Jean-Louis Loday

Van Kampen theorems for diagrams of spaces

where @ means “non-Abelian tensor product” of the two relative homotopy groups, each acting on the other via ;r,C. This new algebraic construction M @ N, which is defined for a pair of groups M, N each of which acts on the other, is studied in

Cohomologie et groupe de Steinberg relatifs

2. As is well-known, a general determination of a triad homotopy group has consequences for certain absolute homotopy groups. Some of these are given in

On the structure of cofree Hopf algebras

3. For example, we prove that for any group G

Order Structure on the Algebra of Permutations and of Planar Binary Trees

Pour tout anneau A les groupes de K-thCorie K&l), i > 1, d&finis par Quillen [9] sont les groupes d’homotopie de l’espace BGL(A)+. Pour tout homomorphisme d’anneaux f: A ---f A’ il est nature1 de dCfinir les groupes de K-thCorie relative l&(f) comme les groupes d’homotopie de la fibre homotopique de BGL(A)+ + BGL(A’)+. L ors q ue i = 1, 2 les groupes &&(A) admettent I’interprCtation algCbrique suivante. Soit vn: B(A) ---f GL(A) l’homomorphisme nature1 du groupe de Steinberg dans le groupe IinCaire; on a alors K,(A) = Ker vn et K,(A) = Coker vn . La connaissance d’une prCsentation par g&-&ateurs et relations de &(A) a permis de calculer explicitement le groupe K,(A) dans de nombreux cas [8]. Le but de cet article est de construire, pour tout homomorphisme surjectif d’anneaux f, un groupe de Steinberg relatif St(f) par gCnCrateurs et relations, ainsi qu’un homomorphisme yr : St(f) + GL(f) tels que K,(f) = Ker vf et &(f) = Coker F~. On obtient ainsi un moyen algCbrique pour calculer K,(f). E n utilisant la longue suite exacte de K-thCorie on peut en dCduire des renseignements sur le groupe I

The Diagonal of the Stasheff Polytope

d’un anneau. Par exemple, si A est un anneau noethCrien rCgulier et si on choisit pour f l’application A[t] --f A x A, P(t) F-+ (P(O), P(l)), on a alors K,(A) = K,(f). La mCthode utilisCe ici est calquCe sur celle de Kervaire [4]. La classification des extensions de groupes est remplacCe par la classification de certains modules croisCs. Un module croisC est la don&e d’un homomorphisme de groupes CL: M+ N et d’une action 9 de N sur M telle que

Parastatistics Algebra, Young Tableaux and the Super Plactic Monoid

Abstract We prove an analogue of the Poincaré-Birkhoff-Witt theorem and of the Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a nondifferential B ∞-algebra. We construct a universal enveloping functor U2 from nondifferential B ∞-algebras to 2-associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra ℋ is of the form U2(Prim ℋ). We take advantage of the description of the free 2as-algebra in terms of planar trees to unravel the operad associated to nondifferential B ∞-algebras.

Completing the Operadic Butterfly

Let Xn be either the symmetric group on n letters, the set of planar binary n-trees or the set of vertices of the (n − 1)-dimensional cube. In each case there exists a graded associative product on ⊕n≥0K[Xn]. We prove that it can be described explicitly by using the weak Bruhat order on Sn, the left-to-right order on planar trees, the lexicographic order in the cube case.

On Restricted Leibniz Algebras

We construct an A-infinity structure on the tensor product of two A-infinity algebras by using the simplicial decomposition of the Stasheff polytope. The key point is the construction of an operad AA-infinity based on the simplicial Stasheff polytope. The operad AA-infinity admits a coassociative diagonal and the operad A-infinity is a retract by deformation of it. We compare these constructions with analogous constructions due to Saneblidze–Umble and Markl–Shnider based on the Boardman–Vogt cubical decomposition of the Stasheff polytope.

Cyclic Homology and Lambda Operations

The parastatistics algebra is a superalgebra with (even) parafermi and (odd) parabose creation and annihilation operators. The states in the parastatistics Fock-like space are shown to be in one-to-one correspondence with the Super Semistandard Young Tableaux (SSYT) subject to further constraints. The deformation of the parastatistics algebra gives rise to a monoidal structure on the SSYT which is a super-counterpart of the plactic monoid.

The symmetric operation in a free pre-Lie algebra is magmatic

Abstract We complete a certain diagram (the operadic butterfly) of the categories of algebras involving Com, As, and Lie by constructing a type of algebras which have 4 generating operations and 16 relations. The associated operad is self-dual for Koszul duality.