Structure constants

Are algebras given by structure constants better?

Magma can work with finite-dimensional algebras, but this can be tricky. Let us start with the quaternion algebra $\langle i,j: i^2=j^2=-1, ij=-ji\rangle$. We can introduce this algebra by generators and relations:

> F<a,b> := FreeAlgebra(Rationals(), 2);
> A<i,j> := quo<F|a^2+1,b^2+1,a*b+b*a>;
> i^2;
-1
> j^2;
-1
> i*j;
-j*i
> Dimension(A);
4

But the problem is that we cannot do very much with this presentation: asking for a basis, the center, or the Jacobson radical produces an error message.

As an alternative, we can construct the algebra using its structure constants. Recall that if $A$ is an $n$-dimensional algebra with basis $e_1,\dots,e_n$, its structure constants are the scalars $a_{ijk}$ such that $e_ie_j=\sum_{k=1}^na_{ijk}e_k$ for all $1\leq i,j\leq n$. Let us construct the array of structure constants of the quaternion algebra. For this, let $n=4$, $e_1=1$, $e_2=i$, $e_3=j$ and $e_4=k$. Since $e_2e_1=e_2$ we get that $a_{211}=a_{213}=a_{214}=0$ and $a_{212}=1$. Similarly, since $e_2^2=-e_1$, $a_{221}=-1$ and $a_{222}=a_{223}=a_{224}=0$. In this way we get the structure constants:

> a := [[[ 0 : k in [1..4]] : j in [1..4]] : i in [1..4]];
> a[1][1][1] := 1;
> a[1][2][2] := 1;
> a[1][3][3] := 1;
> a[1][4][4] := 1;
> a[2][1][2] := 1;
> a[2][2][1] := -1;
> a[2][3][4] := 1;
> a[2][4][3] := -1;
> a[3][1][3] := 1;
> a[3][2][4] := -1;
> a[3][3][1] := -1;
> a[3][4][2] := 1;
> a[4][1][4] := 1;
> a[4][2][3] := 1;
> a[4][3][2] := -1;
> a[4][4][1] := -1;

Now we can construct the quaternion algebra as the algebra with that particular structure constants:

> A := Algebra< Rationals(), 4 | a >;
> Dimension(A);
4
> Basis(A);
[ (1 0 0 0), (0 1 0 0), (0 0 1 0), (0 0 0 1) ]
> JacobsonRadical(A);
Algebra of dimension 0 with base ring Rational Field
> Center(A);
Algebra of dimension 1 with base ring Rational Field
> IsSimple(A);
true

Of course, we could have done this:

> A<i,j> := QuaternionAlgebra<Rationals()|-1, -1>;
> Basis(A);
[ 1, i, j, k ]
> Center(A);
Associative Algebra of dimension 1 with base ring Rational Field
> JacobsonRadical(A);
Associative Algebra of dimension 0 with base ring Rational Field
> IsSimple(A);
true