Division algebras

A division algebra is an algebra with a 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 \({\left\Vert uv\right\Vert =\left\Vert u\right\Vert\left\Vert v\right\Vert }\).

Finite-dimensional 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 \({\mathbb{C}}\).

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. \({\mathbb{C}}\) 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.

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