The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 in the special case.
A maximal torus of rank n − 1 is given by the set of diagonal matrices with determinant 1.
, 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.
In the physics literature, it is common to identify the Lie algebra with the space of trace-zero Hermitian (rather than the skew-Hermitian) matrices.
With this convention, one can then choose generators Ta that are traceless Hermitian complex n × n matrices, where:
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:
denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering
Hence, the restriction of φ to the 3-sphere (since modulus is 1), denoted S3, is an embedding of the 3-sphere onto a compact submanifold of
Quaternions of norm 1 are called versors since they generate the rotation group SO(3): The SU(2) matrix:
Additionally, the determinant of the matrix is the squared norm of the corresponding quaternion.
This representation is routinely used in quantum mechanics to represent the spin of fundamental particles such as electrons.
They also serve as unit vectors for the description of our 3 spatial dimensions in loop quantum gravity.
[10] Its topological structure can be understood by noting that SU(3) acts transitively on the unit sphere
Since the fibers and the base are simply connected, the simple connectedness of SU(3) then follows by means of a standard topological result (the long exact sequence of homotopy groups for fiber bundles).
This can be shown by looking at the induced long exact sequence on homotopy groups.
[12] Descriptions of these representations, from the point of view of its complexified Lie algebra
These λa span all traceless Hermitian matrices H of the Lie algebra, as required.
[15] A Cartan subalgebra then consists of the diagonal matrices with trace zero,[16] which we identify with vectors in
So, SU(n) is of rank n − 1 and its Dynkin diagram is given by An−1, a chain of n − 1 nodes: ....[17] Its Cartan matrix is
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).
In 2005 Gábor Francsics and Peter Lax computed an explicit fundamental domain for the action of this group on HC2.
In physics the special unitary group is used to represent fermionic symmetries.
In theories of symmetry breaking it is important to be able to find the subgroups of the special unitary group.
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.
For both quaternions and coquaternions, all scalar quantities are treated as implicit multiples of I2 and notated as 1.
corresponds to the imaginary units in the algebra so that any point p on this hyperboloid can be used as a pole of a sinusoidal wave according to Euler's formula.
The hyperboloid is stable under SU(1, 1), illustrating the isomorphism with Spin(2, 1).
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.