Generic Operations

Contents

Nonassociative Algebras with Involutions

IsStarAlgebra(A) : AlgGen -> BoolElt
Decides if algebra has an involution, i.e. a *-algebra.
Star(A) : AlgGen -> Map
Returns involution of given *-algebra.

Example AlgNAss_Star_Alg (H96E7)

We demonstrate the functions dealing with involutions of nonassociative algebras. 
> A := OctonionAlgebra(Rationals(),-1,-1,-1);
> IsStarAlgebra(A);
true
>
> s := Star(A);
> A.1; // A.1 is the mult. id.
(1 0 0 0 0 0 0 0)
> A.1 @ s;
(1 0 0 0 0 0 0 0)
>
> A.2;
(0 1 0 0 0 0 0 0)
> A.2 @ s;
( 0 -1  0  0  0  0  0  0)

Operations on Power Associative Algebras

The following operations are defined for nonassociative algebras for which x * (x * x)=(x * x) * x.

GenericMinimalPolynomial(x) : AlgGenElt -> FldElt
The generic minimum polynomial of an element in a power associative algebra.
GenericNorm(x) : AlgGenElt -> FldElt
The generic norm of an element in a power associative algebra.
GenericTrace(x) : AlgGenElt -> FldElt
The generic trace of an element in a power associative algebra.
GenericTracelessSubspaceBasis(A) : AlgGen -> Any
Given a power associative algebra return a basis for the elements of generic trace 0.

Example AlgNAss_Ten_Generic (H96E8)

The trace x + bar(x) of a quaternion doubles the rational component, producing degenerate behavior in characteristic 2. The generic trace avoids this. 
> Q := QuaternionAlgebra(Rationals(), 1,1);
> Trace(Q!1);
2
> GenericTrace(Q!1);
1
> Q := QuaternionAlgebra(GF(2), 1,1);
> Trace(Q!1);
0
> GenericTrace(Q!1);
1
The generic minimum polynomial of an element x in power associative algebra need only be a factor of the minimal polynomial of its right regular matrix yRx:=x * y. 
> J := ExceptionalJordanCSA(GF(5));
> p := GenericMinimumPolynomial(J.3+J.12);
> Rx := AsMatrices(Tensor(J), 2,0);     // yR_x = y*x.
> q := MinimalPolynomial(Rx[3]+Rx[12]);
> Degree(p);
3
> Degree(q);
6
> q mod p;
0
V2.28, 13 July 2023