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 ∈A and m ∈M, a product ma ∈M is defined such that (m + n)a = ma + na, m(a + b) = ma + mb, m(ab) = (ma)b, m1 = m, m(ka) = (ma)k = (mk)a, for all a, b ∈A, m, n ∈M, k ∈K.
Recall that a representation of an algebra A over a field K is an algebra homomorphism of A into HomK(M, M), for some K-module M. Taking M to be an A-module, and defining a mapping ρ: M -> M by ρ(m) := ma (a ∈A, m ∈M) then it is an easy exercise to show that ρis a representation of A. A matrix representation of degree n of the algebra A is an algebra homomorphism of A into Mn(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 ∈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:
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.
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