16 research outputs found
Irreducible Lie-Yamaguti algebras
Lie-Yamaguti algebras (or generalized Lie triple systems) are binary-ternary
algebras intimately related to reductive homogeneous spaces. The Lie-Yamaguti
algebras which are irreducible as modules over their Lie inner derivation
algebra are the algebraic counterpart of the isotropy irreducible homogeneous
spaces. These systems will be shown to split into three disjoint types: adjoint
type, non-simple type and generic type. The systems of the first two types will
be classified and most of them will be shown to be related to a Generalized
Tits Construction of Lie algebras.Comment: 25 page
On the Lie-algebraic origin of metric 3-algebras
Since the pioneering work of Bagger-Lambert and Gustavsson, there has been a
proliferation of three-dimensional superconformal Chern-Simons theories whose
main ingredient is a metric 3-algebra. On the other hand, many of these
theories have been shown to allow for a reformulation in terms of standard
gauge theory coupled to matter, where the 3-algebra does not appear explicitly.
In this paper we reconcile these two sets of results by pointing out the
Lie-algebraic origin of some metric 3-algebras, including those which have
already appeared in three-dimensional superconformal Chern-Simons theories.
More precisely, we show that the real 3-algebras of Cherkis-Saemann, which
include the metric Lie 3-algebras as a special case, and the hermitian
3-algebras of Bagger-Lambert can be constructed from pairs consisting of a
metric real Lie algebra and a faithful (real or complex, respectively) unitary
representation. This construction generalises and we will see how to construct
many kinds of metric 3-algebras from pairs consisting of a real metric Lie
algebra and a faithful (real, complex or quaternionic) unitary representation.
In the real case, these 3-algebras are precisely the Cherkis-Saemann algebras,
which are then completely characterised in terms of this data. In the complex
and quaternionic cases, they constitute generalisations of the Bagger-Lambert
hermitian 3-algebras and anti-Lie triple systems, respectively, which underlie
N=6 and N=5 superconformal Chern-Simons theories, respectively. In the process
we rederive the relation between certain types of complex 3-algebras and metric
Lie superalgebras.Comment: 29 pages (v4: really final version to appear in CMP. Example 7 has
been improved.
Cohomology and relative Rota-Baxter-Nijenhuis structures on LieYRep pairs
A LieYRep pair consists of a Lie-Yamaguti algebra and its representation. In
this paper, we establish the cohomology theory of LieYRep pairs and
characterize their linear deformations by the second cohomology group. Then we
introduce the notion of relative Rota-Baxter-Nijenhuis structures on LieYRep
pairs, investigate their properties, and prove that a relative
Rota-Baxter-Nijenhuis structure gives rise to a pair of compatible relative
Rota-Baxter operators under a certain condition. Finally, we show the
equivalence between -matrix-Nijenhuis structures and Rota-Baxter-Nijenhuis
structures on Lie-Yamaguti algebras
n-ary algebras: a review with applications
This paper reviews the properties and applications of certain n-ary
generalizations of Lie algebras in a self-contained and unified way. These
generalizations are algebraic structures in which the two entries Lie bracket
has been replaced by a bracket with n entries. Each type of n-ary bracket
satisfies a specific characteristic identity which plays the r\^ole of the
Jacobi identity for Lie algebras. Particular attention will be paid to
generalized Lie algebras, which are defined by even multibrackets obtained by
antisymmetrizing the associative products of its n components and that satisfy
the generalized Jacobi identity (GJI), and to Filippov (or n-Lie) algebras,
which are defined by fully antisymmetric n-brackets that satisfy the Filippov
identity (FI). Three-Lie algebras have surfaced recently in multi-brane theory
in the context of the Bagger-Lambert-Gustavsson model. Because of this,
Filippov algebras will be discussed at length, including the cohomology
complexes that govern their central extensions and their deformations
(Whitehead's lemma extends to all semisimple n-Lie algebras). When the
skewsymmetry of the n-Lie algebra is relaxed, one is led the n-Leibniz
algebras. These will be discussed as well, since they underlie the
cohomological properties of n-Lie algebras.
