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Manuscript Title: KILLING - an algebraic computational package for Lie algebras.
Authors: E.S. Bernardes
Program title: KILLING
Catalogue identifier: ADLX_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 130(2000)137
Programming language: Maple V Release 5, Mathematica Release 3.
Keywords: General purpose, Symmetry, Lie algebra, Representation, Symbolic, Algebraic, Computation, Computer algebra.
Classification: 4.2, 5.

Nature of 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.

Solution method:
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:
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.

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).