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So l1 and l1 are equivalent when char f 2 the second

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Unformatted text preview: eld F is not equal to 2 then [L1￿ ] implies that 2[xx] = 0 implies [L1]. So, [L1] and [L1￿ ] are equivalent when char F ￿= 2. The second condition [L2] can be rewritten as follows: [x[yz ]] = [[xy ]z ] + [[zx]y ]. The term [[zx]y ] prevents L from being associative. Since z, x, y are arbitrary we obtain: Proposition 1.1.3. A Lie algebra is associative if and only if [[LL]L] = 0. The notation [[LL]L] indicates the vector subspace of L generated by all expressions [[xy ]z ]. Definition 1.1.4. A (Lie) subalgebra of a Lie algebra L is defined to be a vector subspace K so that [KK ] ⊆ K . For example, [LL] is always a Lie subalgebra of L. Definition 1.1.5. A homom...
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This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.

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