Brace Algebras Research Articles

Enriched pre-Lie operads and freeness theorems

In this paper, we study the \mathcal{C} -enriched pre-Lie operad defined by Calaque and Willwacher for any Hopf cooperad \mathcal{C} to produce conceptual constructions of the operads acting on various deformation complexes. Maps between Hopf cooperads lead to maps between the corresponding enriched pre-Lie operads; we prove criteria for the module action of the domain on the codomain to be free, on the left and on the right. In particular, this implies a new functorial Poincaré–Birkhoff–Witt type theorem for universal enveloping brace algebras of pre-Lie algebras.

Gröbner–Shirshov bases for symmetric brace algebras

In this paper, we develop Gröbner–Shirshov basis theory for symmetric brace algebras. As applications, firstly, we give an alternative proof that the category of symmetric brace algebras is isomorphic to the category of pre-Lie algebras, secondly, we prove that each pre-Lie algebra can be embedded into a simple pre-Lie algebra, and finally, we present a new linear basis for the universal pre-Lie algebra of a Lie algebra.

Post-symmetric braces and integration of post-Lie algebras

The aim of this letter is twofold. Firstly, we introduce the post-Lie analogue of the notion of a symmetric brace algebra, termed in the sequel post-symmetric brace algebra. These brace algebras are defined using a suitable algebraic operad, which turns out to be isomorphic to the operad of the post-Lie algebras. Secondly, using these new brace algebras, together with the so called post-Lie Magnus expansion, we aim both to analyze the enveloping algebra of the corresponding post-Lie algebra and to compare the two Baker-Campbell-Hausdorff series there naturally defined.

The Freiheitssatz and automorphisms for free brace algebras

Over a field of characteristic zero, we prove that the Freiheitssatz holds for brace algebras, the word problem for the brace algebras with a single defining relation is decidable, two-generated subalgebras of free brace algebras are free, and that automorphisms of two-generated free brace algebras are tame.

Gröbner–Shirshov bases for brace algebras

ABSTRACTLet A be a brace algebra. This structure implies that A is also a pre-Lie algebra. In this paper, we establish Composition-Diamond lemma for brace algebras. For each pre-Lie algebra L, we find a Gröbner–Shirshov basis for its universal brace algebra Ub(L). As applications, we determine an explicit linear basis for Ub(L) and prove that L is a pre-Lie subalgebra of Ub(L).

Braces and Poisson additivity

We relate the brace construction introduced by Calaque and Willwacher to an additivity functor. That is, we construct a functor from brace algebras associated to an operad${\mathcal{O}}$to associative algebras in the category of homotopy${\mathcal{O}}$-algebras. As an example, we identify the category of$\mathbb{P}_{n+1}$-algebras with the category of associative algebras in$\mathbb{P}_{n}$-algebras. We also show that under this identification there is an equivalence of two definitions of derived coisotropic structures in the literature.

English

We give a definition of coisotropic morphisms of shifted Poisson (i.e. $P_n$) algebras which is a derived version of the classical notion of coisotropic submanifolds. Using this we prove that an intersection of coisotropic morphisms of shifted Poisson algebras carries a Poisson structure of shift one less. Using an interpretation of Hamiltonian spaces as coisotropic morphisms we show that the classical BRST complex computing derived Poisson reduction coincides with the complex computing coisotropic intersection. Moreover, this picture admits a quantum version using brace algebras and their modules: the quantum BRST complex is quasi-isomorphic to the complex computing tensor product of brace modules.

L-Infinity algebra morphisms and symmetric brace algebras

Morphisms of L-infinity algebras and symmetric brace algebras are studied. This is intended to be an expository note. Much of what is discussed here arose from joint work with M. Markl and from discussions with T. Kadeishvili. We wish to work primarily with symmetric brace algebras but will begin with brace algebras.

