Lie algebra
Graduate (abstract algebra)
A Lie algebra over a field $F$ is a vector space $\mathfrak{g}$ over $F$ with a bilinear map $[\cdot, \cdot] : \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}$, called the Lie bracket, such that, for all $x, y, z \in \mathfrak{g}$,
- $[x, y] = -[y, x]$, and
- $[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0$ (the Jacobi identity).
Properties
- A Lie algebra is an algebra whose multiplication is the Lie bracket. This multiplication is not associative in general, and it is commutative only when every bracket is $0$.
- Every associative algebra, such as the $n \times n$ matrices, becomes a Lie algebra with the commutator $[a, b] = a b - b a$ as its bracket.
- The smooth vector fields on a manifold form a Lie algebra over $\reals$ with the Lie bracket of vector fields.
- For a matrix Lie group $G$, the Lie algebra of $G$ is the set of matrices $X$ such that $e^{t X}$ is in $G$ for every real number $t$, with the commutator as its bracket; the matrix exponential maps the Lie algebra into the group.
Notes
Lie algebras are usually written in lowercase Fraktur letters, such as $\mathfrak{g}$ or $\mathfrak{so}(3)$, and matrix Lie groups in capital letters, such as $G$ or $SO(3)$.