Semisimple

Therefore it is shown that each closed orbit contains an element (i i, B2,0,0) such that both Bi,B2 are semisimple satisfying [ 1,52] = 0. Associating the orbit with the set of simultaneous eigenvalues of (Hi, i 2), we have the claimed isomorphism. ... 

Here, the connection or the gauge potential A assuming values in a(n irreducible) representation R of the compact, semisimple Lie algebra g of the Lie group G is of the form... 

In Section 8.3, we focus on completely reducible modules over associative rings with 1. Section 8.4 deals with irreducible modules over these rings. In Section 8.5, we combine the results obtained in these two sections to obtain the famous (and complete) description of semisimple associative rings with 1 which was first given by Joseph Wedderburn and Emil Artin. 

The interest in semisimple rings is based on Theorem 9.1.5(h), a result in which we shall see that each scheme of finite valency gives rise to semisimple rings, the so-called scheme rings. 

In Section 8.6, we shall provide a few basic facts on characters of semisimple rings. 

An associative ring D with 1 is called semisimple if the D-module D is completely reducible. 

In this section, we shall combine our results about completely reducible modules obtained in Section 8.3 with those about irreducible modules (and artinian simple rings) obtained in Section 8.4 in order to give a complete description of semisimple rings. 

Proposition 8.5.2 Assume D to be semisimple. Then each maximal homogeneous submodule of the D-module D is a minimal ideal of D. 

The following two theorems are the main theorems about semisimple associative rings with 1. They are due to Emil Artin cf. [2]. Less general versions have been given earlier by Joseph Wedderburn cf. [39 Theorem 10] and [39 Theorem 17]. 

Since D is assumed to be semisimple, the D-module D is completely reducible. Thus, by Lemma 8.3.2, H is completely reducible. Thus, by Proposition 8.3.4, H is a direct sum of irreducible submodules. Now, considering that II is a ring with 1, the claim follows from Lemma 8.5.1. 

Theorem 8.5.3 together with Theorem 8.5.4 gives a complete picture about semisimple rings. The following theorem gives a complete picture about the collection of all modules over semisimple rings. 

Looking at Theorem 8.5.3(i), (iv) and Theorem 8.5.6(i) one has a complete picture about the set of all irreducible modules over a semisimple ring. Thus, if all these irreducible modules are finitely generated vector spaces over C,... 

For the remainder of this section, we shall now assume D to be semisimple and to be finitely generated as a vector space over C. 

In the first section of this chapter, we shall prove that CS is semisimple if the characteristic of the field C does not divide any of the integers s with s 6 S. This enables us to refer to some of the results about semisimple rings which we obtained in Section 8.5. We shall also see that C C Z(CS), so that we may speak about characters of CS and refer to results about characters which we obtained in Section 8.6. The results of the first section include the orthogonality relations for fields the characteristic of which does not divide any of the integers s with s S S. 

For the remainder of this section, we assume that, for each element s in S, the characteristic of C does not divide s. According to Theorem 9.1.5(h), this implies that OS is semisimple so that we may refer to results obtained in Section 8.5 and Section 8.6. 

Let x be an irreducible character of CS. Since CS is semisimple, we obtain from Theorem 8.6.2(i) together with Theorem 8.6.4(h) that there exists exactly one maximal homogeneous submodule Hx of the OS -module CS such... 

Consider connected closed subgroups H of G which are normal and solvable. If Ht and H2 are such, so is (the closure of) H H, since the dimensions cannot increase forever, there is actually a largest such subgroup. We denote it by R and call it the radical of G. By (10.3), the unipotent elements in R form a normal subgroup U, the unipotent radical. We call G semisimple if R is trivial, reductive if U is trivial. The theorem then (for char(fc) = 0) is that all representations are sums of irreducibles iff G is reductive. It is not hard to see this condition implies G reductive (cf. Ex. 20) the converse is the hard part. We of course know the result for R, since by (10.3) it is a torus we also know that this R is central (7.7), which implies that the R-eigenspaces in a representation are G-invariant. The heart of the result then is the semisimple case. This can for instance be deduced from the corresponding result on Lie algebras. 

Humphreys, J. Linear Algebraic Groups (New York Springer, 1975). Much like Borel, going on to classify semisimple groups over algebraically closed fields. 

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