6. Commutative versus noncommutative systems

For the integers, it is the case that a × b is equal to b × a: the order of the factors is unimportant. We say that the multiplication is commutative. But when is comes to matrices, the order of the factors is important. For the matrices on the previous page, one finds, with a bit of computation, that A × B is not equal to B × A. Thus, the multiplication is not commutative.

And here we have a fundamental distinction. The theory for commutative algebraic systems is quite different from the theory for noncommutative systems. Commutative algebraic systems, which the polynomials constitute, are closely associated with geometric objects. This relationship is studied in algebraic geometry. Algebraic properties and geometric properties reflect one another and are really just two sides of the same coin.

Noncommutative systems are also related to geometry. But while commutative algebraic systems in algebraic geometry are related to the geometry in an intrinsic way, those in noncommutative geometry are related to the geometry in a more extrinsic way, e.g. operations on geometric objects; often such operations give symmetries of the geometric object. Thus, much of the algebra in modern theoretical physics is noncommutative.

In Bergen and Oslo, the commutative algebra related to algebraic geometry has traditionally been studied. In recent years, we have also studied the interaction between commutative algebra and combinatorics. In Trondheim, the focus has been on noncommutative algebra. There is a large group there that studies Artinian algebra.

Read on about Lie algebras.