Special unitary group
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.
(More general unitary matrices 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 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 Cn.[1] It is itself a subgroup of the general linear group, SU(n) ⊂ U(n) ⊂ GL(n, C).
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.[5]
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}.[2] SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations.
Properties
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.[6] Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below).[7]
The center of SU(n) is isomorphic to the cyclic group Zn, and is composed of the diagonal matrices ζ I for ζ an nth root of unity and I the n×n identity matrix.
Its outer automorphism group, for n ≥ 3, is Z2, while the outer automorphism group of SU(2) is the trivial group.
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).
Lie algebra
Fundamental representation
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:
![{\displaystyle {\begin{aligned}\left\{T_{a},T_{b}\right\}&={\frac {1}{n}}\delta _{ab}I_{n}+\sum _{c=1}^{n^{2}-1}{d_{abc}T_{c}}\\\left[T_{a},T_{b}\right]&=i\sum _{c=1}^{n^{2}-1}f_{abc}T_{c}\,.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77ea996cf93a90b8cd1be15e2315608c17e503e4)
We may also take
as a normalization convention.
Adjoint representation
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:
The group SU(2)
SU(2) is the following group,[9]
where the overline denotes complex conjugation.
There is a 2:1 homomorphism from SU(2) to SO(3).
Diffeomorphism with S3
This is the equation of the 3-sphere S3. This can also be seen using an embedding: the map
![{\displaystyle {\begin{aligned}\varphi \colon \mathbf {C} ^{2}&\to \operatorname {M} (2,\mathbf {C} )\\[5pt]\varphi (\alpha ,\beta )&={\begin{pmatrix}\alpha &-{\overline {\beta }}\\\beta &{\overline {\alpha }}\end{pmatrix}},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf569b1bc4c460ebe1692bda32fd495d12a3d9fa)
where M(2, C) denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering C2 diffeomorphic to R4 and M(2, C) diffeomorphic to R8). 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 M(2, C), namely φ(S3) = SU(2).
Therefore, as a manifold, S3 is diffeomorphic to SU(2), which shows that S3 can be endowed with the structure of a compact, connected Lie group.
Isomorphism with unit quaternions
The complex matrix:
can be mapped to the quaternion:
This map is in fact an isomorphism. Additionally, the determinant of the matrix is the 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.[10]
Lie algebra
The Lie algebra is then generated by the following matrices,
which have the form of the general element specified above.
![{\displaystyle \left[u_{3},u_{1}\right]=2\ u_{2},\quad \left[u_{1},u_{2}\right]=2\ u_{3},\quad \left[u_{2},u_{3}\right]=2\ u_{1}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2da532e2434fd294dbca0834da6f8192eb1dc480)
The Lie algebra serves to work out the representations of SU(2).
The group SU(3)
Topology
Representation theory
Lie algebra
where λ, the Gell-Mann matrices, are the SU(3) analog of the Pauli matrices for SU(2):
![{\displaystyle {\begin{aligned}\lambda _{1}={}&{\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}},&\lambda _{2}={}&{\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}},&\lambda _{3}={}&{\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}},\\[6pt]\lambda _{4}={}&{\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix}},&\lambda _{5}={}&{\begin{pmatrix}0&0&-i\\0&0&0\\i&0&0\end{pmatrix}},\\[6pt]\lambda _{6}={}&{\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}},&\lambda _{7}={}&{\begin{pmatrix}0&0&0\\0&0&-i\\0&i&0\end{pmatrix}},&\lambda _{8}={\frac {1}{\sqrt {3}}}&{\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27272365d50c95972a5a66b020caf63b836131df)
These λa span all traceless Hermitian matrices H of the Lie algebra, as required. Note that λ2, λ5, λ7 are antisymmetric.
They obey the relations
![{\displaystyle {\begin{aligned}\left[T_{a},T_{b}\right]&=i\sum _{c=1}^{8}f_{abc}T_{c},\\\left\{T_{a},T_{b}\right\}&={\frac {1}{3}}\delta _{ab}I_{3}+\sum _{c=1}^{8}d_{abc}T_{c},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c84522e4ae5ab8f237d70c19f62e7b45affdb3c3)
or, equivalently,
- .
