Lie algebra

Summary

A Lie algebra is a vector space with a bilinear product $[x, y]$, the Lie bracket, that is alternating and satisfies the Jacobi identity. The commutator $AB - BA$ of square matrices is the basic example. Every Lie group has a Lie algebra, its tangent space at the identity. This algebra records most of the structure of the group near the identity. Lie algebras are also studied for their own sake, and the semisimple ones over $\C$ have a complete classification.