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Concept
Special unitary group
Summary
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 SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.
The groups SU(2n) are important in quantum computing, as they represent the possible quantum logic gate operations in a quantum circuit with qubits and thus basis states. (Alternatively, the more general unitary group can be used, since
multiplying by a global phase factor does not change the expectation values of a quantum operator.)
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}. SU(2) is also 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 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 n‐th root of unity and I the n×n identity matrix.
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