Editing Lie algebra representation (section)

==Examples ==

===Adjoint representations===
{{main|Adjoint representation of a Lie algebra}}
The most basic example of a Lie algebra representation is the adjoint representation of a Lie algebra <math>\mathfrak{g}</math> on itself:
:<math>\textrm{ad}:\mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}), \quad X \mapsto \operatorname{ad}_X, \quad \operatorname{ad}_X(Y) = [X, Y].</math>
Indeed, by virtue of the [[Jacobi identity]], <math>\operatorname{ad}</math> is a Lie algebra homomorphism.

===Infinitesimal Lie group representations===
A Lie algebra representation also arises in nature. If <math>\phi</math>: ''G'' → ''H'' is a [[homomorphism]] of (real or complex) [[Lie group]]s, and <math>\mathfrak g</math> and <math>\mathfrak h</math> are the [[Lie algebra]]s of ''G'' and ''H'' respectively, then the [[pushforward (differential)|differential]] <math>d_e \phi: \mathfrak g \to \mathfrak h</math> on [[tangent space]]s at the identities is a Lie algebra homomorphism. In particular, for a finite-dimensional vector space ''V'', a [[representation of Lie groups]]
:<math>\phi: G\to \operatorname{GL}(V)\,</math>
determines a Lie algebra homomorphism
:<math>d \phi: \mathfrak g \to \mathfrak{gl}(V)</math>
from <math>\mathfrak g</math> to the Lie algebra of the [[general linear group]] GL(''V''), i.e. the endomorphism algebra of ''V''.

For example, let <math>c_g(x) = gxg^{-1}</math>. Then the differential of <math>c_g: G \to G</math> at the identity is an element of <math>\operatorname{GL}(\mathfrak{g})</math>. Denoting it by <math>\operatorname{Ad}(g)</math> one obtains a representation <math>\operatorname{Ad}</math> of ''G'' on the vector space <math>\mathfrak{g}</math>. This is the [[adjoint representation]] of ''G''. Applying the preceding, one gets the Lie algebra representation <math>d\operatorname{Ad}</math>. It can be shown that <math>d_e\operatorname{Ad} = \operatorname{ad}</math>, the adjoint representation of <math>\mathfrak g</math>.

A partial converse to this statement says that every representation of a finite-dimensional (real or complex) Lie algebra lifts to a unique representation of the associated [[simply connected]] Lie group, so that representations of simply-connected Lie groups are in one-to-one correspondence with representations of their Lie algebras.<ref>{{harvnb|Hall|2015}} Theorem 5.6</ref>

===In quantum physics===
In quantum theory, one considers "observables" that are self-adjoint operators on a [[Hilbert space]]. The commutation relations among these operators are then an important tool. The [[angular momentum operator]]s, for example, satisfy the commutation relations
:<math>[L_x,L_y]=i\hbar L_z, \;\; [L_y,L_z]=i\hbar L_x, \;\; [L_z,L_x]=i\hbar L_y,</math>.
Thus, the span of these three operators forms a Lie algebra, which is isomorphic to the Lie algebra so(3) of the [[rotation group SO(3)]].<ref>{{harvnb|Hall|2013}} Section 17.3</ref> Then if <math>V</math> is any subspace of the quantum Hilbert space that is invariant under the angular momentum operators, <math>V</math> will constitute a representation of the Lie algebra so(3). An understanding of the representation theory of so(3) is of great help in, for example, analyzing Hamiltonians with rotational symmetry, such as the [[Hydrogen-like atom|hydrogen atom]]. Many other interesting Lie algebras (and their representations) arise in other parts of quantum physics. Indeed, the history of representation theory is characterized by rich interactions between mathematics and physics.