Introduction

Let A be an algebra over a field K and let M be a vector space over K. We say that M is a (right) A-module if for each a in A and m in M, a product ma in M is defined such that (m + n)a = ma + aan, m(a + b) = ma + mb, m(ab) = (ma)b, m1 = m, m(ka) = (ma)k = (mk)a, for all a, b in A, m, n in M, k in K.

Recall that a representation of an algebra A over a field K is an algebra homomorphism of A into Hom_K(M, M), for some K-module M. Taking M to be an A-module, and defining a mapping rho: M -> M by rho(m) := ma (a in A, m in M) then it is an easy exercise to show that rho is a representation of A. A matrix representation of degree n of the algebra A is an algebra homomorphism of A into M_n(K), the complete matrix algebra of degree n over K. Suppose M has finite K-dimension n and choose a basis for M. If, for each a in A, we associate the matrix corresponding to the action of a on the basis elements, we obtain a matrix representation of A. Thus, each A-module of finite K-dimension affords a matrix representation of the algebra A. An important special case occurs when A=K[G], the group algebra of a group G. In this case the theory of A-modules coincides with the theory of group representations.

Throughout this chapter we shall use the term A-module when referring to modules as defined above. For Magma V version, A-modules are restricted to one of the following cases:

• A matrix algebra A defined over a field.

• A matrix algebra A defined over a field corresponding to a representation of a group G of finite degree. In this case the user is (implicitly) computing with the group algebra K[G].

• A matrix algebra A defined over an Euclidean Domain R. However, as currently the action of A may be used only in the construction of submodules, discussion will be limited to the case in which the coefficient ring is a field.

Magma provides a range of facilities for defining and computing with A-modules. In particular, extensive machinery is provided for creating K[G]-modules which is described in the following chapter. It should be noted however, that many advanced functions only apply when A is an algebra over a finite field. Since the R-module of n-tuples, R^((n)), underlies an A-module, the operations for R^((n)) are also applicable.