> < ^ Date: Thu, 30 Aug 2001 10:41:16 +0200 (MET DST)
> < ^ From: Marco Costantini <costanti@giove.mat.uniroma1.it >
> < ^ Subject: Lie algebras

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

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,
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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