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Manuscript Title: PASCAL programs for identification of lie algebras, part III: Levi decomposition and canonical basis.
Authors: D.W. Rand
Program title: CANONIK
Catalogue identifier: AAXO_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 46(1987)311
Programming language: Pascal.
Computer: (CDC) CYBER SERIES 170 MODELS 835 AND 855.
Operating system: NOS/BE.
RAM: 320K words
Word size: 64
Keywords: General purpose, Computer algebra, Decomposition, Lie algebra, Levi decomposition, Radical, Nilradical, Ideal.
Classification: 4.2, 5.

Subprograms used:
Cat Id Title Reference
AALB_v1_0 RADICAL CPC 41(1986)105
AAXM_v1_0 SPLIT CPC 46(1987)297

Nature of problem:
Given a Lie algebra L with structure constants which are either integers or polynomials with integer coefficients, LEVI determines a Levi decomposition of L into its radical R(L) and its (semi-)simple part S(L). The program CANONIK uses the methods of all three programs SPLIT, LEVI and RADICAL in order to express L in a standard basis: L is decomposed as a direct sum, each component L' is Levi-decomposed, each S(L') is decomposed as a direct sum, the nilradical of each R(L') is calculated, and finally the basis of each nilradical is ordered according to its upper central series.

Solution method:
LEVI determines the radical R(L) of L using the orthogonal complement, relative to the Killing form, of the derived algebra [L,L]. If the quotient algebra L/R(L) is not closed under Lie multiplication, LEVI modifies its basis in order to close it, thus making it a representative of S(L), the maximal (semi-)simple subalgebra of L. This closure is performed using either a system of linear inhomogeneous equations when R(L) is Abelian, or a recursive method involving quotienting of L by the derived algebra of the radical, hence reducing the problem to one of smaller dimension. The program CANONIK uses this same method as well as those described in Comp. Phys. Commun. 41(1986)105.

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. Such problems are greatest when CANONIK attempts decomposition as a direct sum.

Unusual features:
The computations are exact, free of numerical approximation. When parameters appear in the structure constants, the programs store and print out in summary form all polynomials which are assumed not to vanish --e.g. pivots in systems of linear equations.

Running time:
Varies widely, depending on the complexity of the data. Typically, each algebra requires a few seconds.