PROGRAM SUMMARY
Title of program:
KILLING
Catalogue identifier:
ADLX
Ref. in CPC:
130(2000)137
Distribution format: tar gzip file
Number of lines in distributed program, including test data, etc:
11819
Keywords:
General purpose, Symmetry, Lie algebra, Representation, Symbolic,
Algebraic, Computation, Computer algebra.
Programming language used: Maple V Release 5, Mathematica
Release 3
Nature of physical problem:
Symmetry has been a very important fundamental principle underlying
human knowledge about our physical world. Among several mathematical
formulations of symmetry, the Lie algebras and their corresponding Lie
groups are probably the ones most explored. They were discovered by
Sophus Lie and Wilhelm Killing during the last two decades of the 19th
century. Lie's work on Lie groups was inspired by Galois' work in 1832
in which he discovered the finite groups. Independently, Killing had
started a classification of Lie groups which was the starting point to
the Elie Cartan's doctoral thesis in the beginning of the 20th century.
Cartan was able to make a complete classification of Lie groups. Since
Cartan's classification, the theory of Lie groups has been utilized in
many branches of physics, including molecular physics, atomic physics,
nuclear physics and particle physics.
Method of solution
In spite of the high level of knowledge about the representation theory
of semisimple Lie algebras, the manipulation of elements such as roots,
weights and matrices is very difficult for the non-trivial cases. The
goal in writing this package is to make possible the handling of several
elements of the theory of representation of Lie algebras in a very
convenient way in which the user can easily modify and augment every
code. A great deal of flexibility is achieved by choosing the
algebraic programming scenario in which huge sets of weights and
complicated algebraic matrix elements can be handled in an interactive
way.
Restrictions on the complexity of the problem
Until now, the Gelfand-Tsetlin method has been restricted to classical
orthogonal algebras, and to classical and deformed unitary algebras, and
to the classical symplectic algebra of rank two.
Typical running time
Under one minute for each procedure except for the multiplicities
determination procedures.
References
[1] J.-Q. Chen, Group Representation Theory for Physicists, World
Scientific (1989).
[2] B.G. Wybourne, Classical Groups for Physicists, John Wiley (1974).
[3] A.O. Barut and R. Razcka, Theory of Group Representations and
Applications, World Scientific (1986).
[4] L.C. Biedenharn and M.A. Lohe, Quantum Group Symmetry and q-Tensor
Algebras, World Scientific (1995).
[5] J. Fuchs, Affine Lie Algebras and Quantum Groups, Cambridge (1995).