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 . Lie algebra representations 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. Real and complexified Lie algebras The real Lie algebra has a basis given by (These basis matrices are related to the Pauli matrices by and ) The matrices are a representation of the quaternions: where is the conventional 2×2 identity matrix: Consequently, the commutator brackets of the matrices satisfy It is then convenient to pass to the complexified Lie algebra (Skew self-adjoint matrices with trace zero plus self-adjoint matrices with trace zero gives all matrices with trace zero.) As long as we are working with representations over this passage from real to complexified Lie algebra is harmless. 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 , , and , given by or, explicitly, The non-trivial/non-identical part of the group's multiplication table is where is the 2×2 all-zero matrix. Hence their commutation relations are Up to a