lineq.pl

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lineq.pl

lineq.pl is a Perl wrapper around GAP for solving linear equations on nonassociative algebras. The typical problems are: to compute derivations, or, more generally, low-degree cocycles on an algebra with values in a module, or commutative 2-cocycles, or generalized derivations, or Hom-Lie structures on a Lie algebra, etc.

The script takes as input an algebra and a module over it, specified in GAP syntax, and an equation to solve, and outputs solution in a human-readable form, in terms of (multi)linear maps acting on a basis of the algebra.

Here is the output from lineq.pl --help command:

Usage:   lineq.pl --eq=<file> --alg=<file> [--debug=gap|trans] [--log=<file>]
         lineq.pl --help|--version
Example: lineq.pl --eq=der --alg=W1.gap --log=gap.log --debug=trans

--eq=<file>     <file> contains equation(s) to be solved;
--alg=<file>    <file> contains GAP code with definition of an algebra and a
                module over it;
--debug         print debug messages:
                gap   - related to interaction with GAP
                trans - performing perl transformations of algebraic
                expressions;
                this option can be repeated;
--log=<file>    log GAP session to <file>;
--help          print this help;
--version       print version.

The input file with equation(s) to be solved (option --eq) should contain one entry per line, which are left-hand sides of the equation (left-hand side) = 0 . Comments and blank lines are ignored. The left-hand side should obey the following rules:

the unknown operator is denoted by one upper-case letter;
all of the following: multiplication in the algebra and multiplication by a scalar are denoted by *, and the action of the algebra on the module (left and/or right), are denoted by ^ (in accordance with GAP conventions);
elements of an algebra are denoted by single lower-case letters;
parentheses are used only for functions and for scalar coefficients.

The unknown maps have arguments in the algebra, and values in the module.
Examples:

derivation: D(x*y) + y*D(x) - x*D(y)
derivation assuming two-sided module: D(x*y) - D(x)^y - x^D(y)

2-cocycle:

F(x*y,z) + F(z*x,y) + F(y*z,x)
F(x,y) + F(y,x)

1/2-derivation:
incorrect: D(x*y) - (1/2)*(D(x)*y + x*D(y))
correct: D(x*y) + (1/2)*y*D(x) - (1/2)*x*D(y)
Note that fractions should be surrounded by parentheses.
O(x*y,z) + O(z*x,y) - O(x,y*z) - z*O(x,y) + y*O(x,z)

In the example above, the file W1.gap contains the GAP definition of the Witt Lie algebra over GF(5) and its adjoint module:

A := SimpleLieAlgebra ("W", [1], GF(5));
M := AdjointModule (A);

The file der contains definition of derivations:

D(x*y) + y*D(x) - x*D(y)

The output of the script will be:

finished compose linear system, solving it ...
dimension: 5

#1:
e[1] -> Z(5)*m[4]
e[2] -> Z(5)^0*m[5]
e[3] -> 0
e[4] -> 0
e[5] -> 0

#2:
e[1] -> Z(5)^3*m[3]
e[2] -> Z(5)^0*m[4]
e[3] -> Z(5)^0*m[5]
e[4] -> 0
e[5] -> 0

#3:
e[1] -> Z(5)*m[2]
e[2] -> Z(5)*m[3]
e[3] -> 0
e[4] -> Z(5)^0*m[5]
e[5] -> 0

#4:
e[1] -> 0
e[2] -> Z(5)^0*m[1]
e[3] -> Z(5)^0*m[2]
e[4] -> Z(5)^0*m[3]
e[5] -> Z(5)^0*m[4]

#5:
e[1] -> Z(5)^3*m[1]
e[2] -> 0
e[3] -> Z(5)*m[3]
e[4] -> Z(5)^2*m[4]
e[5] -> Z(5)^0*m[5]

what says that the derivation algebra is 5-dimensional, with the basis consisting of the 5 specified maps.

If the module is not the adjoint module, the action should be specified, according to GAP syntax, by ^. For example, to compute (-2)-derivations of sl(2) with coefficients in the 2-dimensional irreducible module:

L := SimpleLieAlgebra ("A", 1, Rationals);
M := HighestWeightModule (L, [1]);

and

D(x*y) - 2*y^D(x) + 2*x^D(y)

One may wish to modify the location of GAP at the beginning of the script. The whole thing is very messy and certainly full of bugs, but I would like to hear on any problems while using it. YMMV.

Created: Sun Jun 9 2019
Last modified: Thu Apr 01 07:45:38 Central Europe Daylight Time 2021