Composition Algebras

CompositionAlgebra(K, a) : Fld, [FldElt] -> AlgGen
CompositionAlgebra(K, a) : Fld, [RngIntElt] -> AlgGen
Constructs the composition algebra with specified parameters. The algebra returned has an involution.

The method is modestly intentional choosing Magma's favored representation of the individually classified algebras according to Hurwitz's theorem. In the case of fields the type returned is an algebra with involution, possibly the identity.

OctonionAlgebra(K, a, b, c) : Fld, FldElt, FldElt, FldElt -> AlgGen
OctonionAlgebra(K, a, b, c) : Fld, RngIntElt, RngIntElt, RngIntElt -> AlgGen
Octonion algebra with involution given by the specified parameters. This builds the Cayley-Dickson algebra over the quaternion algebra (a/(b)(K)). In particular, Magma's implementation of quaternion algebras is applied.
SplitOctonionAlgebra(K) : Fld -> AlgGen
Returns the split octonion algebra over the field F.

Example AlgNAss_Ten_Triality (H96E1)

The following example demonstrates some of the mechanics by exploring the concept of triality [Sch66, III.8].

The Cartan-Jacobson theorem asserts that for fields of characteristic other than 2 and 3, the derivation algebra of an octonion algebra is of Lie type G2. 

> O := OctonionAlgebra(GF(7),-1,-1,-1);
> L := DerivationAlgebra(O);   // Derivations as an algebra.
> SemisimpleType(L);
G2
Cartan's triality obtains G2 from D4 by relaxing to derivations of the octonions as a generic tensor, rather than as an algebra. This is done computationally by changing the category of the octonion product from an algebra to a tensor. 
> T := Tensor(O);
> T := ChangeTensorCategory(T,HomotopismCategory(3));
> M := DerivationAlgebra(T);  // Derivations as a tensor.
> SemisimpleType(M/SolvableRadical(M));
D4
V2.28, 13 July 2023