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[parent] division algebra (Definition)

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.

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$,
$\displaystyle a1=1a=a.$
(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
$\displaystyle ab=1=ba.$
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$,

$\displaystyle \left(\frac{\alpha,\beta}{K}\right)=\left\{ a_1 1+a_2 i+a_3 j+a_4 k : i^2=\alpha 1, j^2=\beta 1, k^2=-\alpha \beta 1, ij=k=-ji.\right\}$
where $ x^2-\alpha$ and $ x^2-\beta$ are irreducible over $ K$.

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$,

$\displaystyle 1.a=a=a.1.$

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

$\displaystyle a.b=1=b.a$ (1)

and furthermore that for all $ x\in A$,
$\displaystyle b.(a.x)=x=(x.a).b.$ (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)=x L_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. $ \qedsymbol$

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

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.



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See Also: octonion, octonion

Also defines:  division algebra

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Cross-references: algebraically closed, finite field, finite, octonions, quaternions, composition algebras, bijective, surjective, injective, identity map, forces, finite dimensional, sufficient, Schur's lemma, sort, solutions, equations, additive, distributive, endomorphisms, invertible, product, non-associative algebras, irreducible, quaternion algebra, commutative, fields, inverse, identity, unital, algebra, associative, non-associative, algebras, unital ring
There are 13 references to this entry.

This is version 3 of division algebra, born on 2007-03-27, modified 2008-04-15.
Object id is 9117, canonical name is DivisionAlgebra.
Accessed 1248 times total.

Classification:
AMS MSC: 16K99 (Associative rings and algebras :: Division rings and semisimple Artin rings :: Miscellaneous)
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