Algebra II : Noncommutative Rings Identities PDF

The algebra of square matrices of size n ~ 2 over the field of complex numbers is, evidently, the best-known example of a non-commutative alge- 1 bra * Subalgebras and subrings of this algebra (for example, the ring of n x n matrices with integral entries) arise naturally in many areas of mathemat- ics.

Historically however, the study of matrix algebras was preceded by the discovery of quatemions which, introduced in 1843 by Hamilton, found ap- plications in the classical mechanics of the past century.

Later it turned out that quaternion analysis had important applications in field theory.

The al- gebra of quaternions has become one of the classical mathematical objects; it is used, for instance, in algebra, geometry and topology.

We will briefly focus on other examples of non-commutative rings and algebras which arise naturally in mathematics and in mathematical physics.

The exterior algebra (or Grassmann algebra) is widely used in differential geometry - for example, in geometric theory of integration.

Clifford algebras, which include exterior algebras as a special case, have applications in rep- resentation theory and in algebraic topology.

The Weyl algebra (Le. algebra of differential operators with* polynomial coefficients) often appears in the representation theory of Lie algebras.

In recent years modules over the Weyl algebra and sheaves of such modules became the foundation of the so-called microlocal analysis.

The theory of operator algebras (Le.