Первый пример компактной неабелевой группы Ли: SU(2)

First case of a Lie group that is both compact and non abelian
In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non abelian group. The first condition implies the representation theory is discrete: representations are direct sums of a collection of basic irreducible representations (governed by the Peter–Weyl theorem). The second means that there will be irreducible representations in dimensions greater than 1. 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. This underlies the significance of SU(2) for the description of non relativistic spin in theoretical physics; see below for other physical and historical context. As shown below, the finite dimensional irreducible representations of SU(2) are indexed by a non negative integer and have dimension In the physics literature, the representations are labeled by the quantity , where is then either an integer or a half integer, and the dimension is .

The representations of the group are found by considering representations of , the Lie algebra of SU(2). Since the group SU(2) is simply connected, every representation of its Lie algebra can be integrated to a group representation; we will give an explicit construction of the representations at the group level below.

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. That is, for each non negative integer , we let denote the space of homogeneous polynomials of degree in two complex variables. Then the dimension of is There is a natural action of SU(2) on each , 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.)

Note that either all of the weights of the representation are even (if is even) or all of the weights are odd (if is odd). In physical terms, this distinction is important: The representations with even weights correspond to ordinary representations of the rotation group SO(3). By contrast, the representations with odd weights correspond to double valued (spinorial) representation of SO(3), also known as projective representations. In the physics conventions, being even corresponds to being an integer while being odd corresponds to being a half integer. These two cases are described as integer spin and half integer spin, respectively. The representations with odd, positive values of are faithful representations of SU(2), while the representations of SU(2) with non negative, even are not faithful.

See under the example for Borel–Weil–Bott 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. SU(2) symmetry also supports concepts of isobaric spin and weak isospin, collectively known as isospin. The representation with (i. e., in the physics convention) is the 2 representation, the fundamental representation of SU(2). When an element of SU(2) is written as a complex 2 × 2 matrix, it is simply a multiplication of column 2 vectors. It is known in physics as the spin ½ and, historically, as the multiplication of quaternions (more precisely, multiplication by a unit quaternion). This representation can also be viewed as a double valued projective representation of the rotation group SO(3). The representation with (i. e., ) is the 3 representation, the adjoint representation. 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). Note that Hamilton did not use standard group theory terminology since his work preceded Lie group developments. The (i. e. ) representation is used in particle physics for certain baryons, such as the Δ.