Lie Algebras

Abstract

Recall that an algebra (over the reals) is a real vector space A with a bilinear map A × A → A. This map can be thought of as a binary operation giving A a ring structure. We call A a Lie algebra if the binary operation, which we denote by [,], is anticommutative (that is, [a, b] = −[b,a]) and satisfies the Jacobi identity $$\left[ {\left[ {a,b} \right],c} \right] + \left[ {\left[ {b,c} \right],a} \right] + \left[ {\left[ {c,a} \right],b} \right] = 0$$ We call [a, b] the commutator of a and b.KeywordsTangent VectorInvariant SubspaceAdjoint RepresentationReal Vector SpaceAntisymmetric TensorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.