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This problem goes under the name of "addition of angular momentum" or " Clebsch—Gordan theory " in the physics literature. The dual of an irreducible representation is always irreducible,  but may or may not be isomorphic to the original representation. In many cases, it is convenient to study representations of a Lie group by studying representations of the associated Lie algebra.
In general, however, not every representation of the Lie algebra comes from a representation of the group. This fact is, for example, lying behind the distinction between integer spin and half-integer spin in quantum mechanics. On the other hand, if G is a simply connected group, then a theorem  says that we do, in fact, get a one-to-one correspondence between the group and Lie algebra representations.
The Lie correspondence may be employed for obtaining group representations of the connected component of the G. Roughly speaking, this is effected by taking the matrix exponential of the matrices of the Lie algebra representation. A subtlety arises if G is not simply connected. This may result in projective representations or, in physics parlance, multi-valued representations of G. These are actually representations of the universal covering group of G.
The Lie correspondence gives results only for the connected component of the groups, and thus the other components of the full group are treated separately by giving representatives for matrices representing these components, one for each component. These form representatives of the zeroth homotopy group of G. For example, in the case of the four-component Lorentz group , representatives of space inversion and time reversal must be put in by hand. Further illustrations will be drawn from the representation theory of the Lorentz group below.
That is, the exponential map has a local inverse. In most groups, this is only local; that is, the exponential map is typically neither one-to-one nor onto.
Representation of Algebraic Groups and Quantum Groups
It is always possible to pass from a representation of a Lie group G to a representation of its Lie algebra g. It is explicitly computed using . The main result of this section is the following: . In quantum physics, it is natural to allow projective representations in addition to ordinary ones, because states are really defined only up to a constant. In the physics literature, projective representations are often described as multi-valued representations i. This phenomenon is important to the study of fractional spin in quantum mechanics.
We now outline the proof of the main results above. A key question is then this: Is this locally defined map a "local homomorphism"? A pictorial view of how the universal covering group contains all such homotopy classes, and a technical definition of it as a set and as a group is given in geometric view. If G is a connected compact Lie group, its finite-dimensional representations can be decomposed as direct sums of irreducible representations. Let T be a maximal torus in G.
By Schur's lemma , the irreducible representations of T are one dimensional. These representations can be classified easily and are labeled by certain "analytically integral elements" or "weights. The same weight can occur more than once. An important aspect of the representation theory is the associated theory of characters.
Two representations with the same character turn out to be isomorphic. Furthermore, the Weyl character formula gives a remarkable formula for the character of a representation in terms of its highest weight. Not only does this formula gives a lot of useful information about the representation, but it plays a crucial role in the proof of the theorem of the highest weight.
If the Hilbert space V is allowed to be infinite dimensional, the study of unitary representations involves a number of interesting features that are not present in the finite dimensional case. One setting in which the Lie algebra representation is well understood is that of semisimple or reductive Lie groups, where the associated Lie algebra representation forms a g,K -module.
Examples of unitary representations arise in quantum mechanics and quantum field theory, but also in Fourier analysis as shown in the following example. Here are some important examples in which unitary representations of a Lie group have been analyzed. The symmetries of the Hilbert space are then described by unitary operators, but a unitary operator that is a multiple of the identity does not change the physical state of the system.
We have already discussed the irreducible projective unitary representations of the rotation group SO 3 above; considering projective representations allows for fractional spin in addition to integer spin. See the discussion here. This claim follows from Schur's lemma and holds even if the representations are not assumed ahead of time to be finite dimensional.
See also Pontryagin duality for this case. From Wikipedia, the free encyclopedia. Simple Lie groups. Other Lie groups. Lie algebras.
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