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 matrices of the more general unitary group 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 normal 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 C n {\displaystyle \mathbb {C} ^{n}} . It is itself a subgroup of the general linear group, SU ( n ) ⊂ U ( n ) ⊂ GL ( n , C ) . {\displaystyle \operatorname {SU} (n)\subset \operatorname {U} (n)\subset \operatorname {GL} (n,\mathbb {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. 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 (uniquely up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+I, −I}. Since the quaternions can be identified as the even subalgebra of the Clifford algebra Cl(3), SU(2) is 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 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. Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below). The center of SU(n) is isomorphic to the cyclic group Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } , 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 Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } , 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 of SU(n) is the symmetric group Sn, which is represented by signed permutation matrices (the signs being necessary to ensure that the determinant is 1). The Lie algebra of SU(n), denoted by s u ( n ) {\displaystyle {\mathfrak {su}}(n)} , 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.
Lie algebra
The Lie algebra s u ( n ) {\displaystyle {\mathfrak {su}}(n)} of SU(n) consists of n × n skew-Hermitian matrices with trace zero. This (real) Lie algebra has dimension n2 − 1. The complexification of the Lie algebra s u ( n ) {\displaystyle {\mathfrak {su}}(n)} is s l ( n ; C ) {\displaystyle {\mathfrak {sl}}(n;\mathbb {C} )} , the space of all n × n complex matrices with trace zero. A Cartan subalgebra then consists of the diagonal matrices with trace zero, which we identify with vectors in C n {\displaystyle \mathbb {C} ^{n}} whose entries sum to zero. The roots then consist of all the n(n − 1) permutations of (1, −1, 0, ..., 0). A choice of simple roots is
( 1 , − 1 , 0 , … , 0 , 0 ) , ( 0 , 1 , − 1 , … , 0 , 0 ) , ⋮ ( 0 , 0 , 0 , … , 1 , − 1 ) . {\displaystyle {\begin{aligned}(&1,-1,0,\dots ,0,0),\\(&0,1,-1,\dots ,0,0),\\&\vdots \\(&0,0,0,\dots ,1,-1).\end{aligned}}}
So, SU(n) is of rank n − 1 and its Dynkin diagram is given by An−1, a chain of n − 1 nodes: .... Its Cartan matrix is
( 2 − 1 0 … 0 − 1 2 − 1 … 0 0 − 1 2 … 0 ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 0 … 2 ) . {\displaystyle {\begin{pmatrix}2&-1&0&\dots &0\\-1&2&-1&\dots &0\\0&-1&2&\dots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\dots &2\end{pmatrix}}.}
Its Weyl group or Coxeter group is the symmetric group Sn, the symmetry group of the (n − 1)-simplex.
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 i {\displaystyle i} from the mathematicians'. With this convention, one can then choose generators Ta that are traceless Hermitian complex n × n matrices, where:
T a T b = 1 2 n δ a b I n + 1 2 ∑ c = 1 n 2 − 1 ( i f a b c + d a b c ) T c {\displaystyle T_{a}\,T_{b}={\tfrac {1}{\,2n\,}}\,\delta _{ab}\,I_{n}+{\tfrac {1}{2}}\,\sum _{c=1}^{n^{2}-1}\left(if_{abc}+d_{abc}\right)\,T_{c}}
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 commutator is:
[ T a , T b ] = i ∑ c = 1 n 2 − 1 f a b c T c , {\displaystyle ~\left[T_{a},\,T_{b}\right]~=~i\sum _{c=1}^{n^{2}-1}\,f_{abc}\,T_{c}\;,}
