Lie group
Representations

One important aspect of the study of Lie groups is their representations, that is, the way they can act (linearly) on vector spaces. In physics, Lie groups often encode the symmetries of a physical system. The way one makes use of this symmetry to help analyze the system is often through representation theory. Consider, for example, the time-independent Schrödinger equation in quantum mechanics, H ^ ψ = E ψ {\displaystyle {\hat {H}}\psi =E\psi }. Assume the system in question has the rotation group SO(3) as a symmetry, meaning that the Hamiltonian operator H ^ {\displaystyle {\hat {H}}} commutes with the action of SO(3) on the wave function ψ {\displaystyle \psi }. (One important example of such a system is the Hydrogen atom, which has a single spherical orbital.) This assumption does not necessarily mean that the solutions ψ {\displaystyle \psi } are rotationally invariant functions. Rather, it means that the space of solutions to H ^ ψ = E ψ {\displaystyle {\hat {H}}\psi =E\psi } is invariant under rotations (for each fixed value of E {\displaystyle E}). This space, therefore, constitutes a representation of SO(3). These representations have been classified and the classification leads to a substantial simplification of the problem, essentially converting a three-dimensional partial differential equation to a one-dimensional ordinary differential equation.

The case of a connected compact Lie group K (including the just-mentioned case of SO(3)) is particularly tractable. In that case, every finite-dimensional representation of K decomposes as a direct sum of irreducible representations. The irreducible representations, in turn, were classified by Hermann Weyl. The classification is in terms of the "highest weight" of the representation. The classification is closely related to the classification of representations of a semisimple Lie algebra.

One can also study (in general infinite-dimensional) unitary representations of an arbitrary Lie group (not necessarily compact). For example, it is possible to give a relatively simple explicit description of the representations of the group SL(2,R) and the representations of the Poincaré group.
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