Special unitary group
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.
The special unitary group SU(n) is a real Lie group (though not a complex Lie group). Its dimension as a real manifold is n2 − 1. Topologically, it is compact and simply connected. Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below).
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
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.
Then, all such transition functions are classified by homotopy classes of maps
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:
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, n − p > 1 ,
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,
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