lecture notes1

The set of all derivations on a is denoted dera

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: th coefficients in F . 1.3. Derivations. Definition 1.3.1. Suppose that A is a nonassociative algebra over F . Then a derivation on A is a linear function δ : A → A so that δ (xy ) = δ (x)y + xδ (y ) for all x, y ∈ A. The set of all derivations on A is denoted Der(A). Proposition 1.3.2. Der(A) is a Lie subalgebra of gl(A). ￿ Proof. Go back to the definition of a Lie algebra. Using the skew symmetry condition [L1￿ ], Condition [L2] can be rephrased as: [z [xy ]] = [[zx]y ] + [x[zy ] In other words, the bracket by z operation adz (·) = [z (·)] satisfies: adz [xy ] = [adz (x)y ] + [xadz (y )] So any Lie algebra acts on itself by derivations. This gives a homomorphism: called the adjoint representation. ad : L → Der(A) 1.4. Abstract Lie algebras. We could simply start with the definition and try to construct all possible Lie algebras. Take L = F n . n = 1: Show that all one dimensional Lie algebras are abelian. n = 2: If L = F 2 there are, up to isomorphism, exactly two examples. n = 3: Example: Take L = R3 and take the Lie bracket to be the cross product. ijk [xy ] = det x1 x2 x3 y1 y2 y3 Verify the Jacobi identiy. Which Lie algebra is this?...
View Full Document

This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.

Ask a homework question - tutors are online