# Hermann Göring

In mathematics, a Lie algebra (pronounced "Lee") is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket, an alternating bilinear map g × g → g , ( x , y ) ↦ [ x , y ] {\displaystyle {\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}},\ (x,y)\mapsto [x,y]} , that satisfies the Jacobi identity. The vector space g {\displaystyle {\mathfrak {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.