18 research outputs found
Ternary algebras and groups
We construct explicitly groups associated to specific ternary algebras which
extend the Lie (super)algebras (called Lie algebras of order three). It turns
out that the natural variables which appear in this construction are variables
which generate the three-exterior algebra. An explicit matrix representation of
a group associated to a peculiar Lie algebra of order three is constructed
considering matrices with entry which belong to the three exterior algebra.Comment: 11 pages contribution to the 5th International Symposium on Quantum
Theory and Symmetries (QTS5
Generalization of n-ary Nambu algebras and beyond
The aim of this paper is to introduce -ary Hom-algebra structures
generalizing the -ary algebras of Lie type enclosing -ary Nambu algebras,
-ary Nambu-Lie algebras, -ary Lie algebras, and -ary algebras of
associative type enclosing -ary totally associative and -ary partially
associative algebras. Also, we provide a way to construct examples starting
from an -ary algebra and an -ary algebras endomorphism. Several examples
could be derived using this process
Cohomology of Filippov algebras and an analogue of Whitehead's lemma
We show that two cohomological properties of semisimple Lie algebras also
hold for Filippov (n-Lie) algebras, namely, that semisimple n-Lie algebras do
not admit non-trivial central extensions and that they are rigid i.e., cannot
be deformed in Gerstenhaber sense. This result is the analogue of Whitehead's
Lemma for Filippov algebras. A few comments about the n-Leibniz algebras case
are made at the end.Comment: plain latex, no figures, 29 page
k-Leibniz algebras from lower order ones: from Lie triple to Lie l-ple systems
Two types of higher order Lie -ple systems are introduced in this
paper. They are defined by brackets with arguments satisfying
certain conditions, and generalize the well known Lie triple systems. One of
the generalizations uses a construction that allows us to associate a
-Leibniz algebra \fL with a metric -Leibniz algebra \tilde{\fL}
by using a -linear Kasymov trace form for \tilde{\fL}. Some specific
types of -Leibniz algebras, relevant in the construction, are introduced as
well. Both higher order Lie -ple generalizations reduce to the standard
Lie triple systems for .Comment: 22 pages, no figure
Manin products, Koszul duality, Loday algebras and Deligne conjecture
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.Comment: Final version, a few references adde
Topics on n-ary algebras
We describe the basic properties of two n-ary algebras, the Generalized Lie
Algebras (GLAs) and, particularly, the Filippov (or n-Lie) algebras (FAs), and
comment on their n-ary Poisson counterparts, the Generalized Poisson (GP) and
Nambu-Poisson (N-P) structures. We describe the Filippov algebra cohomology
relevant for the central extensions and infinitesimal deformations of FAs. It
is seen that semisimple FAs do not admit central extensions and, moreover, that
they are rigid. This extends the familiar Whitehead's lemma to all
FAs, n=2 being the standard Lie algebra case. When the n-bracket of the FAs is
no longer required to be fully skewsymmetric one is led to the n-Leibniz (or
Loday's) algebra structure. Using that FAs are a particular case of n-Leibniz
algebras, those with an anticommutative n-bracket, we study the class of
n-Leibniz deformations of simple FAs that retain the skewsymmetry for the first
n-1 entires of the n-Leibniz bracket.Comment: 11 page
On Deformations of n-Lie algebras
The aim of this paper is to review the deformation theory of -Lie
algebras. We summarize the 1-parameter formal deformation theory and provide a
generalized approach using any unital commutative associative algebra as a
deformation base. Moreover, we discuss degenerations and quantization of
-Lie algebras.Comment: Proceeding of the conference Dakar's Workshop in honor of Pr Amin
Kaidi. arXiv admin note: text overlap with arXiv:hep-th/9602016 by other
author
On the generalizations of Poisson structures
The characterization of the Nambu-Poisson n-tensors as a subfamily of the
Generalized-Poisson ones recently introduced (and here extended to the odd
order case) is discussed. The homology and cohomology complexes of both
structures are compared, and some physical considerations are made.Comment: Latex file. 12 pages. Trivial changes, a misprint (subindices)
corrected. To appear in J. Phys. A (letters