Definition:Unitary Division Algebra

Definition

Let $\struct {A_F, \oplus}$ be a division algebra.

Then $\struct {A_F, \oplus}$ is a unitary division algebra if and only if it has an identity element $1_{A_F}$ called a unit for $\oplus$, that is:

$\exists 1_{A_F} \in A_F: \forall a \in A_F: a \oplus 1_{A_F} = 1_{A_F} \oplus a = a$

The unit is usually denoted $1$ when there is no source of confusion with the identity elements of the underlying structures of the division algebra.

Also known as

Some sources use this as the definition of division algebra.

That is, its unitary nature is subsumed.

Also see

• Results about unitary division algebras can be found here.

Sources

• John C. Baez: The Octonions (2002): 1.1 Preliminaries