division algebra

Let $K$ be a unital ring and $A$ a $K$ -algebra. Defining “division” requires special considerations when the algebras are non-associative so we introduce the definition in stages.

1 Associative division algebras

If $A$ is an associative algebra then we say $A$ is a division algebra if

(i)

$A$ is unital with identity $1$ . So for all $a\in A$ ,

$a1=1a=a.$
(ii)

Also every non-zero element of $A$ has an inverse. That is $a\in A$ , $a\neq 0$ , then there exists a $b\in A$ such that

$ab=1=ba.$

We denote $b$ by $a^{-1}$ and we may prove $a^{-1}$ is unique to $a$ .

The standard examples of associative division algebras are fields, which are commutative, and the non-split quaternion algebra: $\alpha,\beta\in K$ ,

\left(\frac{\alpha,\beta}{K}\right)=\left\{a_{1}1+a_{2}i+a_{3}j+a_{4}k:i^{2}=% \alpha 1,j^{2}=\beta 1,k^{2}=-\alpha\beta 1,ij=k=-ji.\right\}

where $x^{2}-\alpha$ and $x^{2}-\beta$ are irreducible over $K$ .

2 Non-associative division algebras

For non-associative algebras $A$ , the notion of an inverse is not immediate. We use $x . y$ for the product of $x,y\in A$ .

Invertible as endomorphisms: Let $a\in A$ . Then define $L_{a}:x\mapsto a.x$ and $R_{a}:x\mapsto x.a$ . As the product of $A$ is distributive, both $L_{a}$ an $R_{a}$ are additive endomorphisms of $A$ . If $L_{a}$ is invertible then we may call $a$ “left invertible” and similarly, when $R_{a}$ is invertible we may call $a$ “right invertible” and “invertible” if both $L_{a}$ and $R_{a}$ are invertible.

In this model of invertible, $A$ is a division algebra if, and only if, for each non-zero $a\in A$ , both $L_{a}$ and $R_{a}$ invertible. Equivalently: the equations $a.x=b$ and $y.a=b$ have unique solutions for nonzero $a,b\in A$ . However, $x$ and $y$ need not be equal.

A common method to produce non-associative division algebras of this sort is through Schur’s Lemma.

Invertible in the product: In some instances, the notion of invertible via endomorphisms is not sufficient. Instead, assume $A$ has an identity, that is, an element $1\in A$ such that for all $a\in A$ ,

1.a=a=a.1.

Next if $a\in A$ , we say $a$ is invertible if there exists a $b\in A$ such that

a.b=1=b.a

(1)

and furthermore that for all $x\in A$ ,

b.(a.x)=x=(x.a).b.

(2)

Evidently (1) can be inferred from (2). This added assumption substitutes for the need of associativity in the proofs of uniqueness of inverses and in solving equations with non-associative products.

Proposition 1.

If $A$ is a finite dimensional algebra over a field, then invertible in this sense forces both $L_{a}$ and $R_{a}$ to be invertible as well.

Proof.

Let $x\in A$ . Then $xL_{1}=1.x=x=b.(a.x)=xL_{a}L_{b}$ . So $L_{1}=L_{a}L_{b}$ . As $L_{1}$ is the identity map, $L_{a}$ is injective and $L_{b}$ is surjective. As $A$ is finite dimensional, injective and surjective endomorphisms are bijective. ∎

In this model, a non-associative algebra is a division algebra $A$ if it is unital and every non-zero element is invertible.

3 Alternative division algebras

The standard examples of non-associative division algebras are actually alternative alegbras, specfically, the composition algebras of fields, non-split quaternions and non-split octonions – only the latter are actually not associative. Invertible in the octonions is interpreted in the second stronger form.

Theorem 2 (Bruck-Klienfeld).

Every alternative division algebra is either associative or a non-split octonion.

This result is usually followed by two useful results which serve to omit the need to consider non-associative examples.

Theorem 3 (Artin-Zorn, Wedderburn).

A finite alternative division algebra is associative and commutative, so it is a finite field.

Theorem 4.

An alternative division algebra over an algebraically closed field is the field itself.

Title	division algebra
Canonical name	DivisionAlgebra
Date of creation	2013-03-22 16:52:03
Last modified on	2013-03-22 16:52:03
Owner	Algeboy (12884)
Last modified by	Algeboy (12884)
Numerical id	6
Author	Algeboy (12884)
Entry type	Definition
Classification	msc 16K99
Related topic	octonion
Related topic	Octonion
Defines	division algebra