Lie algebra

Vector Space $V$ over a field $\mathbb{F}$ equipped with a bilinear alternating map $V\times V\to V$ that satisfies the Jacobi identity. More specifically, $\forall X,Y,Z\in V$, we have:

1. $\lambda X+Y\in V\forall \lambda \in \mathbb{F}$
2. $XY\in V$
3. $X(YZ)+Y(ZX)+Z(XY)=0$ ( Jacobi identity)

Historically, Lie Algebras were studied in relation to Lie group: A lie algebra is the tangent space of a Lie group at the Identity. This is known as the Lie group - Lie algebra correspondence.