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The center of SU(n) is isomorphic to the cyclic group, and is composed of the diagonal matrices ? I for ? an nth root of unity and I the n×n identity matrix.
A maximal torus, of rank n - 1, is given by the set of diagonal matrices with determinant 1. The Weyl group is the symmetric groupSn, which is represented by signed permutation matrices (the signs being necessary to ensure the determinant is 1).
The Lie algebra of SU(n), denoted by , can be identified with the set of tracelessantiHermitiann×n complex matrices, with the regular commutator as a Lie bracket. Particle physicists often use a different, equivalent representation: The set of traceless Hermitiann×n complex matrices with Lie bracket given by -i times the commutator.
Lie algebra
The Lie algebra of consists of skew-Hermitian matrices with trace zero.[4] This (real) Lie algebra has dimension . More information about the structure of this Lie algebra can be found below in the section "Lie algebra structure."
Fundamental representation
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. That is to say, the physicists' Lie algebra differs by a factor of from the mathematicians'. With this convention, one can then choose generators Ta that are tracelessHermitian complex n×n matrices, where:
where the f are the structure constants and are antisymmetric in all indices, while the d-coefficients are symmetric in all indices.
As a consequence, the anticommutator and commutator are:
The factor of in the commutation relations arises from the physics convention and is not present when using the mathematicians' convention.
We may also take
as a normalization convention.
Adjoint representation
In the -dimensional adjoint representation, the generators are represented by × matrices, whose elements are defined by the structure constants themselves:
If we consider as a pair in where and , then the equation becomes
This is the equation of the 3-sphere S3. This can also be seen using an embedding: the map
where denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering diffeomorphic to and diffeomorphic to ). Hence, the restriction of ? to the 3-sphere (since modulus is 1), denoted S3, is an embedding of the 3-sphere onto a compact submanifold of , namely ?(S3) = SU(2).
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]
Relation to spatial rotations
Every unit quaternion is naturally associated to a spatial rotation in 3 dimensions, and the product of two quaternions is associated to the composition of the associated rotations. Furthermore, every rotation arises from exactly two unit quaternions 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 SU(2)/{±I}, the manifold underlying SO(3) is obtained by identifying antipodal points of the 3-sphere S3 , and SU(2) is the universal cover of SO(3).
The group is a simply-connected, compact Lie group.[8] Its topological structure can be understood by noting that SU(3) acts transitively on the unit sphere in . The stabilizer of an arbitrary point in the sphere is isomorphic to SU(2), which topologically is a 3-sphere. It then follows that SU(3) is a fiber bundle over the base with fiber . 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).[9]
The -bundles over are classified by since any such bundle can be constructed by looking at trivial bundles on the two hemispheres and looking at the transition function on their intersection which is homotopy equivalent to , so
Then, all such transition functions are classified by homotopy classes of maps
and as rather than , cannot be the trivial bundle , and therefore must be the unique nontrivial (twisted) bundle. This can be shown by looking at the induced long exact sequence on homotopy groups.
while all other fabc not related to these by permutation are zero. In general, they vanish unless they contain an odd number of indices from the set {2, 5, 7}.[c]
The symmetric coefficients d take the values
They vanish if the number of indices from the set {2, 5, 7} is odd.
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]
where
Lie algebra structure
As noted above, the Lie algebra of consists of skew-Hermitian matrices with trace zero.[12]
The complexification of the Lie algebra is , the space of all complex matrices with trace zero.[13] A Cartan subalgebra then consists of the diagonal matrices with trace zero,[14] which we identify with vectors in whose entries sum to zero. The roots then consist of all the permutations of (1, -1, 0, ..., 0).
Specifically, fix a Hermitian matrixA of signature p q in , then all
satisfy
Often one will see the notation SU(p, q) without reference to a ring or field; in this case, the ring or field being referred to is and this gives one of the classical Lie groups. The standard choice for A when is
However, there may be better choices for A for certain dimensions which exhibit more behaviour under restriction to subrings of .
Example
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 HC2.[16]
A further example is , which is isomorphic to .
Important subgroups
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 ,
Since the rank of SU(n) is 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,
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.
This group is locally isomorphic to SO(2,1) and SL(2,R)[17] where the numbers separated by a comma refer to the signature of the quadratic form preserved by the group. The expression in the definition of SU(1,1) is an Hermitian form which becomes an isotropic quadratic form when u and v are expanded with their real components. An early appearance of this group was as the "unit sphere" of coquaternions, introduced by James Cockle in 1852. Let
Then the 2×2 identity matrix, and and the elements i, j, and k all anticommute, like regular quaternions. Also is still a square root of -I2 (negative of the identity matrix), whereas are not, unlike the quaternions. For both quaternions and coquaternions, all scalar quantities are treated as implicit multiples of I2 , called the unit (co)quaternion, and occasionally explicitly notated as 1 .
The coquaternion with scalar w, has conjugate similar to Hamilton's quaternions. The quadratic form is
Note that the 2-sheet hyperboloid 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 SO(2,1). The variability of the pole of a wave, as noted in studies of polarization, might view elliptical polarization as an exhibit of the elliptical shape of a wave with The Poincaré sphere model used since 1892 has been compared to a 2-sheet hyperboloid model.[18]
^Francsics, Gabor; Lax, Peter D. (September 2005). "An explicit fundamental domain for the Picard modular group in two complex dimensions". arXiv:math/0509708.
^Gilmore, Robert (1974). Lie Groups, Lie Algebras and some of their Applications. John Wiley & Sons. pp. 52, 201-205. MR1275599.
^Siegel, C.L. (1971). Topics in Complex Function Theory. 2. Translated by Shenitzer, A.; Tretkoff, M. Wiley-Interscience. pp. 13-15. ISBN0-471-79080 X.
References
Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN978-3319134666
Iachello, Francesco (2006), Lie Algebras and Applications, Lecture Notes in Physics, 708, Springer, ISBN3540362363