Converting an algebra to a tensor enables Magma to compute standard invariants of any algebra.
We note that there are known errors for R and C due to the numerical stability of the linear algebra involved in the computations.
We will construct a representation of (SL)2(9) in (Mat)4(GF(3)).
First we construct (GL)2(9) from (Mat)4(GF(3)).
> M := MatrixAlgebra(GF(3), 4);
> f := ConwayPolynomial(3, 2);
> C := CompanionMatrix(f);
> I := IdentityMatrix(GF(3), 2);
> A := sub< M | [InsertBlock(M!0, X, i, j) :
> X in [I, C], i in [1, 3], j in [1, 3]] >;
> T := CommutatorTensor(A);
> T;
Tensor of valence 3, U2 x U1 >-> U0
U2 : Full Vector space of degree 8 over GF(3)
U1 : Full Vector space of degree 8 over GF(3)
U0 : Full Vector space of degree 8 over GF(3)
> GL2 := HeisenbergAlgebra(T);
> GL2;
Algebra of dimension 8 with base ring GF(3)
Our Lie algebra is not simple as it has a nontrivial center, so we will obtain
(SL)2 by factoring out the center. Note that our algebras are over the
prime field GF(3), so the center is 2-dimensional (over GF(3)). Notice
that (SL)2(9) has a trivial center but has a 2-dimensional centroid.
> SL2 := GL2/Center(GL2);
> SL2;
Algebra of dimension 6 with base ring GF(3)
> Center(SL2);
Algebra of dimension 0 with base ring GF(3)
> Centroid(SL2);
Matrix Algebra of degree 6 with 2 generators over GF(3)
RightNucleus(A) : Alg -> AlgMat
MidNucleus(A) : Alg -> AlgMat
Returns the nucleus of the algebra A as a subalgebra of the enveloping algebra of right multiplication (R)(A).
Returns the derivation algebra of the algebra A as a Lie subalgebra of (End)K(A).
We will compute the derivation algebra of the (rational) octonions O and also the 27 dimension exceptional Jordan algebra (H)3( O ).
Because the intrinsics use exact linear algebra, we do not use the more familiar field C in this context.
First we consider O .
We verify that (Der)( O ) isomorphic to G2.
> A := OctonionAlgebra(Rationals(), -1, -1, -1);
> A;
Algebra of dimension 8 with base ring Rational Field
> D := DerivationAlgebra(A);
> D;
Matrix Lie Algebra of degree 8 over Rational Field
> SemisimpleType(D);
G2
Now we will just briefly perform a sanity check and verify that D acts as it should.
> a := Random(Basis(A));
> b := Random(Basis(A));
> del := Random(Basis(D));
> (a*b)*del eq (a*del)*b + a*(b*del);
true
Finally, we construct (H)3( O ) the 3 x 3 Hermitian matrices, and we verify that (Der)((H)3( O ) isomorphic to F4.
> J := ExceptionalJordanCSA(A);
> J;
Algebra of dimension 27 with base ring Rational Field
> D_J := DerivationAlgebra(J);
> Dimension(D_J);
52
> SemisimpleType(D_J);
F4
We demonstrate further how to use these functions to get invariants of nonassociative algebras.
First, we will obtain the derivation Lie algebra of the Octonions, which are of type G
2.
> A := OctonionAlgebra(GF(7),-1,-1,-1);
> A;
Algebra of dimension 8 with base ring GF(7)
> D := DerivationAlgebra(A);
> D.1;
[0 0 0 0 0 0 0 0]
[0 0 6 0 6 3 2 1]
[0 1 0 3 4 1 1 3]
[0 0 4 0 6 4 2 3]
[0 1 3 1 0 6 2 0]
[0 4 6 3 1 0 6 2]
[0 5 6 5 5 1 0 4]
[0 6 4 4 0 5 3 0]
> Dimension(D);
14
> SemisimpleType(D);
G2
Now we will show that the left, mid, and right nuclei are all one dimensional.
All of which are generated by R
1, multiplication by 1
A.
> Z := Center(A);
> Z;
Algebra of dimension 1 with base ring GF(7)
>
> L := LeftNucleus(A);
> L;
Matrix Algebra of degree 8 with 1 generator over GF(7)
> L.1;
[1 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0]
[0 0 0 0 1 0 0 0]
[0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 1]
>
> L eq MidNucleus(A);
true
> L eq RightNucleus(A);
true
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