The transformation algebra of a division algebra

The transformation algebra of an algebra A is the subalgebra of the

algebra of linear endomorphisms of A generated by all left and right

multiplications with elements in A. It was introduced by Albert as a part

of an effort to create a unified structure theory for non-associative

algebras.

One problem with the transformation algebra is that it is a very crude

invariant for general algebras. In the talk, I shall suggest a way to

compensate for this and show that by adding certain information, the

transformation algebra can be used to give a complete picture of the

category of unital division algebras of fixed (finite) dimension over a

field.