Editing Lie algebra (section)

=== Infinite dimensions ===
* The Lie algebra of vector fields on a smooth manifold of positive dimension is an infinite-dimensional Lie algebra over <math>\mathbb{R}</math>.
* The [[Kac–Moody algebra]]s are a large class of infinite-dimensional Lie algebras, say over <math>\mathbb{C}</math>, with structure much like that of the finite-dimensional simple Lie algebras (such as <math>\mathfrak{sl}(n,\C)</math>).
* The [[Moyal bracket|Moyal algebra]] is an infinite-dimensional Lie algebra that contains all the [[Classical Lie groups#Relationship with bilinear forms|classical Lie algebra]]s as subalgebras.
* The [[Virasoro algebra]] is important in [[string theory]].
* The functor that takes a Lie algebra over a field ''F'' to the underlying vector space has a [[left adjoint]] <math>V\mapsto L(V)</math>, called the ''[[free Lie algebra]]'' on a vector space ''V''. It is spanned by all iterated Lie brackets of elements of ''V'', modulo only the relations coming from the definition of a Lie algebra. The free Lie algebra <math>L(V)</math> is infinite-dimensional for ''V'' of dimension at least 2.<ref>{{harvnb|Serre|2006|loc=Part I, Chapter IV.}}</ref>