chapter1 - Chapter 1 Lie Groups and Algebraic Groups...

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Unformatted text preview: Chapter 1 Lie Groups and Algebraic Groups Hermann Weyl, in his famous book (Weyl [1946]), gave the name classical groups to certain families of matrix groups. In this chapter we introduce these groups and develop the basic ideas of Lie groups, Lie algebras, and linear algebraic groups. We show how to put a Lie group structure on a closed subgroup of the general linear group and determine the Lie algebras of the classical groups. We develop the theory of complex linear algebraic groups far enough to obtain the basic results on their Lie algebras, rational representations, and Jordan–Chevalley decompositions (we defer the deeper results about algebraic groups to Chapter 11). We show that linear al- gebraic groups are Lie groups, introduce the notion of a real form of an algebraic group (considered as a Lie group), and show how the classical groups introduced at the beginning of the chapter appear as real forms of linear algebraic groups. 1.1 The Classical Groups 1.1.1 General and Special Linear Groups Let F denote either the real numbers R or the complex numbers C , and let V be a finite-dimensional vector space over F . The set of all invertible linear transforma- tions from V to V will be denoted as GL( V ) . This set has a group structure under composition of transformations, with identity element the identity transformation Id( x ) = x for all x ∈ V . The group GL( V ) is the first of the classical groups. To study it in more detail, we recall some standard terminology related to linear transformations and their matrices. Let V and W be finite-dimensional vector spaces over F . Let { v 1 , . . ., v n } and { w 1 , . . ., w m } be bases for V and W , respectively. If T : V d47 W is a linear map 1 2 CHAPTER 1. LIE GROUPS AND ALGEBRAIC GROUPS then Tv j = m summationdisplay i =1 a ij w i for j = 1 , . . ., n with a ij ∈ F . The numbers a ij are called the matrix coefficients or entries of T with respect to the two bases, and the m × n array A = a 11 a 12 ··· a 1 n a 21 a 22 ··· a 2 n . . . . . . . . . . . . a m 1 a m 2 ··· a mn is the matrix of T with respect to the two bases. When the elements of V and W are identified with column vectors in F n and F m using the given bases, then action of T becomes multiplication by the matrix A . Let S : W d47 U be another linear transformation, with U an l-dimensional vector space with basis { u 1 , . . ., u l } , and let B be the matrix of S with respect to the bases { w 1 , . . ., w m } and { u 1 , . . ., u l } . Then the matrix of S ◦ T with respect to the bases { v 1 , . . ., v n } and { u 1 , . . ., u l } is given by BA —the product being the usual product of matrices....
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chapter1 - Chapter 1 Lie Groups and Algebraic Groups...

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