and the corresponding anticommutator is:
{ T a , T b } = 1 n δ a b I n + ∑ c = 1 n 2 − 1 d a b c T c . {\displaystyle \left\{T_{a},\,T_{b}\right\}~=~{\tfrac {1}{n}}\,\delta _{ab}\,I_{n}+\sum _{c=1}^{n^{2}-1}{d_{abc}\,T_{c}}~.}
The factor of i in the commutation relation arises from the physics convention and is not present when using the mathematicians' convention. The conventional normalization condition is
∑ c , e = 1 n 2 − 1 d a c e d b c e = n 2 − 4 n δ a b . {\displaystyle \sum _{c,e=1}^{n^{2}-1}d_{ace}\,d_{bce}={\frac {\,n^{2}-4\,}{n}}\,\delta _{ab}~.}
The generators satisfy the Jacobi identity:
[ T a , [ T b , T c ] ] + [ T b , [ T c , T a ] ] + [ T c , [ T a , T b ] ] = 0. {\displaystyle [T_{a},[T_{b},T_{c}]]+[T_{b},[T_{c},T_{a}]]+[T_{c},[T_{a},T_{b}]]=0.}
By convention, in the physics literature the generators T a {\displaystyle T_{a}} are defined as the traceless Hermitian complex matrices with a 1/2 factor: for SU(2), the generators are chosen as 1 2 σ 1 {\displaystyle {\tfrac {1}{2}}\sigma _{1}} , 1 2 σ 2 {\displaystyle {\tfrac {1}{2}}\sigma _{2}} , 1 2 σ 3 {\displaystyle {\tfrac {1}{2}}\sigma _{3}} where σ a {\displaystyle \sigma _{a}} are the Pauli matrices, while for the case of SU(3) one defines T a = 1 2 λ a {\displaystyle T_{a}={\tfrac {1}{2}}\lambda _{a}} where λ a {\displaystyle \lambda _{a}} are the Gell-Mann matrices. With these definitions, the generators satisfy the following normalization condition:
T r ( T a T b ) = 1 2 δ a b . {\displaystyle Tr(T_{a}T_{b})={\frac {1}{2}}\delta _{ab}.}
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:
( T a ) j k = − i f a j k . {\displaystyle \left(T_{a}\right)_{jk}=-if_{ajk}.}
SU(2)
Using matrix multiplication for the binary operation, SU(2) forms a group,
SU ( 2 ) = { ( α − β ¯ β α ¯ ) : α , β ∈ C , | α | 2 + | β | 2 = 1 } , {\displaystyle \operatorname {SU} (2)=\left\{{\begin{pmatrix}\alpha &-{\overline {\beta }}\\\beta &{\overline {\alpha }}\end{pmatrix}}:\ \ \alpha ,\beta \in \mathbb {C} ,|\alpha |^{2}+|\beta |^{2}=1\right\}~,}
where the overline denotes complex conjugation.
Diffeomorphism with the 3-sphere S3 If we consider α , β {\displaystyle \alpha ,\beta } as a pair in C 2 {\displaystyle \mathbb {C} ^{2}} where α = a + b i {\displaystyle \alpha =a+bi} and β = c + d i {\displaystyle \beta =c+di} with real numbers a , b , c , d {\displaystyle a,b,c,d} , then the equation | α | 2 + | β | 2 = 1 {\displaystyle |\alpha |^{2}+|\beta |^{2}=1} becomes
a 2 + b 2 + c 2 + d 2 = 1. {\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=1.}
This is the equation of the 3-sphere S3. This can also be seen using an embedding: the map
φ : C 2 →
M ( 2 , C ) φ ( α , β ) =
( α − β ¯ β α ¯ ) , {\displaystyle {\begin{aligned}\varphi \colon \mathbb {C} ^{2}\to {}&\operatorname {M} (2,\mathbb {C} )\\[5pt]\varphi (\alpha ,\beta )={}&{\begin{pmatrix}\alpha &-{\overline {\beta }}\\\beta &{\overline {\alpha }}\end{pmatrix}},\end{aligned}}}
where M ( 2 , C ) {\displaystyle \operatorname {M} (2,\mathbb {C} )} denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering C 2 {\displaystyle \mathbb {C} ^{2}} isomorphic to R 4 {\displaystyle \mathbb {R} ^{4}} and M ( 2 , C ) {\displaystyle \operatorname {M} (2,\mathbb {C} )} isomorphic to R 8 {\displaystyle \mathbb {R} ^{8}} ). Hence, the restriction of φ to the 3-sphere S3 is an embedding of the 3-sphere onto a compact submanifold of M ( 2 , C ) {\displaystyle \operatorname {M} (2,\mathbb {C} )} , namely φ(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.
