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Dear gap folks,

I would like to work with Lie algebras, and I wonder whether there are

lists of them available. There is one list in the article J. Patera, R. T.

Sharp, P. Winternitz, "Invariants of real low dimensional Lie algebras". I

am looking also for a list of the (low dimensional) Lie algebras over

finite fields, of the (low dimensional) restricted Lie algebras, and of

the (low dimensional) graded Lie algebras. Are there also such list

already available as gap input?

By the way, waiting for the lists, I enter some Lie algebras by hand,

giving the entries in the structure constant table, finding a strange

behavior of gap. Chapter 59.3, "Constructing Algebras by Structure

Constants" of the reference manual says: "For convenience, these entries

may also be rational numbers that are automatically replaced by the

corresponding elements in the appropriate prime field in finite

characteristic if necessary."

I want to enter the 3-dimensional Lie algebras with the relations

[a1,a2]=a3, [a1,a3]=[a2,a3]=0, considering it both over rationals and over

a finite field.

GAP4, Version: 4r2 fix8 of 7-June-2001, sparc-sun-solaris2.7-gcc

gap> t:=EmptySCTable(3,0,"antisymmetric");

[ [ [ [ ], [ ] ], [ [ ], [ ] ], [ [ ], [ ] ] ],

[ [ [ ], [ ] ], [ [ ], [ ] ], [ [ ], [ ] ] ],

[ [ [ ], [ ] ], [ [ ], [ ] ], [ [ ], [ ] ] ], -1, 0 ]

gap> SetEntrySCTable(t,1,2,[1,3]);

gap> TestJacobi(t);

true

gap> lie_rational:=LieAlgebraByStructureConstants(Rationals,t);

<Lie algebra of dimension 3 over Rationals>

gap> lie_gf:=LieAlgebraByStructureConstants(GF(3),t);

<Lie algebra of dimension 3 over GF(3)>

gap> Size(last);

27

gap> LieLowerCentralSeries(lie_rational);

[ <Lie algebra of dimension 3 over Rationals>,

<Lie algebra of dimension 1 over Rationals>,

<Lie algebra of dimension 0 over Rationals> ]

gap> LieLowerCentralSeries(lie_gf);

Error family of <coeffs> does not fit to <Fam> at

Error( "family of <coeffs> does not fit to <Fam>" );

ObjByExtRep( F, SCTableProduct( F!.sctable, x![1], y![1] ) ) called from

BasisVectors( Basis( U ) ) called from

MutableBasisOfProductSpace( U, V ) called from

ProductSpace( L, L ) called from

LieDerivedSubalgebra( L ) called from

...

Entering break read-eval-print loop, you can 'quit;' to quit to outer

loop,

or you can return to continue

brk>

Of course starting with

t_rational:=EmptySCTable(3,0,"antisymmetric");

and

t_gfp:=EmptySCTable(3,Zero(GF(p)),"antisymmetric");

for each p prime, we don't get this error any more, but we have to use a

different structure constant table for each prime. It would be more

convenient if it were possible to enter also the zero of the structure

constant table as rational, and to have it automatically replaced by the

corresponding elements in the appropriate prime field in finite

characteristic if necessary.

Best regards,

Marco Costantini

costanti@mat.uniroma1.it

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