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Manuscript Title: PASCAL programs for identification of Lie algebras. Part 1. RADICAL: a program to calculate the radical and nil radical of parameter-free and parameter-dependent Lie algebras.
Authors: D.W. Rand
Program title: RADICAL
Catalogue identifier: AALB_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 41(1986)105
Programming language: Pascal.
Computer: CDC CYBER 170 MODEL 835.
Operating system: NOS/BE.
RAM: 200K words
Word size: 64
Keywords: General purpose, Lie algebras, Ideal, Radical, Nil radical, Algebraic manipulation, Greatest common divisor, Subresultant.
Classification: 4.2.

Revision history:
Type Tit le Reference
correction 000A CORRECTION 02/06/87 See below

Nature of problem:
Given a Lie algebra L with structure constants which are either integers or polynomials with integer coefficients, RADICAL determines its radical (maximal solvable ideal) and nil radical (maximal nilpotent ideal).

Solution method:
The radical is determined as the solution of a system of at most dim(L) linear homogeneous equations (LHE) in dim(L) unknowns. The nil radical is obtained using a new recursive algorithm conceived by Zassenhaus. Numeric calculations are done in fixed-point to maintain exactness, and parameter-dependent calculations are performed using linked-list representations of polynomials and the subresultant algorithm for greatest common divisor (g.c.d.) computation.

Restrictions:
The program has been successfully used on parameter-free Lie algebras of dimension as high as 30 and for algebras of dimension 15 depending on five independent parameters. The complexity of the calculations depends greatly on the number of non-zero structure constants and on the number of parameters. As these increase, either integer overflow (for which spot-checks are done) or lack of storage may occur.

Unusual features:
Zassenhaus' nil radical algorithm, unlike that of Beck et al., is rational in that it requires no irrational operations such as eigenvalue calculation. Hence the results are exact. Polynomials and structure constant arrays are represented efficiently as linked lists. The major subprograms are present in two parallel versions: one performing integer calculations for the parameter-free case, the other polynomial calculations for the parameter-dependent case. The latter is a form of algebraic manipulation, involving polynomial addition, multiplication, division, pseudo-division, g.c.d. computation, etc. While solving the numerous systems of LHEs the program stores all polynomials which may not vanish, e.g. pivots, divisors, etc., and prints them in summary form so that the user may note any special parameter values for which the generic results may be invalid.

Running time:
Ranges from a fraction of a second (for algebras of dimension <=5) to several seconds (for the examples mentioned above).

CORRECTION SUMMARY
Manuscript Title: PASCAL programs for indentification of Lie algebras. Part 1. RADICAL: a program to calculate the radical and nil radical of parameter-free and parameter-dependent Lie algebras. (C.P.C. 41(1986)105).
Authors: D.W. Rand
Program title: 000A CORRECTION 02/06/87
Catalogue identifier: AALB_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 47(1987)369
Classification: 4.2.