction were in fact already indicated by R. Umble and S. Saneblidze in [9]; we will try to give a more satisfactory presentation. We also prove that there does not exist an associative tensor product of AinftyA_\inftyAinfty-algebras.