Representation of One Endomorphism
We deal here with one endomorphism of a module, actually a free module, and especially a finite dimensional vector space over a field k. We obtain the Jordan canonical form for a representing matrix, which has a particularly simple shape when k is algebra
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XIV
Representation of One Endomorphism
We deal here with one endomorphism of a module , actually a free module, and especially a finite dimensional vector space over a field k. We obtain the Jordan canonical form for a representing matrix , which has a particularly simple shape when k is algebraically closed. This leads to a discussion of eigenvalues and the characteristic polynomial. The main theorem can be viewed as giving an example for the general structure theorem of modules over a principal ring. In the present case, the principal ring is the polynomial ring k[X] in one variable .
§1 .
REPRESENTATIONS
Let k be a commutative ring and E a module over k. As usual, we denote by Endk(E) the ring of k-endomorphisms of E, i.e. the ring of k-linear maps of E into itself. Let R be a k-algebra (given by a ring-homomorphism k -> R which allows us to consider R as a k-module). Bya representation of R in E one means a kalgebra homomorphism R -> Endk(E), that is a ring-homomorphism
which makes the following diagram commutative :
~ / k
553 S. Lang, Algebra © Springer Science+Business Media LLC 2002
554
REPRESENTATION OF ONE ENDOMORPHISM
XIV, §1
[As usual , we view Endk(E) as a k-algebra ; if I denotes the identity map of E, we have the homomorphism of k into Endk(E) given by a f--+ aI. We shall also use I to denote the unit matrix if bases have been chosen. The context will always make our meaning clear.] We shall meet several example s of representations in the sequel, with various types of rings (both commutative and non-commutative). In this chapter, the rings will be commutative. We observe that E may be viewed as an Endk(E) module. Hence E may be viewed as an R-module, defining the operation of R on E by letting (x, v) f--+ p(x) v
for x E R and VEE. We usually write xv instead of p(x) v. A subgroup F of E such that RF c F will be said to be an invariant submodule of E. (It is both R-invariant and k-invariant.) We also say that it is invariant under the representation. We say that the representation is irreducible, or simple, if E =1= 0, and if the only invariant submodules are and E itself. The purpose of representation theories is to determine the structure of all representations of various interesting rings, and to classify their irreducible representations. In most cases, we take k to be a field, which mayor may not be algebraically closed. The difficulties in proving theorems about representations may therefore lie in the complication of the ring R, or the complication of the field k, or the complication of the module E, or all three. A representation p as above is said to be completely reducible or semi-simple if E is an R-direct sum of R-submodules E i ,
°
E = E 1 EEl .• . EEl Em
such that each E, is irreducible. We also say that E is completely reducible. It is not true that all representations are completely reducible, and in fact those considered in this chapter will not be in general. Certain types of completely reducible representations will be studied later. There is a special type of repre
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