CENTRAL SIMPLE ALGEBRA

In ring theory and related areas of mathematics a 'central simple algebra' ('CSA') over a field ''K'' (also called a 'Brauer algebra' after Richard Brauer), is a finite-dimensional associative algebra ''A'', which is simple, and for which the center is exactly ''K''. In other words, any simple algebra is a central simple algebra over its center.
For example, the complex numbers 'C' form a CSA over themselves, but not over the real numbers 'R' (the center of 'C' is all of 'C', not just 'R'). The quaternions 'H' form a 4 dimensional CSA over 'R'.
According to the Artin–Wedderburn theorem a simple algebra ''A'' is isomorphic to ''M''(''n'',''S'') for some division ring ''S''. Given two central simple algebras ''A'' ~ ''M''(''n'',''S'') and ''B'' ~ ''M''(''m'',''T'') over the same field ''F'' , ''A'' and ''B'' are called 'similar' (or 'Brauer equivalent') if their division rings ''S'' and ''T'' are isomorphic. The set of all equivalence classes of central simple algebras over a given field ''F'', under this equivalence relation, can be equipped with a group operation given by the tensor product of algebras. The resulting group is called the Brauer group Br(''F'') of the field ''F''.

Contents

Properties

See also

Properties

★ Every automorphism of a central simple algebra is an inner automorphism (follows from Skolem-Noether theorem)

★ If ''S'' is a simple subalgebra of a central simple algebra ''A'' then dim_''F''''S'' divides dim_''F''''A''

★ Every 4 dimensional central simple algebra over a field ''F'' is isomorphic to a quaternion algebra; in fact, it is either a two-by-two matrix algebra, or a division algebra

See also

★ Brauer group

★ Severi-Brauer variety