26 research outputs found
Poincar\'e and sl(2) algebras of order 3
In this paper we initiate a general classification for Lie algebras of order
3 and we give all Lie algebras of order 3 based on
and the Poincar\'e algebra in four-dimensions. We then
set the basis of the theory of the deformations (in the Gerstenhaber sense) and
contractions for Lie algebras of order 3.Comment: Title and presentation change
Formal deformations, contractions and moduli spaces of Lie algebras
Jump deformations and contractions of Lie algebras are inverse concepts, but
the approaches to their computations are quite different. In this paper, we
contrast the two approaches, showing how to compute jump deformations from the
miniversal deformation of a Lie algebra, and thus arrive at the contractions.
We also compute contractions directly. We use the moduli spaces of real
3-dimensional and complex 3 and 4-dimensional Lie algebras as models for
explaining a deformation theory approach to computation of contractions.Comment: 27 page
Expanding Lie (super)algebras through abelian semigroups
We propose an outgrowth of the expansion method introduced by de Azcarraga et
al. [Nucl. Phys. B 662 (2003) 185]. The basic idea consists in considering the
direct product between an abelian semigroup S and a Lie algebra g. General
conditions under which relevant subalgebras can systematically be extracted
from S \times g are given. We show how, for a particular choice of semigroup S,
the known cases of expanded algebras can be reobtained, while new ones arise
from different choices. Concrete examples, including the M algebra and a
D'Auria-Fre-like Superalgebra, are considered. Finally, we find explicit,
non-trace invariant tensors for these S-expanded algebras, which are essential
ingredients in, e.g., the formulation of Supergravity theories in arbitrary
space-time dimensions.Comment: 42 pages, 8 figures. v2: Improved figures, updated notation and
terminolog
Quasi-classical Lie algebras and their contractions
After classifying indecomposable quasi-classical Lie algebras in low
dimension, and showing the existence of non-reductive stable quasi-classical
Lie algebras, we focus on the problem of obtaining sufficient conditions for a
quasi-classical Lie algebras to be the contraction of another quasi-classical
algebra. It is illustrated how this allows to recover the Yang-Mills equations
of a contraction by a limiting process, and how the contractions of an algebra
may generate a parameterized families of Lagrangians for pairwise
non-isomorphic Lie algebras.Comment: 17 pages, 2 Table
Non-solvable contractions of semisimple Lie algebras in low dimension
The problem of non-solvable contractions of Lie algebras is analyzed. By
means of a stability theorem, the problem is shown to be deeply related to the
embeddings among semisimple Lie algebras and the resulting branching rules for
representations. With this procedure, we determine all deformations of
indecomposable Lie algebras having a nontrivial Levi decomposition onto
semisimple Lie algebras of dimension , and obtain the non-solvable
contractions of the latter class of algebras.Comment: 21 pages. 2 Tables, 2 figure
Galilei invariant theories. I. Constructions of indecomposable finite-dimensional representations of the homogeneous Galilei group: directly and via contractions
All indecomposable finite-dimensional representations of the homogeneous
Galilei group which when restricted to the rotation subgroup are decomposed to
spin 0, 1/2 and 1 representations are constructed and classified. These
representations are also obtained via contractions of the corresponding
representations of the Lorentz group. Finally the obtained representations are
used to derive a general Pauli anomalous interaction term and Darwin and
spin-orbit couplings of a Galilean particle interacting with an external
electric field.Comment: 23 pages, 2 table
Expansions of algebras and superalgebras and some applications
After reviewing the three well-known methods to obtain Lie algebras and
superalgebras from given ones, namely, contractions, deformations and
extensions, we describe a fourth method recently introduced, the expansion of
Lie (super)algebras. Expanded (super)algebras have, in general, larger
dimensions than the original algebra, but also include the Inonu-Wigner and
generalized IW contractions as a particular case. As an example of a physical
application of expansions, we discuss the relation between the possible
underlying gauge symmetry of eleven-dimensional supergravity and the
superalgebra osp(1|32).Comment: Invited lecture delivered at the 'Deformations and Contractions in
Mathematics and Physics Workshop', 15-21 January 2006, Mathematisches
Forschungsinstitut Oberwolfach, German
Extensions, expansions, Lie algebra cohomology and enlarged superspaces
After briefly reviewing the methods that allow us to derive consistently new
Lie (super)algebras from given ones, we consider enlarged superspaces and
superalgebras, their relevance and some possible applications.Comment: 9 pages. Invited talk delivered at the EU RTN Workshop, Copenhagen,
Sep. 15-19 and at the Argonne Workshop on Branes and Generalized Dynamics,
Oct. 20-24, 2003. Only change: wrong number of a reference correcte
An integrable discretization of the rational su(2) Gaudin model and related systems
The first part of the present paper is devoted to a systematic construction
of continuous-time finite-dimensional integrable systems arising from the
rational su(2) Gaudin model through certain contraction procedures. In the
second part, we derive an explicit integrable Poisson map discretizing a
particular Hamiltonian flow of the rational su(2) Gaudin model. Then, the
contraction procedures enable us to construct explicit integrable
discretizations of the continuous systems derived in the first part of the
paper.Comment: 26 pages, 5 figure
Contractions and deformations of quasi-classical Lie algebras preserving a non-degenerate quadratic Casimir operator
By means of contractions of Lie algebras, we obtain new classes of
indecomposable quasi-classical Lie algebras that satisfy the Yang-Baxter
equations in its reformulation in terms of triple products. These algebras are
shown to arise naturally from non-compact real simple algebras with non-simple
complexification, where we impose that a non-degenerate quadratic Casimir
operator is preserved by the limiting process. We further consider the converse
problem, and obtain sufficient conditions on integrable cocycles of
quasi-classical Lie algebras in order to preserve non-degenerate quadratic
Casimir operators by the associated linear deformations.Comment: 12 pages. LATEX with revtex4; Proceedings of the XII International
Conference on Symmetry Methods in Physics, (Yerevan, 2006) eds. G.S. Pogosyan
et al