Bernard Kolman

COMPUTING THE STRUCTURE OF A LIE ALGEBRA

Publisher Summary This chapter presents algorithms for locating in a finite-dimensional Lie algebra L items of structural interest. This structural analysis includes calculating the various radicals of L, the various series of L, a Cartan subalgebra of L, and the derivation algebra of L and also determining whether L is nilpotent, soluble, or semisimple. The algorithms presented allow the construction of most of the fundamental objects occurring in the structure theory such as the nil or soluble radical, a Cartan subalgebra, a Borei subalgebra. Once these are known, it is easy to decide whether an algebra is semisimple and if so to list its simple direct summands and multiplicities and to identify their isomorphism types in the standard list by way of Dynkin diagrams. The methods required are often different from the standard theorems.

Computers in Lie Algebras. II: Calculation of Outer Multiplicities

The tensor product of two irreducible representations

MATRICES AND DETERMINANTS

\rho _1

ROOTS OF POLYNOMIALS

and

Construction of nilpotent Lie algebras over arbitrary fields

\rho _2

Algorithms for central extensions of Lie algebras

of a complex simple Lie algebra L is a completely reducible representation of L, but it is often not irreducible. Associated with each irreducible representation

Computer Approaches to the Representations of Lie Algebras

\rho

EXPONENTIAL AND LOGARITHMIC FUNCTIONS

of L is its outer multiplicity with respect to

2 – The Simplex Method

\rho _1 \otimes \rho _2

1 – Introduction to Linear Programming

. The outer multiplicity of