Generic Operations

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: Star Alg (ex-ffb750)

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)

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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: Ten Generic (ex-099024)

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

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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 \(yR_x:=x*y\).

> 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

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