Quadri-algebras

In this paper, we study Rota-Baxter operators and super O-operator of associative superalgebras, Lie superalgebras, pre-Lie superalgebras and L-dendriform superalgebras. Then we give some properties of pre-Lie superalgebras constructed from associative superalgebras, Lie superalgebras and L-dendriform superalgebras. Moreover, we provide all Rota-Baxter operators of weight zero on complex pre-Lie superalgebras of dimensions 2 and 3.

show abstract

“…The pre-Lie superalgebra (C 2 h ,4 , •): 1 • e 2 = e 3 , e 2 • e 1 = e 2 + e 3 , e 3 • e 1 = he 3 , h = ±1.…”

Section: Caseunclassified

“…Recently, classifications of complex pre-Lie superalgebras in dimensions two and three were given by Zhang and Bai [15]. See [3,21,38,39,55] about further results.…”

Section: Introductionmentioning

confidence: 99%

Rota–Baxter Operators on Pre-Lie Superalgebras

Abdaoui

Mabrouk

Makhlouf

2017

Bull. Malays. Math. Sci. Soc.

View full text Add to dashboard Cite

show abstract

“…The operad Quad and its Koszul dual. In this section, we study the example of black square-product Quad = Dend Dend introduced by M. Aguiar and J.-L. Loday in [AL04]. We prove that the Koszul dual of Quad is the operad Quad !…”

Section: Corollary 38 ([Efg05]mentioning

confidence: 99%

“…Koszulity of Quad and other operads defined by square products. Aguiar and Loday made in [AL04] the conjecture that the operad Quad is Koszul. We show this statement using poset's method of [Val2].…”

Section: Corollary 48 the Dimensions Of The Components Of The Koszulmentioning

confidence: 99%

Manin products, Koszul duality, Loday algebras and Deligne conjecture

Vallette¹

2008

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

134

View full text Add to dashboard Cite

Abstract. In this article we give a conceptual definition of Manin products in any category endowed with two coherent monoidal products. This construction can be applied to associative algebras, non-symmetric operads, operads, colored operads, and properads presented by generators and relations. These two products, called black and white, are dual to each other under Koszul duality functor. We study their properties and compute several examples of black and white products for operads. These products allow us to define natural operations on the chain complex defining cohomology theories. With these operations, we are able to prove that Deligne's conjecture holds for a general class of operads and is not specific to the case of associative algebras. Finally, we prove generalized versions of a few conjectures raised by M. Aguiar and J.-L. Loday related to the Koszul property of operads defined by black products. These operads provide infinitely many examples for this generalized Deligne's conjecture.

show abstract

“…This identity is called the Rota-Baxter relation and will be introduced in the next section. See [6,7] for reviews and [8,9,10,11,12] for recent developments relating this identity to other fields in mathematics.…”

Section: Introductionmentioning

confidence: 99%

Integrable renormalization I: The ladder case

Ebrahimi‐Fard

Guo

Kreimer

2004

Journal of Mathematical Physics

View full text Add to dashboard Cite

In recent years a Hopf algebraic structure underlying the process of renormalization in quantum field theory was found. It led to a Birkhoff factorization for (regularized) Hopf algebra characters, i.e. for Feynman rules. In this work we would like to show that this Birkhoff factorization finds its natural formulation in terms of a classical r-matrix, coming from a Rota-Baxter structure underlying the target space of the regularized Hopf algebra characters. Working in the rooted tree Hopf algebra, the simple case of the Hopf subalgebra of ladder trees is treated in detail. The extension to the general case, i.e. the full Hopf algebra of rooted trees or Feynman graphs is briefly outlined. * kurusch@ihes.fr † liguo@newark.rutgers.edu ‡ kreimer@ihes.fr and dkreimer@bu.edu. Center for Math. Phys., Boston University.

show abstract

Quadri-algebras

Cited by 71 publications

References 19 publications

Rota–Baxter Operators on Pre-Lie Superalgebras

Rota–Baxter Operators on Pre-Lie Superalgebras

Manin products, Koszul duality, Loday algebras and Deligne conjecture

Integrable renormalization I: The ladder case

Contact Info

Product

Resources

About