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.
Returns the center of the algebra A.
Returns the centroid of the K-algebra A as a subalgebra of (End)K(A).
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)
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
> 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); G2Now we will show that the left, mid, and right nuclei are all one dimensional. All of which are generated by R1, multiplication by 1A.
> 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