# 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

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

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

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 $x{L}_{1}=1.x=x=b.(a.x)=x{L}_{a}{L}_{b}$. 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^{}.
∎

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 |