PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (205)
Orphanage
Unclass'd
Unproven (490)
Corrections (8)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
[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.



"division algebra" is owned by Algeboy.
(view preamble)

View style:

See Also: octonion, octonion

Also defines:  division algebra

This object's parent.
Log in to rate this entry.
(view current ratings)

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 1437 times total.

Classification:
AMS MSC: 16K99 (Associative rings and algebras :: Division rings and semisimple Artin rings :: Miscellaneous)
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)