Special unitary group

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 more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.

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.[1]

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 n2 − 1 . Topologically, it is compact and simply connected.[2] Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below).[3]

A maximal torus, of rank n − 1, is given by the set of diagonal matrices with determinant 1. The Weyl group is the symmetric group Sn, which is represented by signed permutation matrices (the signs being necessary to ensure the determinant is 1).

where the f are the structure constants and are antisymmetric in all indices, while the d-coefficients are symmetric in all indices.

In the (n2 − 1) -dimensional adjoint representation, the generators are represented by (n2 − 1) × (n2 − 1) matrices, whose elements are defined by the structure constants themselves:

This is the equation of the 3-sphere S3. This can also be seen using an embedding: the map

Therefore, as a manifold, S3 is diffeomorphic to SU(2), which shows that SU(2) is simply connected and that S3 can be endowed with the structure of a compact, connected Lie group.

This map is in fact an isomorphism. Additionally, the determinant of the matrix is the square norm of the corresponding quaternion. Clearly any matrix in SU(2) is of this form and, since it has determinant 1, the corresponding quaternion has norm 1. Thus SU(2) is isomorphic to the unit quaternions.[6]

Then, all such transition functions are classified by homotopy classes of maps

where λ, the Gell-Mann matrices, are the SU(3) analog of the Pauli matrices for SU(2):

These λa span all traceless Hermitian matrices H of the Lie algebra, as required. Note that λ2, λ5, λ7 are antisymmetric.

A generic SU(3) group element generated by a traceless 3×3 Hermitian matrix H, normalized as tr(H2) = 2, can be expressed as a second order matrix polynomial in H:[11]

So, SU(n) is of rank n − 1 and its Dynkin diagram is given by An−1, a chain of n − 1 nodes: Dyn-node.pngDyn-3.pngDyn-node.pngDyn-3.pngDyn-node.pngDyn-3.png...Dyn-3.pngDyn-node.png.[15] Its Cartan matrix is

Its Weyl group or Coxeter group is the symmetric group Sn, the symmetry group of the (n − 1)-simplex.

For a field F, the generalized special unitary group over F, SU(p, q; F), is the group of all linear transformations of determinant 1 of a vector space of rank n = p + q over F which leave invariant a nondegenerate, Hermitian form of signature (p, q). This group is often referred to as the special unitary group of signature p q over F. The field F can be replaced by a commutative ring, in which case the vector space is replaced by a free module.

In physics the special unitary group is used to represent bosonic symmetries. In theories of symmetry breaking it is important to be able to find the subgroups of the special unitary group. Subgroups of SU(n) that are important in GUT physics are, for p > 1, np > 1 ,

where × denotes the direct product and U(1), known as the circle group, is the multiplicative group of all complex numbers with absolute value 1.

For completeness, there are also the orthogonal and symplectic subgroups,

Since the rank of SU(n) is n − 1 and of U(1) is 1, a useful check is that the sum of the ranks of the subgroups is less than or equal to the rank of the original group. SU(n) is a subgroup of various other Lie groups,

There are also the accidental isomorphisms: SU(4) = Spin(6) , SU(2) = Spin(3) = Sp(1) ,[d] and U(1) = Spin(2) = SO(2) .

One may finally mention that SU(2) is the double covering group of SO(3), a relation that plays an important role in the theory of rotations of 2-spinors in non-relativistic quantum mechanics.

An early appearance of this group was as the "unit sphere" of coquaternions, introduced by James Cockle in 1852. Let

When an element of SU(1,1) is interpreted as a Möbius transformation, it leaves the unit disk stable, so this group represents the motions of the Poincaré disk model of hyperbolic plane geometry. Indeed, for a point [ z, 1 ] in the complex projective line, the action of SU(1,1) is given by