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This is the vector space of all n n matrices with

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Unformatted text preview: . So, our finite dimensional examples are all linear. What are the finite dimensional linear Lie algebras? If V = F n then gl(V ) is denoted gl(n, F ). This is the vector space of all n × n matrices with coefficients in F with Lie bracket given by commutator: [xy ] = xy − yx. A subalgebra is given by a subset of gl(n, F ) which is closed under this bracket and under addition and scalar multiplication. Example 1.2.4. Let sl(n, F ) ⊆ gl(n, F ) denote the set of all n × n matrices with trace equal to zero. (1) Tr([xy ]) = Tr(xy ) − Tr(yx) = 0. So, sl(n, F ) is closed under [ ]. (2) Tr(x + y ) = Tr(x) + Tr(y ) = 0. (3) Tr(ax) = a Tr(x) = 0 Therefore, sl(n, F ) is a linear Lie algebra. Proposition 1.2.5. Suppose that f : V ×...
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