Georgi E. Shilov - Linear Algebra

A.11. Recently the concept of a category and certain related ideas have begun to play an important role in various branches of mathematics. An example of a category is a collection of sets together with mappings of the sets into one another. A collection of linear spaces or algebras together with their morphisms is another example of a category.

The exact definition of a category is as follows: Let be a set of indices , and let be a set of elements X ( called objects of the category . Suppose that for every pair of objects X , and X there is a set of other elements A called mappings of the object X into the object X such that the product of the mappings A and A is defined for arbitrary , , and belongs to , where multiplication is associative, i.e.,

A ( A A ) = ( A A ) A )

for arbitrary , , , . ln particular, the set of mappings of the objects X into themselves is defined, and (associative) multiplication of mappings is defined in . Finally, it is required that the set contain the unit element , which has the property that

A = A , A = A

for arbitrary , and . Instead of we will usually write simply .

A set of objects X and mappings A with the properties just enumerated is called a category. A category is called linear if in the set of mappings A (with arbitrary fixed and ) there are defined operations of addition of mappings and multiplication of mappings by numbers (from the field K). This makes the set into a linear space over the field K. Thus in a linear category the set becomes an algebra with a unit (over the field K ).

A.12 . In this appendix we will consider linear categories whose elements are finite-dimensional linear spaces (of dimension 1) over the field C of complex numbers, while the mappings are linear mappings (morphisms) of one such space into another.

Thus we start with the following definition: Let X ( be a set of finite-dimensional complex linear spaces, and for every let be an algebra of linear operators carrying X into itself. Moreover, suppose that for every pair of indices and there is a set of linear operators A carrying X into X such that I) if contains the operators A and B , then contains the operator sum A + B , and 2) if contains the operator A , then contains the product A where is an arbitrary complex number. A family of linear operators with these two properties will be called a linear family. In particular, the linear family coincides with the algebra . It is also assumed that

(1)

for arbitrary , and , i.e., that every product

A A ( A , A )

belongs to . Such a set of spaces X together with algebras and linear families will be called a category of finite-dimensional spaces or simply a category , and will be denoted by .

If we choose a basis in every space X, then the algebras and linear families can be identified with the algebras and linear families of the corresponding matrices, a fact which will henceforth be exploited systematically.

In what follows, we will find the categories of linear spaces corresponding to given algebras , confining ourselves to the case where the are semisimple algebras containing the unit matrix. According to Sec. 11.86, for such an algebra the space X can be decomposed into a direct sum of subspaces X, invariant under all the operators A, where in each subspace Xj the algebra is a simple algebra containing the unit matrix, i.e., is described in some basis by the set of all quasi-diagonal matrices of the form