59.4 Some Special Algebras

is a quaternion algebra over the field F with parameters a and b in F, i.e., a four-dimensional associative F-algebra with basis (e,i,j,k) and multiplication defined by e e = e, e i = i e = i, e j = j e = j, e k = k e = k, i i = a e, i j = - j i = k, i k = - k i = a j, j j = b e, j k = - k j = b i, k k = - a b e. The default values for a and b are -1 in F.

The embedding of the field GaussianRationals into a quaternion algebra A over Rationals is not uniquely determined. One can specify one as a vector space homomorphism that maps 1 to the first algebra generator of A, and E(4) to one of the others.

gap> QuaternionAlgebra( Rationals );
<algebra-with-one of dimension 4 over Rationals>

Let A = e F Åi F Åj F Åk F be a quaternion algebra over the field F of cyclotomics, with basis (e,i,j,k).

If v = v₁ + v₂ j is a row vector over A with v₁ = e w₁ + i w₂ and v₂ = e w₃ + i w₄ then ComplexificationQuat( v ) is the concatenation of w₁ + E(4) w₂ and w₃ + E(4) w₄.

If M = M₁ + M₂ j is a matrix over A with M₁ = e N₁ + i N₂ and M₂ = e N₃ + i N₄ then ComplexificationQuat( M ) is the block matrix A over e F Åi F such that A(1,1) = N₁ + E(4) N₂, A(2,2) = N₁ - E(4) N₂, A(1,2) = N₃ + E(4) N₄ and A(2,1) = - N₃ + E(4) N₄.

Then ComplexificationQuat(v)*ComplexificationQuat(M)= ComplexificationQuat(v*M), since

The algebra of octonions over F.

gap> OctaveAlgebra( Rationals );
<algebra of dimension 8 over Rationals>

is the full matrix algebra R ^{n ×n} , for a ring R and a nonnegative integer n.

gap> A:=FullMatrixAlgebra( Rationals, 20 );
( Rationals^[ 20, 20 ] )
gap> Dimension( A );
400

The zero-dimensional algebra over R.

gap> A:= NullAlgebra( Rationals );
<algebra over Rationals>
gap> Dimension( A );
0

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