Editing Lie algebra representation (section)

===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.