SU(2) is the universal covering group of SO(3), and so its representation theory includes that of the latter, by dint of a surjective homomorphism to it.
As shown below, the finite-dimensional irreducible representations of SU(2) are indexed by a non-negative integer
) The matrices are a representation of the quaternions: where I is the conventional 2×2 identity matrix:
this passage from real to complexified Lie algebra is harmless.
[3] The reason for passing to the complexification is that it allows us to construct a nice basis of a type that does not exist in the real Lie algebra
The complexified Lie algebra is spanned by three elements
, given by or, explicitly, The non-trivial/non-identical part of the group's multiplication table is where O is the 2×2 all-zero matrix.
The following elementary result[4] is a key step in the analysis.
is an irreducible, finite-dimensional representation of the complexified Lie algebra.
is clearly invariant under the action of the complexified Lie algebra.
We thus obtain a complete description of what an irreducible representation must look like; that is, a basis for the space and a complete description of how the generators of the Lie algebra act.
we can construct a representation by simply using the above formulas and checking that the commutation relations hold.
as an element of the universal enveloping algebra or as an operator in each irreducible representation.
is then computed as Since SU(2) is simply connected, a general result shows that every representation of its (complexified) Lie algebra gives rise to a representation of SU(2) itself.
It is desirable, however, to give an explicit realization of the representations at the group level.
The group representations can be realized on spaces of polynomials in two complex variables.
, given by The associated Lie algebra representation is simply the one described in the previous section.
(See here for an explicit formula for the action of the Lie algebra on the space of polynomials.)
given by Characters plays an important role in the representation theory of compact groups.
The character is easily seen to be a class function, that is, invariant under conjugation.
In the SU(2) case, the fact that the character is a class function means it is determined by its value on the maximal torus
consisting of the diagonal matrices in SU(2), since the elements are orthogonally diagonalizable with the spectral theorem.
, it is easy to see that the associated character satisfies This expression is a finite geometric series that can be simplified to This last expression is just the statement of the Weyl character formula for the SU(2) case.
In this approach, the Weyl character formula plays an essential part in the classification, along with the Peter–Weyl theorem.
Representations of SU(2) describe non-relativistic spin, due to being a double covering of the rotation group of Euclidean 3-space.
Relativistic spin is described by the representation theory of SL2(C), a supergroup of SU(2), which in a similar way covers SO+(1;3), the relativistic version of the rotation group.
When an element of SU(2) is written as a complex 2 × 2 matrix, it is simply a multiplication of column 2-vectors.
It describes 3-d rotations, the standard representation of SO(3), so real numbers are sufficient for it.
Physicists use it for the description of massive spin-1 particles, such as vector mesons, but its importance for spin theory is much higher because it anchors spin states to the geometry of the physical 3-space.
This representation emerged simultaneously with the 2 when William Rowan Hamilton introduced versors, his term for elements of SU(2).