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".

Properties

* Every automorphism of a central simple algebra is an inner automorphism (follows from Skolem-Noether theorem)
* The dimension of a central simple algebra as a vector space over its centre is always a square
* 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