120 research outputs found
Determinantal formulae for the Casimir operators of inhomogeneous Lie algebras
Contractions of Lie algebras are combined with the classical matrix method of
Gel'fand to obtain matrix formulae for the Casimir operators of inhomogeneous
Lie algebras. The method is presented for the inhomogeneous pseudo-unitary Lie
algebras . This procedure is extended to contractions of
isomorphic to an extension by a derivation of the
inhomogeneous special pseudo-unitary Lie algebras ,
providing an additional analytical method to obtain their invariants. Further,
matrix formulae for the invariants of other inhomogeneous Lie algebras are
presented.Comment: Final ammended versio
Unitary representations of three dimensional Lie groups revisited: An approach via harmonic functions
Harmonic functions of the three dimensional Lie groups defined on certain
manifolds related to the Lie groups themselves and carrying all their unitary
representations are explicitly constructed. The realisations of these Lie
groups are shown to be related with each other by either natural operations as
real forms or In\"on\"u-Wigner contractions.Comment: The title was changed; More details are given for the constuction of
harmonic functions 19 page
Color Lie algebras and Lie algebras of order F
The notion of color algebras is generalized to the class of F-ary algebras,
and corresponding decoloration theorems are established. This is used to give a
construction of colored structures by means of tensor products with
Clifford-like algebras. It is moreover shown that color algebras admit
realisations as q=0 quon algebras.Comment: LaTeX, 16 page
Parafermions, ternary algebras and their associated superspace
Parafermions of order two are shown to be the fundamental tool to construct
ternary superspaces related to cubic extensions of the Poincar\'e algebraComment: Talk given at the VIII. International Workshop Lie Theory and its
applications in physics, 15 - 21 June 2009, Varna, Bulgari
Obtainment of internal labelling operators as broken Casimir operators by means of contractions related to reduction chains in semisimple Lie algebras
We show that the In\"on\"u-Wigner contraction naturally associated to a
reduction chain of semisimple Lie algebras
induces a decomposition of the Casimir operators into homogeneous polynomials,
the terms of which can be used to obtain additional mutually commuting missing
label operators for this reduction. The adjunction of these scalars that are no
more invariants of the contraction allow to solve the missing label problem for
those reductions where the contraction provides an insufficient number of
labelling operators
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
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