# Definition A Lie Algebra is a [[vector space]] $\mathfrak{g}$ equipped with a bilinear [[map]] $[\cdot, \cdot]: \mathfrak{g} \times \mathfrak{g} \rightarrow \mathfrak{g}$ called the *Lie bracket*, which satisfies: 1. *Antisymmetry*: $[X,Y] = -[Y,X]$ 2. *[[Jacobi Identity]]*: $[[X,Y],Z] + [[Y,Z],X] + [[Z,X],Y] = 0\quad \forall X, Y, Z \in \mathfrak{g}$ ## Matrix Lie Groups Given a [[matrix Lie group]] $G \subset GL(n, \mathbb{C})$ (where $GL(n, \mathbb{C})$ is the [[general linear group]]), we can define the Lie algebra $\mathfrak{g}$ as: $ \mathfrak{g} \equiv \left\{X \in M_n(\mathbb{C}) \, | \, e^{tX} \in G\, \forall t \in \mathbb{R}\right\} $ where $M_n(\mathbb{C})$ denotes the [[vector space]] of all [[matrix|matrices]] of dimension $n\times n$ over the complex numbers (see [[Matrix#Vector Space|here]]). Note that the Lie Algebras of Matrix Lie Groups satisfy the definition of the abstract Lie Algebra given [[#Definition|above]] as long as the Lie bracket is the [[commutator]]. An important fact is that the correspondence between matrix Lie groups and Lie algebras is *not* one-to-one; two different matrix Lie groups might have isomorphic Lie algebras. For example, even though $SU(2)$ [[Relationship Between SO(3) and SU(2)|is the double cover of]] $SO(3)$, they have [[Lie Algebra Isomorphism|isomorphic]] Lie algebras $\mathfrak{su}(2) \cong \mathfrak{so}(3)$. # Elements Elements of the Lie Algebra are often called "infinitesimal generators", especially by physicists. In some sense, they generate whatever the group operation is via the identification $g = e^{t X}$. However, it is more rigorous to think of them as derivatives of the group elements.