1 Associative division algebras
2 Non-associative division algebras
Invertible as endomorphisms: Let . Then define and . As the product of is distributive, both an are additive endomorphisms of . If is invertible then we may call “left invertible” and similarly, when is invertible we may call “right invertible” and “invertible” if both and are invertible.
In this model of invertible, is a division algebra if, and only if, for each non-zero , both and invertible. Equivalently: the equations and have unique solutions for nonzero . However, and need not be equal.
A common method to produce non-associative division algebras of this sort is through Schur’s Lemma.
Next if , we say is invertible if there exists a such that
and furthermore that for all ,
If is a finite dimensional algebra over a field, then invertible in this sense forces both and to be invertible as well.
In this model, a non-associative algebra is a division algebra 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.
An alternative division algebra over an algebraically closed field is the field itself.
|Date of creation||2013-03-22 16:52:03|
|Last modified on||2013-03-22 16:52:03|
|Last modified by||Algeboy (12884)|