Encyclopedia of types of algebras 2010

  title={Encyclopedia of types of algebras 2010},
  author={Guillaume W. Zinbiel},
  journal={arXiv: Rings and Algebras},
This is a cornucopia of types of algebras with some of their properties from the operadic point of view. 
Some problems in operad theory
This is a list of some problems and conjectures related to various types of algebras, that is to algebraic operads. Some comments and hints are included.
Dual Alternative Algebras in Characteristic Three
We prove that the dimension of the arity n component of the operad of dual alternative algebras over a field of characteristic three is equal to 2 n − n, and describe the structure of the
On some basic properties of Leibniz algebras
This paper gives an overview of some basic properties of Leibniz algebras. Some of the results were known earlier, but in the article they are accompanied by new simple proofs. Some of the results
A trick to compute certain Manin products of operads
We describe simple tricks to compute the Manin black products with the operads $\mathcal{A}ss$, $\mathcal{C}om$ and $pre\mathcal{L}ie$.
Five interpretations of Fa\`a di Bruno's formula
In these lectures we present five interpretations of the Fa' di Bruno formula which computes the n-th derivative of the composition of two functions of one variable: in terms of groups, Lie algebras
Universal enveloping commutative Rota-Baxter algebras of precommutative and postcommutative algebras
Universal enveloping commutative Rota—Baxter algebras of preand postcommutative algebras are constructed. The pair of varieties (RBλCom,postCom) is proved to be a PBW-pair and the pair (RBCom,preCom)
Derived bracket construction up to homotopy and Schroder numbers
We will introduce the notion of higher derived bracket construction in the category of operads and prove that the higher derived bracket construction of Lie operad is equivalent to the cobar
A Non-crossing Word Cooperad for Free Homotopy Probability Theory
We construct a cooperad which extends the framework of homotopy probability theory to free probability theory. The cooperad constructed, which seems related to the sequence and cactus operads, may be
On liezation of the Leibniz algebras and its applications
We consider some the fundamental properties of the Leibniz algebras. Some results were known before, but in the paper they are proved by a single method of liezation—the transition to a Lie algebra,
Universal Deformation Formulas
We give a conceptual explanation of universal deformation formulas for unital associative algebras and prove some results on the structure of their moduli spaces. We then generalize universal


Completing the Operadic Butterfly
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
Dialgebras and Related Operads
Dialgebras.- Dialgebra (co)homology with coefficients.- Un endofoncteur de la categorie des operades.- Un theoreme de Milnor-Moore pour les algebres de Leibniz.
A Koszul duality for props
The notion of prop models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the
Compatible associative products and trees
We compute dimensions of graded components for free algebras with two compatible associative products, and give a combinatorial interpretation of these algebras in terms of planar rooted trees.
Jordan triple systems by the grid approach
Special families of compatible tripotents are identified and classification of grids is explained.
QED Hopf algebras on planar binary trees
Abstract In this paper we describe the Hopf algebras on planar binary trees used to renormalize the Feynman propagators of quantum electrodynamics, and the coaction which describes the
Moufang loops and alternative algebras
Let O be the algebra O of classical real octonions or the (split) algebra of octonions over the finite field GF(p 2 ), p > 2. Then the quotient loop O*/Z* of the Moufang loop O* of all invertible
Operads and Moduli Spaces of Genus 0 Riemann Surfaces
In this paper, we study two dg (differential graded) operads related to the homology of moduli spaces of pointed algebraic curves of genus 0. These two operads are dual to each other, in the sense of
Homology of generalized partition posets
Abstract We define a family of posets of partitions associated to an operad. We prove that the operad is Koszul if and only if the posets are Cohen–Macaulay. On the one hand, this characterization
The General Theory
The concept of invariance with respect to transformations of a group is one of the most important and successful ideas of nineteenth century mathematics. After the use of coordinates had dominated