scholarly journals Involutive and oriented dendriform algebras

2021 ◽  
Vol 581 ◽  
pp. 63-91
Author(s):  
Apurba Das ◽  
Ripan Saha
Keyword(s):  
Dendriform Algebras

scholarly journals 3-L-dendriform algebras and generalized derivations

Filomat ◽  
2021 ◽  
Vol 35 (6) ◽  
pp. 1949-1961
Author(s):  
Taoufik Chtioui ◽  
Sami Mabrouk
Keyword(s):  
Lie Algebras ◽  
Algebraic Structures ◽  
Generalized Derivations ◽  
Dendriform Algebras

The main goal of this paper is to introduce the notion of 3-L-dendriform algebras which are the dendriform version of 3-pre-Lie algebras. In fact they are the algebraic structures behind the O-operator of 3-pre-Lie algebras. They can be also regarded as the ternary analogous of L-dendriform algebras. Moreover, we study the generalized derivations of 3-L-dendriform algebras. Finally, we explore the spaces of quasi-derivations, the centroids and the quasi-centroids and give some properties.

scholarly journals Free dendriform algebras. Part I. A parenthesis setting

2006 ◽  
Vol 2006 ◽  
pp. 1-16
Author(s):  
Philippe Leroux
Keyword(s):  
Free Probability ◽  
Dendriform Algebra ◽  
Dendriform Algebras

We propose both a reformulation of some known results on the free dendriform algebra on one generator from a parenthesis setting instead of using permutations and some developments as well. Moreover, by introducing the concept of NCP-operad, we show how to use the free dendriform algebra on one generator to reformulate some results obtained by Speicher in free probability.

scholarly journals Nijenhuis operators on pre-Lie algebras

2019 ◽  
Vol 21 (07) ◽  
pp. 1850050 ◽  
Author(s):  
Qi Wang ◽  
Yunhe Sheng ◽  
Chengming Bai ◽  
Jiefeng Liu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Regular Representation ◽  
Complex Structure ◽  
Geometric Interpretation ◽  
Dual Representation ◽  
New Approach ◽  
Lie Bracket ◽  
Graded Lie Algebra ◽  
Dendriform Algebras

First we use a new approach to define a graded Lie algebra whose Maurer–Cartan elements characterize pre-Lie algebra structures. Then using this graded Lie bracket, we define the notion of a Nijenhuis operator on a pre-Lie algebra which generates a trivial deformation of this pre-Lie algebra. There are close relationships between [Formula: see text]-operators, Rota–Baxter operators and Nijenhuis operators on a pre-Lie algebra. In particular, a Nijenhuis operator “connects” two [Formula: see text]-operators on a pre-Lie algebra whose any linear combination is still an [Formula: see text]-operator in certain sense and hence compatible [Formula: see text]-dendriform algebras appear naturally as the induced algebraic structures. For the case of the dual representation of the regular representation of a pre-Lie algebra, there is a geometric interpretation by introducing the notion of a pseudo-Hessian–Nijenhuis structure which gives rise to a sequence of pseudo-Hessian and pseudo-Hessian–Nijenhuis structures. Another application of Nijenhuis operators on pre-Lie algebras in geometry is illustrated by introducing the notion of a para-complex structure on a pre-Lie algebra and then studying para-complex quadratic pre-Lie algebras and para-complex pseudo-Hessian pre-Lie algebras in detail. Finally, we give some examples of Nijenhuis operators on pre-Lie algebras.

scholarly journals Twisted dendriform algebras and the pre-Lie Magnus expansion

2011 ◽  
Vol 215 (11) ◽  
pp. 2615-2627 ◽  
Author(s):  
Kurusch Ebrahimi-Fard ◽  
Dominique Manchon
Keyword(s):  
Magnus Expansion ◽  
Dendriform Algebras

scholarly journals Combinatorial proofs of freeness of some P-algebras

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Vincent Vong
Keyword(s):  
Hilbert Series ◽  
Nous Permet ◽  
International Audience ◽  
Dendriform Algebras ◽  
Combinatorial Methods

International audience We present new combinatorial methods for solving algebraic problems such as computing the Hilbert series of a free $P$-algebra over one generator, or proving the freeness of a $P$-algebra. In particular, we apply these methods to the cases of dendriform algebras, quadrialgebras and tridendriform algebras, which leads us to prove a conjecture of Aguiar and Loday about the freeness of the quadrialgebra generated by the permutation 12. Nous présentons de nouvelles méthodes combinatoires permettant de résoudre des problèmes algébriques concernant les $P$-algèbres, comme déterminer la série de Hilbert de la $P$-algèbre libre sur un générateur, ou de prouver qu’une $P$-algèbre est libre. Nous les appliquons aux cas des algèbres dendriformes, des quadrialgèbres, et des algèbres tridendriformes. Cette approche nous permet en particulier de résoudre une conjecture de Aguiar et Loday à propos de la liberté de la quadrialgèbre engendrée par la permutation 12.

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