340 research outputs found
Graded Contractions of Affine Kac-Moody Algebras
The method of graded contractions, based on the preservation of the
automorphisms of finite order, is applied to the affine Kac-Moody algebras and
their representations, to yield a new class of infinite dimensional Lie
algebras and representations. After the introduction of the horizontal and
vertical gradings, and the algorithm to find the horizontal toroidal gradings,
I discuss some general properties of the graded contractions, and compare them
with the In\"on\"u-Wigner contractions. The example of is discussed
in detail.Comment: 23 pages, Ams-Te
On Deformations and Contractions of Lie Algebras
In this contributed presentation, we discuss and compare the mutually opposite procedures of deformations and contractions of Lie algebras. We suggest that with appropriate combinations of both procedures one may construct new Lie algebras. We first discuss low-dimensional Lie algebras and illustrate thereby that whereas for every contraction there exists a reverse deformation, the converse is not true in general. Also we note that some Lie algebras belonging to parameterized families are singled out by the irreversibility of deformations and contractions. After reminding that global deformations of the Witt, Virasoro, and affine Kac-Moody algebras allow one to retrieve Lie algebras of Krichever-Novikov type, we contract the latter to find new infinite dimensional Lie algebras
Graded contractions of bilinear invariant forms of Lie algebras
We introduce a new construction of bilinear invariant forms on Lie algebras,
based on the method of graded contractions. The general method is described and
the -, -, and -contractions are
found. The results can be applied to all Lie algebras and superalgebras (finite
or infinite dimensional) which admit the chosen gradings. We consider some
examples: contractions of the Killing form, toroidal contractions of ,
and we briefly discuss the limit to new WZW actions.Comment: 15 page
Casimir invariants for the complete family of quasi-simple orthogonal algebras
A complete choice of generators of the center of the enveloping algebras of
real quasi-simple Lie algebras of orthogonal type, for arbitrary dimension, is
obtained in a unified setting. The results simultaneously include the well
known polynomial invariants of the pseudo-orthogonal algebras , as
well as the Casimirs for many non-simple algebras such as the inhomogeneous
, the Newton-Hooke and Galilei type, etc., which are obtained by
contraction(s) starting from the simple algebras . The dimension of
the center of the enveloping algebra of a quasi-simple orthogonal algebra turns
out to be the same as for the simple algebras from which they come by
contraction. The structure of the higher order invariants is given in a
convenient "pyramidal" manner, in terms of certain sets of "Pauli-Lubanski"
elements in the enveloping algebras. As an example showing this approach at
work, the scheme is applied to recovering the Casimirs for the (3+1)
kinematical algebras. Some prospects on the relevance of these results for the
study of expansions are also given.Comment: 19 pages, LaTe
Central extensions of the families of quasi-unitary Lie algebras
The most general possible central extensions of two whole families of Lie
algebras, which can be obtained by contracting the special pseudo-unitary
algebras su(p,q) of the Cartan series A_l and the pseudo-unitary algebras
u(p,q), are completely determined and classified for arbitrary p,q. In addition
to the su(p,q) and u({p,q}) algebras, whose second cohomology group is well
known to be trivial, each family includes many non-semisimple algebras; their
central extensions, which are explicitly given, can be classified into three
types as far as their properties under contraction are involved. A closed
expression for the dimension of the second cohomology group of any member of
these families of algebras is given.Comment: 23 pages. Latex2e fil
Possible contractions of quantum orthogonal groups
Possible contractions of quantum orthogonal groups which correspond to
different choices of primitive elements of Hopf algebra are considered and all
allowed contractions in Cayley--Klein scheme are obtained. Quantum deformations
of kinematical groups have been investigated and have shown that quantum analog
of (complex) Galilei group G(1,3) do not exist in our scheme.Comment: 10 pages, Latex. Report given at XXIII Int. Colloquium on Group
Theoretical Methods in Physics, July 31- August 5, 2000, Dubna (Russia
On the electrodynamics of moving bodies at low velocities
We discuss the seminal article in which Le Bellac and Levy-Leblond have
identified two Galilean limits of electromagnetism, and its modern
implications. We use their results to point out some confusion in the
literature and in the teaching of special relativity and electromagnetism. For
instance, it is not widely recognized that there exist two well defined
non-relativistic limits, so that researchers and teachers are likely to utilize
an incoherent mixture of both. Recent works have shed a new light on the choice
of gauge conditions in classical electromagnetism. We retrieve Le
Bellac-Levy-Leblond's results by examining orders of magnitudes, and then with
a Lorentz-like manifestly covariant approach to Galilean covariance based on a
5-dimensional Minkowski manifold. We emphasize the Riemann-Lorenz approach
based on the vector and scalar potentials as opposed to the Heaviside-Hertz
formulation in terms of electromagnetic fields. We discuss various applications
and experiments, such as in magnetohydrodynamics and electrohydrodynamics,
quantum mechanics, superconductivity, continuous media, etc. Much of the
current technology where waves are not taken into account, is actually based on
Galilean electromagnetism
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