Initial definitions

Let \(K\) be a field. A central simple \(K\)-algebra is a \(K\)-algebra \(B\) (not necessarily commutative) with centre \(K\) such that \(B\) has exactly two two-sided ideals, namely \({0}\) and \(B\) (or \(\bot\) and \(\top\), as Lean would call them). We will be concerned only with central simple \(K\)-algebras which are finite-dimensional as \(K\)-vector spaces, and when \(K\) is clear we will just refer to them as central simple algebras. We remark that a 4-dimensional central simple algebra is called a quaternion algebra; we will have more to say about these later on.

Matrix algebras \(M_n(K)\) are examples of finite-dimensional central simple \(K\)-algebras. If \(K=\mathbb{C}\) (or more generally if \(K\) is algebraically closed) then matrix algebras are the only finite-dimensional examples up to isomorphism. There are other examples over the reals: for example Hamilton's quaternions \(\mathbb{H}:=\mathbb{R}\oplus\mathbb{R} i\oplus\mathbb{R} j\oplus\mathbb{R} k\) with the usual rules \(i^2=j^2=k^2=-1\), \(ij=-ji=k\) etc, are an example of a central simple \(\mathbb{R}\)-algebra (and a quaternion algebra), and matrix algebras over \(\mathbb{H}\) are other central simple \(\mathbb{R}\)-algebras. For a general field \(K\) one can make an analogue of Hamilton's quaternions \(K\oplus Ki\oplus Kj\oplus Kk\) with the same multiplication rules (\(i^2=-1\) and so on) to describe the multiplication, and if the characteristic of \(K\) isn't 2 then this is a quaternion algebra (which may or may not be isomorphic to \(M_2(K)\) in this generality).

Some central simple algebras \(B\) are division algebras, meaning that they are division rings, or equivalently that every nonzero \(b\in B\) has a two-sided inverse. For example Hamilton's quaternions are a division algebra over \(\mathbb{R}\), because \((x+yi+zj+tk)(x-yi-zj-tk)=x^2+y^2+z^2+t^2\), so the inverse of a nonzero \(x+yi+zj+tk\) is \((x-yi-zj-tk)/(x^2+y^2+z^2+t^2)\). However \(2\times 2\) matrices over a field \(K\), whilst being a central simple algebra over \(K\), are never a division algebra (even if \(K=\mathbb{C}\)) because a nonzero matrix with determinant zero such as \(\begin{pmatrix}1&0\\0&0\end{pmatrix}\) has no inverse.

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