PUMaC 2017 · 加试 · 第 2 题

Problem 2.3.1. (3 points) Show that if X and Y are matrices that commute and X is nilpotent then tr( XY ) = 0. We are now ready to begin investigating Lie algebras. Despite their simple definition, we will soon see a host of powerful results that aid in their classification. 3 Lie Algebras I (77 points) 3.1 What is a Lie algebra? (26 points) Definition 3.1.A. A Lie algebra is a vector space L with, besides addition, the bilinear operation [ · , · ] : L × L → L . This bracket (sometimes the Lie bracket ) is subject to the following properties: PUMaC 2017 Power Round Page 12 • [ x, x ] = 0 for x ∈ L • [ x, [ y, z ]] + [ y, [ z, x ]] + [ z, [ x, y ]] = 0 for any x, y, z ∈ L (the Jacobi identity ) n n Linear maps from F → F are actually a very good way to think about many Lie algebras. (You’ll show in Problem 3.1.1 that these functions do indeed form a Lie algebra.) Lemma 3.1.1 ( anticommutativity of the bracket ) . [ x, y ] = − [ y, x ] for x, y elements of a Lie algebra L . Proof. Let z = x + y . 0 = [ z, z ] = [ x + y, x + y ] = [ x, x ]+[ x, y ]+[ y, x ]+[ y, y ] = [ x, y ]+[ y, x ] . Definition 3.1.B. A Lie algebra for which the bracket is degenerate, i.e. [ x, y ] = 0 for all x, y ∈ L , is termed an abelian Lie algebra. Note that the dimension of a Lie algebra is its dimension as a vector space under addition. In general, we do not have a meaningful way to treat the bracket operation as an operation giving rise to a vector space. We also have a natural extension to the concept of vector space homomorphism and isomorphism: Definition 3.1.C. A homomorphism of Lie algebras is a function ϕ : L → L where L 1 2 1 and L are lie algebras over the same field, and for which ϕ ([ x, y ]) = [ ϕ ( x ) , ϕ ( y )] for any 2 x, y ∈ L . 1 Definition 3.1.D. An isomorphism of Lie algebras is a bijective homomorphism of Lie algebras. The following problems will serve to hone your intuition.