The f are the structure constants of the Lie algebra, given by
- ,,,
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}.[3]
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[15]:
![{\displaystyle {\begin{aligned}\exp(i\theta H)={}&\left[-{\frac {1}{3}}I\sin \left(\varphi +{\frac {2\pi }{3}}\right)\sin \left(\varphi -{\frac {2\pi }{3}}\right)-{\frac {1}{2{\sqrt {3}}}}~H\sin(\varphi )-{\frac {1}{4}}~H^{2}\right]{\frac {\exp \left({\frac {2}{\sqrt {3}}}~i\theta \sin(\varphi )\right)}{\cos \left(\varphi +{\frac {2\pi }{3}}\right)\cos \left(\varphi -{\frac {2\pi }{3}}\right)}}\\[6pt]&{}+\left[-{\frac {1}{3}}~I\sin(\varphi )\sin \left(\varphi -{\frac {2\pi }{3}}\right)-{\frac {1}{2{\sqrt {3}}}}~H\sin \left(\varphi +{\frac {2\pi }{3}}\right)-{\frac {1}{4}}~H^{2}\right]{\frac {\exp \left({\frac {2}{\sqrt {3}}}~i\theta \sin \left(\varphi +{\frac {2\pi }{3}}\right)\right)}{\cos(\varphi )\cos \left(\varphi -{\frac {2\pi }{3}}\right)}}\\[6pt]&{}+\left[-{\frac {1}{3}}~I\sin(\varphi )\sin \left(\varphi +{\frac {2\pi }{3}}\right)-{\frac {1}{2{\sqrt {3}}}}~H\sin \left(\varphi -{\frac {2\pi }{3}}\right)-{\frac {1}{4}}~H^{2}\right]{\frac {\exp \left({\frac {2}{\sqrt {3}}}~i\theta \sin \left(\varphi -{\frac {2\pi }{3}}\right)\right)}{\cos(\varphi )\cos \left(\varphi +{\frac {2\pi }{3}}\right)}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3911db5c2d7bd1d44edad1dff8db8099b077ed6)
where
![{\displaystyle \varphi \equiv {\frac {1}{3}}\left[\arccos \left({\frac {3{\sqrt {3}}}{2}}\det H\right)-{\frac {\pi }{2}}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3b54c3b30a637ef4902d486f9d86c0a7e46db78)
Lie algebra structure
A choice of simple roots is
So, SU(n) is of rank n − 1 and its Dynkin diagram is given by An−1, a chain of n − 1 nodes: [[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/8/8b/Dyn-node.png|undefined|Dyn-node.png|h24|w9]][[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/b/b3/Dyn-3.png|undefined|Dyn-3.png|h24|w6]][[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/8/8b/Dyn-node.png|undefined|Dyn-node.png|h24|w9]][[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/b/b3/Dyn-3.png|undefined|Dyn-3.png|h24|w6]][[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/8/8b/Dyn-node.png|undefined|Dyn-node.png|h24|w9]][[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/b/b3/Dyn-3.png|undefined|Dyn-3.png|h24|w6]]...[[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/b/b3/Dyn-3.png|undefined|Dyn-3.png|h24|w6]][[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/8/8b/Dyn-node.png|undefined|Dyn-node.png|h24|w9]].[19] Its Cartan matrix is
Its Weyl group or Coxeter group is the symmetric group S**n, the symmetry group of the (n − 1)-simplex.
Generalized special unitary group
For a field F, the 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 p qF. The field F can be replaced by a commutative ring, in which case the vector space is replaced by a free module.
Specifically, fix a Hermitian matrix A of signature p q in GL(n, R), 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 C and this gives one of the classical Lie groups. The standard choice for A when F = C is
However, there may be better choices for A for certain dimensions which exhibit more behaviour under restriction to subrings of C.
Example
An important example of this type of group is the Picard modular group SU(2, 1; Z[i]) which acts (projectively) on complex hyperbolic space of degree two, in the same way that SL(2,9;Z) 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.[20]
A further example is SU(1, 1; C), which is isomorphic to SL(2,R).
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 p > 1, n − p > 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,
See spin group, and simple Lie groups for E6, E7, and G2.
There are also the accidental isomorphisms: SU(4) = Spin(6) , SU(2) = Spin(3) = Sp(1) ,[4] 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.
The group SU(1,1)
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 pole p ≠ ± i. The Poincaré sphere model used since 1892 has been compared to a 2-sheet hyperboloid model.[22]
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
- wheredenotes the similarity ofprojective coordinates.
![{\displaystyle [z,1]{\begin{pmatrix}u&v\\v^{*}&u^{*}\end{pmatrix}}\ =\ [uz+v^{*},\ vz+u^{*}]\ \thicksim \ \left[{\frac {uz+v^{*}}{vz+u^{*}}},\ 1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cfc05f2c18cb8475941f81e57ed176fab612a51)
See also
Unitary group
Projective special unitary group, PSU(n)
Generalizations of Pauli matrices
Representation theory of SU(2)