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Lie algebra

In mathematics, a Lie algebra is a vector space g together with an operation called the Lie bracket, an alternating bilinear map g × g → g, that satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted [ x , y ]. The vector space g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. (Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Lie_algebra)

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