Lie Algebras and Bilinear Forms

If \(J\) is the matrix of a bilinear form, the Lie algebra of derivations of \(J\) consists of the matrices \(X\) such that \(XJ + JX^\mathrm{tr}= 0\).

DerivationAlgebra(J): AlgMatElt -> AlgLie

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".

Example: lie C3 (ex-3caa04)

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

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Example: lie G2 (ex-196405)

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 \({\bf F}_{8}\) 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

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Another way to construct a Lie algebra from an alternating form \(\beta\) with matrix \(J\) defined on a vector space \(V\) of dimension \(n\) over a field \(F\) is to set \(L = V\oplus F\) and define the multiplication by \([e_i,e_j] = \beta(e_i,e_j)\), where \(e_1\), \(e_2\), …, \(e_n\) is a basis for \(V\). (All other structure constants are 0.) This is the (generalised) Heisenberg algebra.

HeisenbergAlgebra(J): AlgMatElt -> AlgLie

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".

Example: heisenberg (ex-aa6842)

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

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