# Division algebras

A (finite-dimensional unital) division algebra is an algebra with multiplicative identity where unique right and left inverses exist for every non-zero element. For an associative division algebra, these inverses are equal, turning the non-zero vectors into a group under multiplication. Ignoring scalar multiplication, an associative algebra is a ring, and a commutative associative division algebra is a field; thus an associative division algebra with scalar multiplication ignored is sometimes called a division ring or skew field. A module over a non-commutative skew field (such as $${\mathbb{H}}$$) can be seen to have much of the same features as a vector space, including a basis.

In a division algebra, the existence of the left inverse $${u_{L}^{-1}}$$ of $${u}$$ allows us to “divide” elements in the sense that for any non-zero $${u}$$ and $${v}$$, $${xu=v}$$ has the solution $${x=vu_{L}^{-1}}$$, which we can regard as the “left” version of $${v/u}$$; similarly, $${ux=v}$$ has the solution $${x=u_{R}^{-1}v}$$. This is equivalent to requiring that there be no zero divisors, i.e. $${uv=0\Rightarrow u=0}$$ or $${v=0}$$. A normed division algebra has a vector space norm that additionally satisfies $${\left\Vert uv\right\Vert =\left\Vert u\right\Vert\left\Vert v\right\Vert }$$.

 Δ Division algebras are often defined more generally, with possibly infinite dimension and no unique multiplicative identity assumed; the above definitions then do not apply.

Real division algebras are highly constrained: all have dimension 1, 2, 4, or 8; the commutative ones all have dimension 1 or 2; the only associative ones are $${\mathbb{R}}$$, $${\mathbb{C}}$$, and $${\mathbb{H}}$$; and the only normed ones are $${\mathbb{R}}$$, $${\mathbb{C}}$$, $${\mathbb{H}}$$, and $${\mathbb{O}}$$. Here we review these division algebras:

• $${\mathbb{C}}$$, the complex numbers, has basis $${\left\{ 1,i\right\} }$$ where $${i^{2}\equiv-1}$$
• $${\mathbb{H}}$$, the quaternions, has basis $${\left\{ 1,i,j,k\right\} }$$ where $${i^{2}=j^{2}=k^{2}=ijk\equiv-1}$$
• $${\mathbb{O}}$$, the octonions, has basis $${\left\{ 1,i,j,k,l,li,lj,lk\right\} }$$, all anti-commuting square roots of $${-1}$$; we will not describe the full multiplication table here

We can define the quaternionic conjugate by reversing the sign of the $${i}$$, $${j}$$, and $${k}$$ components, with the octonionic conjugate defined similarly. The norm is then defined by $${\left\Vert v\right\Vert =\sqrt{vv^{*}}}$$ in these algebras, as it is in $${\mathbb{C}}$$. Note that this implies two sided inverses for all normed real division algebras, namely $${v^{-1}=v^{*}/\left\Vert v\right\Vert ^{2}}$$.

We lose a property of the real numbers each time we increase dimension in the above algebras: $${\mathbb{C}}$$ is not ordered; $${\mathbb{H}}$$ is not commutative; and $${\mathbb{O}}$$ is not associative. It turns out that any subalgebra of $${\mathbb{O}}$$ generated by two elements, however, is in fact associative.

$${\mathbb{C}}$$ is a field (ignoring real scalar multiplication) and so can be used as the scalars in a vector space $${\mathbb{C}^{n}}$$. One could imagine then trying to find a multiplication on $${\mathbb{C}^{n}}$$ to obtain complex division algebras, but the only finite-dimensional complex division algebra is $${\mathbb{C}}$$ itself.

The quaternions form a non-commutative ring, and so can be used as the scalars in a left module $${\mathbb{H}^{n}}$$, but there is no obvious definition of $${\mathbb{O}^{n}}$$ since the octonions are not associative. However, we can form all of the algebras $${\mathbb{R}\left(n\right)}$$, $${\mathbb{C}\left(n\right)}$$, $${\mathbb{H}\left(n\right)}$$, and $${\mathbb{O}\left(n\right)}$$, where $${\mathbb{K}\left(n\right)}$$ denotes the algebra of $${n\times n}$$ matrices with entries in $${\mathbb{K}}$$. $${\mathbb{H}\left(n\right)}$$ can even be viewed as the group of linear transformations on $${\mathbb{H}^{n}}$$, if $${\mathbb{H}^{n}}$$ is defined as a right module while matrix multiplication takes place from the left as usual.