13 research outputs found
An Explicit Construction of Casimir Operators and Eigenvalues : I
We give a general method to construct a complete set of linearly independent
Casimir operators of a Lie algebra with rank N. For a Casimir operator of
degree p, this will be provided by an explicit calculation of its symmetric
coefficients . It is seen that these coefficients can be
descibed by some rational polinomials of rank N. These polinomials are also
multilinear in Cartan sub-algebra indices taking values from the set . The crucial point here is that for each degree one needs, in
general, more than one polinomials. This in fact is related with an observation
that the whole set of symmetric coefficients is
decomposed into sum subsets which are in one to one correspondence with these
polinomials. We call these subsets clusters and introduce some indicators with
which we specify different clusters. These indicators determine all the
clusters whatever the numerical values of coefficients
are. For any degree p, the number of clusters is independent of rank N. This
hence allows us to generalize our results to any value of rank N.
To specify the general framework explicit constructions of 4th and 5th order
Casimir operators of Lie algebras are studied and all the polinomials
which specify the numerical value of their coefficients are given explicitly.Comment: 14 pages, no figures, revised version, to appear in Jour.Math.Phy
On the Calculation of Group Characters
It is known that characters of irreducible representations of finite Lie
algebras can be obtained using theWeyl character formula including Weyl group
summations which make actual calculations almost impossible except for a few
Lie algebras of lower rank. By starting from the Weyl character formula, we
show that these characters can be re-expressed without referring to Weyl group
summations. Some useful technical points are given in detail for the
instructive example of G2 Lie algebra.Comment: 6 pages, no figure, Plain Te
Fundamental Weights, Permutation Weights and Weyl Character Formula
For a finite Lie algebra of rank N, the Weyl orbits
of strictly dominant weights contain number of
weights where is the dimension of its Weyl group . For any
, there is a very peculiar subset for
which we always have For
any dominant weight , the elements of are called
{\bf Permutation Weights}.
It is shown that there is a one-to-one correspondence between elements of
and where is the Weyl vector of .
The concept of signature factor which enters in Weyl character formula can be
relaxed in such a way that signatures are preserved under this one-to-one
correspondence in the sense that corresponding permutation weights have the
same signature. Once the permutation weights and their signatures are specified
for a dominant , calculation of the character for
irreducible representation will then be provided by
multiplicity rules governing generalized Schur functions. The main idea is
again to express everything in terms of the so-called {\bf Fundamental Weights}
with which we obtain a quite relevant specialization in applications of Weyl
character formula.Comment: 6 pages, no figures, TeX, as will appear in Journal of Physics
A:Mathematical and Genera