[GAP Forum] Help with recovering Lie brackets

[Willem Adriaan De Graaf]

Dear Alan,

You can use * to multiply elements of a Lie algebra. In the example you
mention:

gap> L:= SimpleLieAlgebra( "F", 4, Rationals );
<Lie algebra of dimension 52 over Rationals>
gap> c:= ChevalleyBasis(L);;
gap> c[1][1]*c[1][2];
0*v.1
gap> c[1][1]*c[1][3];
(-1)*v.5

StructureConstantsTable is applied to a basis of L:

gap> T:= StructureConstantsTable( Basis(L) );;
gap> T[1][3];
[ [ 5 ], [ -1 ] ]

The latter means that e1*e3 is equal to -e5 (where e1,e2,... denote the
standard basis vectors).

Best wishes,

Willem


On Sun, 25 Apr 2021 at 19:38, Alan Hylton <agh314 at lehigh.edu> wrote:

> Hi everyone,
>
> I am brand new to GAP. I am trying to follow along the Lie algebra
> documentation here:
> https://www.gap-system.org/Manuals/doc/ref/chap64_mj.html
>
> Right now I can create my Lie algebra:
>
> gap> L := SimpleLieAlgebra( "F", 4, Rationals );
>
> and I can my positive/negative roots and Cartan:
>
> gap> ChevalleyBasis(L);
> [ [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8, v.9, v.10, v.11, v.12, v.13,
> v.14, v.15, v.16, v.17, v.18, v.19,
>       v.20, v.21, v.22, v.23, v.24 ], [ v.25, v.26, v.27, v.28, v.29, v.30,
> v.31, v.32, v.33, v.34, v.35, v.36,
>       v.37, v.38, v.39, v.40, v.41, v.42, v.43, v.44, v.45, v.46, v.47,
> v.48 ], [ v.49, v.50, v.51, v.52 ] ]
>
> But I cannot figure out how to do basic computations - like [e1, e2]. There
> is a function called LieBracket, and the documentation suggests that *
> works, but I can't get either to work.
>
> I would also like to see if I can return a list of non-trivial bracket
> relations, or structure constants. I found there is a function called
> StructureConstantsTable, but it does not seem to work with L.
>
> I'm sorry to ask what is surely simple stuff - I just don't see it yet.
>
> Thanks!
> agh
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