The standard Poisson structure may also be extended to the n-ary case. We
shall review here the even generalized Poisson structures, whose GJI reproduces
the pattern of the generalized Lie algebras, and the Nambu-Poisson structures,
which satisfy the FI and determine Filippov algebras. Finally, the recent work
of Bagger-Lambert and Gustavsson on superconformal Chern-Simons theory will be
briefly discussed. Emphasis will be made on the appearance of the 3-Lie algebra
structure and on why the A_4 model may be formulated in terms of an ordinary
Lie algebra, and on its Nambu bracket generalization.Comment: Invited topical review for JPA Math.Theor. v2: minor changes,
references added. 120 pages, 318 reference
Automorphisms of extensions of Lie-Yamaguti algebras and Inducibility problem
Lie-Yamaguti algebras generalize both the notions of Lie algebras and Lie
triple systems. In this paper, we consider the inducibility problem for
automorphisms of Lie-Yamaguti algebra extensions. More precisely, given an
abelian extension of Lie-Yamaguti algebra , we are interested in finding the pairs
, which are inducible by
an automorphism in . We connect the inducibility
problem to the cohomology of Lie-Yamaguti algebra. In particular, we
show that the obstruction for a pair of automorphisms in to be inducible lies in a cohomology class of the
cohomology group . We develop the Wells exact sequence
for Lie-Yamaguti algebra extensions, which relates the space of derivations,
automorphism groups, and -cohomology groups of Lie-Yamaguti algebras.
Finally, we consider nilpotent Lie-Yamaguti algebras of index with a
one-dimensional center. We give two infinite families of such nilpotent
Lie-Yamaguti algebras and characterize the inducible pairs for extensions
arising from these examples. Finally, We give an algorithm to characterize
inducible pairs of automorphisms for extensions arising from nilpotent
Lie-Yamaguti algebras (index ) with a one-dimensional center.Comment: Any comments/suggestions are welcom
Metric 3-Leibniz algebras and M2-branes
This thesis is concerned with superconformal Chern-Simons theories with
matter in 3 dimensions. The interest in these theories is two-fold. On the one
hand, it is a new family of theories in which to test the AdS/CFT
correspondence and on the other, they are important to study one of the main
objects of M-theory (M2-branes). All these theories have something in common:
they can be written in terms of 3-Leibniz algebras. Here we study the structure
theory of such algebras, paying special attention to a subclass of them that
gives rise to maximal supersymmetry and that was the first to appear in this
context: 3-Lie algebras.
In chapter 2, we review the structure theory of metric Lie algebras and their
unitary representations. In chapter 3, we study metric 3-Leibniz algebras and
show, by specialising a construction originally due to Faulkner, that they are
in one to one correspondence with pairs of real metric Lie algebras and unitary
representations of them. We also show a third characterisation for six extreme
cases of 3-Leibniz algebras as graded Lie (super)algebras. In chapter 4, we
study metric 3-Lie algebras in detail. We prove a structural result and also
classify those with a maximally isotropic centre, which is the requirement that
ensures unitarity of the corresponding conformal field theory. Finally, in
chapter 5, we study the universal structure of superpotentials in this class of
superconformal Chern-Simons theories with matter in three dimensions. We
provide a uniform formulation for all these theories and establish the
connection between the amount of supersymmetry preserved and the gauge Lie
algebra and the appropriate unitary representation to be used to write down the
Lagrangian. The conditions for supersymmetry enhancement are then expressed
equivalently in the language of representation theory of Lie algebras or the
language of 3-Leibniz algebras.Comment: PhD thesis of Elena M\'endez-Escobar in the University of Edinburgh
(supervised by Jos\'e Figueroa-O'Farrill and Joan Sim\'on