Matrix algebra as Clifford algebra

Question

Many kinds of Clifford algebras have corresponding sub-algebras of matrix algebras in sense of isomorphism. Say, quaternion, spacetime algebra and also Dirac algebra. Generally, Clifford algebra has matrix representation. Is the converse still true? Is there any sub-algebra of Clifford algebra isomorphic to matrix algebra of $n$ dimension?

Answer 1

It depends to your Clifford algebra. The Clifford algebra of a hyperbolic quadratic form $q$ of dimension $2n$ over a field $k$ is isomorphic to the full matrix algebra $M_{2^{2n}}(k)$. By contrast it may happen that a Clifford algebra be a division algebra, so it cannot contain a full matrix subalgebra.It also depends on your base field, the Clifford algebra of the form $q=-x^2-y^2$ over the field of real numbers is isomorphic to the division algebra of hamiltonian quaternion. The Clifford algebra of the same form considered over the field of complex numbers is isomorphic to the full matrix algebra $M_2(\mathbb{C})$. Also if you are just interested to investigate the Clifford algebra of of the standard quadratic forms $q=x_1^2+\cdots+x_n^2$ (or $q=-x_1^2-\cdots-x_n^2$) over $\mathbb{R}$ you can use Bott periodicity.

Answer 2

The conventional real Clifford algebras are matrix algebras over the real numbers, complex numbera, or quaternions, and are simple algebras in the sense of ring theory. Any subalgebra that is a simple algebra will also be such a matrix algebra. And conversely: those types of rings are all the simple algebras, up to isomorphism, that are finite dimensional over the reals.