A simple symmetry generating operads related to rooted planar -ary trees and polygonal numbers.
An introduction to some novel applications of Lie algebra cohomology in mathematics and physics.
En esta nota se presenta en primer lugar una introducción autocontenida a la cohomología de álgebras de Lie, y en segundo lugar algunas de sus aplicaciones recientes en matemáticas y física.
Classification of all associative mono--ary algebras with 2 elements.
Cohomology ring of n-Lie algebras.
Natural graded Lie brackets on the space of cochains of n-Leibniz and n-Lie algebras are introduced. It turns out that these brackets agree under the natural embedding introduced by Gautheron. Moreover, n-Leibniz and n-Lie algebras turn to be canonical structures for these brackets in a similar way in which associative algebras (respectively, Lie algebras) are canonical structures for the Gerstenhaber bracket (respectively, Nijenhuis-Richardson bracket).
Commentary to: Generalized derivations of Lie triple systems
Commutator algebras arising from splicing operations
We prove that the variety of Lie algebras arising from splicing operation coincides with the variety CM of centreby-metabelian Lie algebras. Using these Lie algebras we find the minimal dimension algebras generated the variety CM and the variety of its associative envelope algebras. We study the splicing n-ary operation. We show that all n-ary (n > 2) commutator algebras arising from this operation are nilpotent of index 3. We investigate the generalization of the splicing n-ary operation, and...
Edomorphisms and Null subalgeras of finite dimensional algebras.
Free associative algebras, noncommutative Gröbner bases, and universal associative envelopes for nonassociative structures
First, we provide an introduction to the theory and algorithms for noncommutative Gröbner bases for ideals in free associative algebras. Second, we explain how to construct universal associative envelopes for nonassociative structures defined by multilinear operations. Third, we extend the work of Elgendy (2012) for nonassociative structures on the 2-dimensional simple associative triple system to the 4- and 6-dimensional systems.
Irreducible identities of -algebras.
Localization in semicommutative (m,n)-rings
We give a construction for (m,n)-rings of quotients of a semicommutative (m,n)-ring, which generalizes the ones given by Crombez and Timm and by Paunić for the commutative case. We also study various constructions involving reduced rings and rings of quotients and give some functorial interpretations.
-Lie structures that are generated by Wronskians.
-арные алгебры Мальцева
n-лиевы алгебры.
Operads for -ary algebras – calculations and conjectures
In [8] we studied Koszulity of a family of operads depending on a natural number and on the degree of the generating operation. While we proved that, for , the operad is Koszul if and only if is even, and while it follows from [4] that is Koszul for even and arbitrary , the (non)Koszulity of for odd and remains an open problem. In this note we describe some related numerical experiments, and formulate a conjecture suggested by the results of these computations.
Operads of decorated trees and their duals
This is an extended version of a talk presented by the second author on the Third Mile High Conference on Nonassociative Mathematics (August 2013, Denver, CO). The purpose of this paper is twofold. First, we would like to review the technique developed in a series of papers for various classes of di-algebras and show how the same ideas work for tri-algebras. Second, we present a general approach to the definition of pre- and post-algebras which turns out to be equivalent to the construction of dendriform...
Quasi-trace functions on Lie algebras and their applications to 3-Lie algebras
We introduce the notion of quasi-trace functions on Lie algebras. As applications we study realizations of 3-dimensional and 4-dimensional 3-Lie algebras. Some comparison results on cohomologies of 3-Lie algebras and Leibniz algebras arising from quasi-trace functions are obtained.
Restricted and quasi-toral restricted Lie-Rinehart algebras
In this paper, we introduce the definition of restrictable Lie-Rinehart algebras, the concept of restrictability is by far more tractable than that of a restricted Lie-Rinehart algebra. Moreover, we obtain some properties of p-mappings and restrictable Lie-Rinehart algebras. Finally, we give some sufficient conditions for the commutativity of quasi-toral restricted Lie-Rinehart algebras and study how a quasi-toral restricted Lie-Rinehart algebra with zero center and of minimal dimension should be....
Root decomposition of the ternary Mal'tsev algebra .
Some necessary and sufficient conditions for nilpotent -Lie superalgebras
The paper studies nilpotent -Lie superalgebras over a field of characteristic zero. More specifically speaking, we prove Engel’s theorem for -Lie superalgebras which is a generalization of those for -Lie algebras and Lie superalgebras. In addition, as an application of Engel’s theorem, we give some properties of nilpotent -Lie superalgebras and obtain several sufficient conditions for an -Lie superalgebra to be nilpotent by using the notions of the maximal subalgebra, the weak ideal and the...
The -Lie property of the Jacobian as a condition for complete integrability.