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Group of unitary matrices with determinant of 1

In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 in the special case. The group operation is matrix multiplication. The special unitary group is a normal subgroup of the unitary group U(n), consisting of all n×n unitary matrices. As a compact classical group, U(n) is the group that preserves the standard inner product on It is itself a subgroup of the general linear group,

The SU(n) groups find wide application in the Standard Model of particle physics, especially [[#The group SU(2) in the electroweak interaction and [[#SU(3) in quantum chromodynamics. The simplest case, SU(1), is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3 sphere. Since unit quaternions can be used to represent rotations in 3 dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+I, −I}. Since the quaternions can be identified as the even subalgebra of the Clifford Algebra Cl(3), SU(2) is in fact identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations.

The special unitary group SU(n) is a strictly real Lie group (vs. a more general complex Lie group). Its dimension as a real manifold is n^(2) − 1. Topologically, it is compact and simply connected. Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below). The center of SU(n) is isomorphic to the cyclic group , and is composed of the diagonal matrices ζ I for ζ an nth root of unity and I the n × n identity matrix. Its outer automorphism group for n ≥ 3 is while the outer automorphism group of SU(2) is the trivial group. A maximal torus of rank n − 1 is given by the set of diagonal matrices with determinant 1. The Weyl group of SU(n) is the symmetric group Sn, which is represented by signed permutation matrices (the signs being necessary to ensure that the determinant is 1). The Lie algebra of SU(n), denoted by , can be identified with the set of traceless anti‑Hermitian n × n complex matrices, with the regular commutator as a Lie bracket. Particle physicists often use a different, equivalent representation: The set of traceless Hermitian n × n complex matrices with Lie bracket given by −i times the commutator.

The Lie algebra of consists of n × n skew Hermitian matrices with trace zero. This (real) Lie algebra has dimension n^(2) − 1. More information about the structure of this Lie algebra can be found below in .

Every versor is naturally associated to a spatial rotation in 3 dimensions, and the product of versors is associated to the composition of the associated rotations. Furthermore, every rotation arises from exactly two versors in this fashion. In short: there is a 2:1 surjective homomorphism from SU(2) to SO(3); consequently SO(3) is isomorphic to the quotient group , the manifold underlying SO(3) is obtained by identifying antipodal points of the 3 sphere S^(3), and SU(2) is the universal cover of SO(3).

The group SU(3) is an 8 dimensional simple Lie group consisting of all 3 × 3 unitary matrices with determinant 1.

The representation theory of SU(3) is well understood. Descriptions of these representations, from the point of view of its complexified Lie algebra , may be found in the articles on Lie algebra representations or the Clebsch–Gordan coefficients for SU(3).

An important example of this type of group is the Picard modular group which acts (projectively) on complex hyperbolic space of degree two, in the same way that acts (projectively) on real hyperbolic space of dimension two. In 2005 Gábor Francsics and Peter Lax computed an explicit fundamental domain for the action of this group on HC^(2). A further example is , which is isomorphic to .