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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?...
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This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.
- Fall '11