Tag - Operads

Maria Ofelia Ronco: Generalization of dendriform algebras

9th September, 2021

In a joint work with D. López N. and L.-F. Préville-Ratelle in 2015 we introduce a family of non-symmetric operads Dyckm, which satisfies that:

1. Dyck0 is the operad of associative algebras,

2. Dyck1 is the operad Dend of dendriform algebras, introduced by J.-L. Loday,

3. the vector space spanned by the set of m-Dyck paths has a natural structure of free Dyckm algebra over one element,

4. for any k ≥ 1, there exist degeneracy operators si : Dyckm → Dyckm-1 and face operators dj: Dyckm → Dyckm+1, which defines a simplicial complex in the category of non-symmetric operads.

The main examples of Dyckm algebra are the vector spaces spanned by the m-simplices of certain combinatorial Hopf algebras, like the Malvenuto-Reutenauer algebras and the algebra of packed words.

A well-known result on associative algebras states that, as an 𝒮-module, the operad of Ass of associative algebras is the composition Ass = Com ∘ Lie, where Com is the operad of commutative algebras and Lie is the operad of Lie algebras. The version of this result for dendriform algebras is that Dend = Ass ∘ Brace, where Brace is the operad of brace algebras.

Our goal is to introduce the notion of m-brace algebra, for m ≥ 2, and prove that there exists a Poincaré-Birkoff-Witt Theorem in this context, stating that Dyckm = Ass ∘ m-Brace.

Askar Dzhumadil’daev: Dimension formula for Koszul operads

26th August, 2021

We give recurrence formula for dimensions of Koszul operads. For example, dimensions of multi-linear parts of Lie-admissible operad satisfy the following recurrence relations dn=∑i=1n-1 μk Bn-1,k(d1…dn-1), where Bn,k are Bell polynomials and μk = k! ∑i=0k (k–i+1)i /i !. If p ≥ 5 is prime, then dp-1 ≡ 1 (mod p), dp ≡ -1 (mod p), dp+1 ≡ -1 (mod p), dp+2 ≡ -6 (mod p), dp+3 ≡ -56 (mod p), dp+4 ≡ -725 (mod p).

This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.

Pavel Kolesnikov: Derived algebras and their identities

9th July, 2020

In this talk we will consider a "differential counterpart" of the dendriform splitting procedure for operads. This problem has a very natural interpretation in the language of non-associative algebras. It is well-known that a (non-associative, in general) algebra equipped with a Rota-Baxter operator (a formalization of integration) gives rise to a system in a class of splitting algebras. The latter include dendriform (pre-associative), pre-Lie (left-symmetric), pre-Poisson, Zinbiel (pre-commutative) algebras, etc. What happens if we replace a Rota-Baxter operator with a derivation? The answer is well known for associative commutative algebras: the resulting class of systems obtained in this way coincides with the variety Nov of Novikov algebras. We will show in general that for an arbitrary binary operad Var the variety of derived Var-algebras coincides with the Manin white product of operads Var and Nov. If we allow the initial multiplication(s) to leave in the language of a derived algebra then the same sort of description can be obtained just by replacement of Nov with GD!, the Koszul dual to the operad of Gelfand-Dorfman algebras. We will also discuss similar statements for the "integral" case of Rota-Baxter operators.