Tridendriform structure on combinatorial Hopf algebras

We extend the definition of tridendriform bialgebra by introducing a parameter 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 q-Gerstenhaber–Voronov algebras. We prove the equivalence between the categories of conilpotent q-tridendriform bialgebras and of q-Gerstenhaber–Voronov algebras. The space spanned by surjective maps between finite sets, as well as the space spanned by parking functions, have a natural structure of q-tridendriform bialgebra, 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 of M -permutations defined by T. Lam and P. Pylyavskyy comes from a q-tridendriform algebra which is a quotient of ST ( q ) .

Free Brace Algebras are Free Pre-Lie Algebras

Let 𝔤 be a free brace algebra. This structure implies that 𝔤 is also a pre-Lie algebra and a Lie algebra. It is already known that 𝔤 is a free Lie algebra. We prove here that 𝔤 is also a free pre-Lie algebra, using a description of 𝔤 with the help of planar rooted trees, a permutative product, and manipulations on the Poincaré–Hilbert series of 𝔤.

On the Lie enveloping algebra of a pre-Lie algebra

AbstractWe construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. It turns S(L) into a Hopf algebra which is isomorphic to the enveloping algebra of LLie. Then we prove that in the case of rooted trees our construction gives the Grossman-Larson Hopf algebra, which is known to be the dual of the Connes-Kreimer Hopf algebra. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, we give a similar interpretation of the Hopf algebra of planar rooted trees.

Dyer–Lashof–Cohen operations in Hochschild cohomology

We give explicit formulae for operations in Hochschild cohomology which are analogous to the operations in the homology of double loop spaces. As a corollary we obtain that any brace algebra in finite characteristic is always a restricted Lie algebra.

Symmetric Brace Algebras

We develop a symmetric analog of brace algebras and discuss the relation of such algebras to L∞-algebras. We give an alternate proof that the category of symmetric brace algebras is isomorphic to the category of pre-Lie algebras. As an application, symmetric braces are used to describe transfers of strongly homotopy structures. We then explain how these symmetric brace algebras may be used to examine the L∞-algebras that result from a particular gauge theory for massless particles of high spin.

Sur l'algèbre enveloppante d'une algèbre pré-Lie

We construct an associative product on the symmetric module S ( L ) of any pre-Lie algebra L. It turns S ( L ) into a Hopf algebra which is isomorphic to the envelopping algebra of L Lie . Then we prove that in the case of rooted trees our construction is dual to that of Connes and Kreimer. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, we give a similar interpretation of the Hopf algebra of planar rooted trees. To cite this article: J.-M. Oudom, D. Guin, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Eulerian idempotents and Milnor–Moore theorem for certain non-cocommutative Hopf algebras

The purpose of this paper is to prove a Milnor–Moore style theorem for a particular kind of non-cocommutative Hopf algebras: the dendriform algebras. A dendriform Hopf algebra is a Hopf algebra, such that the product ∗ is the sum of two operations ≺ and ≻, verifying certain conditions between them and with the coproduct Δ. The role of Lie algebras is played by brace algebras, which are defined by n-ary operations (one for each n⩾2) satisfying some relations. We show that a dendriform Hopf algebra is isomorphic to the enveloping algebra of its brace algebra of primitive elements. One of the ingredients of the proof is the construction of Eulerian idempotents in this context.

Un théorème de Cartier–Milnor–Moore–Quillen pour les bigèbres dendriformes et les algèbres braces

The classical theorem of Cartier–Milnor–Moore–Quillen gives an equivalence between the category of connected cocommutative bialgebras and the category of Lie algebras. We establish an analogous equivalence between the category of connected dendriform bialegebras and the category of brace algebras. It is given by the primitive elements functor and the “enveloping dendriform algebra” of a brace algebra.

A Milnor–Moore theorem for dendriform Hopf algebras

A dendriform Hopf algebra is a Hopf algebra, such that the product ∗ is the sum of two operations ≺ and ≻, verifying certain conditions between them and with the coproduct Δ. The purpose of this Note is to announce a Milnor–Moore style theorem for these algebras. The role of Lie algebras is played by brace algebras, which are defined by n-ary operations (one for each n≥2) satisfying some relations. We show that a dendriform Hopf algebra is isomorphic to the envelopping algebra of its brace algebra of primitive elements. One of the ingredients of the proof is the construction of Eulerian idempotents in this context.