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RESEARCH PRODUCT

Semisimple Lie Algebras

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Let F be the field of real or complex numbers. A Lie algebra is a vector space g over F with a Lie product (or commutator) [·,·]: g × g → g such that $$x \mapsto \left[ {x,y} \right]\;is\;linear\;for\;any\;y \in g,$$ (1) $$\left[ {x,y} \right] =- \left[ {y,x} \right],$$ (2) $$\left[ {x,\left[ {y,z} \right]} \right] + \left[ {y,\left[ {z,x} \right]} \right] + \left[ {z,\left[ {x,y} \right]} \right] = 0.$$ (3) The last condition is called the Jacobi identity. From (1) and (2) it follows that also y ↦ [x,y] is linear for any x ∈ g. In this chapter we shall consider only fini te-dimensional Lie algebras. In any vector space g one can always define a trivial Lie product [x,y] = 0. A Lie algebra with this commutator is Abelian.