|
If J is the matrix of a bilinear form, the Lie algebra of derivations of
J consists of the matrices X such that XJ + JXtr = 0.
Another way to construct a Lie algebra from an alternating form
β with matrix J defined on a vector space V of dimension
n over a field F is to set L = V direct-sum F and define the
multiplication by [ei, ej] = β(ei, ej), where e1, e2, ..., en is a basis for V. (All other structure constants are 0.)
This is the (generalised) Heisenberg algebra.
Rep: MonStgElt Default: "Sparse"
Check: BoolElt Default: false
The Lie algebra of derivations of the bilinear form with matrix J.
The possible values for Rep are "Dense", "Sparse"
and "Partial" with the default being "Sparse".
Construct the Lie algebra preserving the standard alternating form
of rank 6 over the field of 7 elements and check that it is a simple
algebra of type C 3.
> J := StandardAlternatingForm(6,7);
> L := DerivationAlgebra(J);
> IsSimple(L);
true
> SemisimpleType(L);
C3
Over a field of characteristic 2, the Lie algebra of an alternating
form is no longer simple. In this example the Lie algebra L of the
standard alternating form of rank 6 over GF(8) is of symplectic
type C 3 but has an ideal I of type G 2. The dimension of the
centre of L is 1 and so the ideal I will appear as either
the first or second composition factor.
> J := StandardAlternatingForm(6,8);
> L := DerivationAlgebra(J);
> SemisimpleType(L);
C3
> Dimension(Centre(L));
1
> CF := CompositionFactors(L);
> CF;
[
Lie Algebra of dimension 1 with base ring GF(2^3),
Lie Algebra of dimension 14 with base ring GF(2^3),
Lie Algebra of dimension 1 with base ring GF(2^3),
Lie Algebra of dimension 1 with base ring GF(2^3),
Lie Algebra of dimension 1 with base ring GF(2^3),
Lie Algebra of dimension 1 with base ring GF(2^3),
Lie Algebra of dimension 1 with base ring GF(2^3),
Lie Algebra of dimension 1 with base ring GF(2^3)
]
> exists(I){I : I in CF | Dimension(I) eq 14 };
true
> IsSimple(I);
true
> SemisimpleType(I);
G2
Rep: MonStgElt Default: "Sparse"
Check: BoolElt Default: false
The nilpotent Lie algebra whose structure constants are obtained
from the alternating form with matrix J as described above.
The possible values for Rep are "Dense", "Sparse"
and "Partial" with the default being "Sparse".
Continuing the previous example we construct an explicit isomorphism
between the quotient of L by its G 2 ideal and the Heisenberg
algebra of the form.
> Q := CS[1];
> W := L/Q;
> W;
Lie Algebra of dimension 7 with base ring GF(2^3)
> Z := Centre(W);
> z := Z.1;
> exists(u1,v1){ <u,v> : u,v in W | u*v ne 0 and u*v eq z };
true
> W1 := Centraliser(W,sub<W|u1,v1>);
true
> exists(u2,v2){ <u,v> : u,v in W1 | u*v ne 0 and u*v eq z };
true
> W2:= Centraliser(W1,sub<W1|u2,v2>);
> exists(u3,v3){ <u,v> : u,v in W2 | u*v ne 0 and u*v eq z };
> H := HeisenbergAlgebra(J);
> f := hom< H -> W | u1,u2,u3,v3,v2,v1,z >;
> forall{ <u,v> : u,v in Basis(H) | f(u*v) eq f(u)*f(v) };
true
> Kernel(f);
Lie Algebra of dimension 0 with base ring GF(2^3)
> Image(f) eq W;
true
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
|