Construction
A central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A, which is a simple ring, and for which the center is exactly K. Note that CSAs are in general not division algebras, though CSAs can be used to classify division algebras.
For example, the complex numbers C form a CSA over themselves, but not over R (the center is C itself, hence too large to be CSA over R). The finite-dimensional division algebras with center R (that means the dimension over R is finite) are the real numbers and the quaternions by a theorem of Frobenius, while any matrix ring over the reals or quaternions – M(n,R) or M(n,H) – is a CSA over the reals, but not a division algebra (if ).
We obtain an equivalence relation on CSAs over K by the Artin–Wedderburn theorem (Wedderburn's part, in fact), to express any CSA as a M(n,D) for some division algebra D. If we look just at D, that is, if we impose an equivalence relation identifying M(m,D) with M(n,D) for all integers m and n at least 1, we get the Brauer equivalence and the Brauer classes.
Given central simple algebras A and B, one can look at the their tensor product A ⊗ B as a K-algebra (see tensor product of R-algebras). It turns out that this is always central simple. A slick way to see this is to use a characterisation: a central simple algebra over K is a K-algebra that becomes a matrix ring when we extend the field of scalars to an algebraic closure of K.
Given this closure property for CSAs, they form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse class to that of an algebra A is the one containing the opposite algebra Aop (the opposite ring with the same action by K since the image of K → A is in the center of A). In other words, for a CSA A we have A ⊗ Aop = M(n2,K), where n is the degree of A over K. (This provides a substantial reason for caring about the notion of an opposite algebra: it provides the inverse in the Brauer group.)
Read more about this topic: Brauer Group
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