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Manuscript Title: PASCAL programs for identification of lie algebras, part II: SPLIT, a program to decompose parameter-free and parameter-dependent lie algebras into direct sums.
Authors: D.W. Rand, P. Winternitz, H. Zassenhaus
Program title: SPLIT
Catalogue identifier: AAXM_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 46(1987)297
Programming language: Pascal.
Computer: (CDC) CYBER SERIES 170 MODELS 835 AND 855.
Operating system: NOS/BE.
RAM: 260K words
Word size: 64
Keywords: General purpose, Computer algebra, Lie algebra, Direct sum, Decomposition, Idempotent.
Classification: 4.2, 5.

Subprograms used:
Cat Id Title Reference
AALB_v1_0 RADICAL CPC 41(1986)105

Nature of problem:
Given a Lie algebra L with structure constants which are either integers or polynomials with integer coefficients, SPLIT determines whether L can be expressed as a direct sum of Lie algebras of smaller dimension and, if so, calculates the structure constants of its components.

Solution method:
SPLIT first removes from L any central component CC(L), i.e. a maximal Abelian subalgebra of L contained in the centre of L and whose intersection with [L,L] is zero. The program then attempts decomposition using the method of idempotents, i.e. by seeking an idempotent element in the algebra of matrices which commute with adjoint representation of L. Complete decomposition is obtained by recursive application of the same method to each component.

The program fails to complete the decomposition when the matrix of change of basis contains irrational numbers or expressions or when highly non-trivial polynomial factorization is required. Integer overflow or lack of memory may occur if dim(L) is large, if the structure constant array is very dense, and/or if the structure constants depend on many parameters.

Unusual features:
All calculations are performed in fixed-point to maintain exactness. Computations for parameter-dependent algebras involve polynomial manipulation, with several algorithms adapted. Both polynomials and structure constants arrays are represented compactly as linked lists, using the data structures decribed in Comp. Phys. Commun. 41(1986)105. The search for an idempotent involves use of the Smith canonical form to determine the invariant factors of a square matrix.

Running time:
Varies widely, depending on the complexity of the data. See discussion of test run.