Isomorphism with group of versors and relation to spatial rotations
The ring of quaternions can be associated with a subring of 2-by-2 complex matrices by the ring isomorphism mapping
a 1 ^ + b i ^ + c j ^ + d k ^ {\displaystyle a\,{\hat {1}}+b\,{\hat {i}}+c\,{\hat {j}}+d\,{\hat {k}}}
to
( a + b i c + d i − c + d i a − b i ) ( a , b , c , d ∈ R ) . {\displaystyle {\begin{pmatrix}a+bi&c+di\\-c+di&a-bi\end{pmatrix}}\quad (a,b,c,d\in \mathbb {R} ).}
Additionally, the determinant of the matrix is the squared 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, restricting the above map gives a group isomorphism between SU(2) and the group of quaternions with norm 1. Quaternions of norm 1 are called versors since they can be associated with rotations in 3-dimensional space: Every versor is naturally associated to a spatial rotation in 3 dimensions, and the product of versors is associated to the composition of the associated rotations. Furthermore, every rotation arises from exactly two versors in this fashion. In short: there is a 2:1 surjective homomorphism from SU(2) to the rotation group 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).
Lie algebra The Lie algebra of SU(2) consists of 2 × 2 skew-Hermitian matrices with trace zero. Explicitly, this means
s u ( 2 ) = { ( i a − z ¯ z − i a ) : a ∈ R , z ∈ C } . {\displaystyle {\mathfrak {su}}(2)=\left\{{\begin{pmatrix}i\ a&-{\overline {z}}\\z&-i\ a\end{pmatrix}}:\ a\in \mathbb {R} ,z\in \mathbb {C} \right\}~.}
The Lie algebra is then generated by the following matrices,
u 1 = ( 0 i i 0 ) , u 2 = ( 0 − 1 1 0 ) , u 3 = ( i 0 0 − i ) , {\displaystyle u_{1}={\begin{pmatrix}0&i\\i&0\end{pmatrix}},\quad u_{2}={\begin{pmatrix}0&-1\\1&0\end{pmatrix}},\quad u_{3}={\begin{pmatrix}i&0\\0&-i\end{pmatrix}}~,}
which have the form of the general element specified above. This can also be written as s u ( 2 ) = span { i σ 1 , i σ 2 , i σ 3 } {\displaystyle {\mathfrak {su}}(2)=\operatorname {span} \left\{i\sigma _{1},i\sigma _{2},i\sigma _{3}\right\}} using the Pauli matrices. These satisfy the quaternion relationships u 2 u 3 = − u 3 u 2 = u 1 {\displaystyle u_{2}u_{3}=-u_{3}u_{2}=u_{1}} , u 3 u 1 = − u 1 u 3 = u 2 {\displaystyle u_{3}u_{1}=-u_{1}u_{3}=u_{2}} , and u 1 u 2 = − u 2 u 1 = u 3 {\displaystyle u_{1}u_{2}=-u_{2}u_{1}=u_{3}} . The commutator bracket is therefore specified by
[ u 3 , u 1 ] = 2 u 2 , [ u 1 , u 2 ] = 2 u 3 , [ u 2 , u 3 ] = 2 u 1 . {\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}~.}
The above generators are related to the Pauli matrices by u 1 = i σ 1 , u 2 = − i σ 2 {\displaystyle u_{1}=i\ \sigma _{1},u_{2}=-i\ \sigma _{2}} and m u 3 = + i σ 3 {\displaystyle mu_{3}=+i\ \sigma _{3}} . 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. They also correspond to the Pauli X, Y, and Z gates, which are standard generators for the single qubit gates, corresponding to 3D rotations about the axes of the Bloch sphere. The Lie algebra serves to work out the representations of SU(2).
SU(3)
The group SU(3) is an 8-dimensional simple Lie group consisting of all 3 × 3 unitary matrices with determinant 1.
Topology The group SU(3) is a simply-connected, compact Lie group. Its topological structure can be understood by noting that SU(3) acts transitively on the unit sphere S 5 {\displaystyle S^{5}} in C 3 ≅ R 6 {\displaystyle \mathbb {C} ^{3}\cong \mathbb {R} ^{6}} . 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 S5 with fiber S3. 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). The SU(2)-bundles over S5 are classified by π 4 ( S 3 ) = Z 2 {\displaystyle \pi _{4}{\mathord {\left(S^{3}\right)}}=\mathbb {Z} _{2}} since any such bundle can be constructed by looking at trivial bundles on the two hemispheres S N 5 , S S 5 {\displaystyle S_{\text{N}}^{5},S_{\text{S}}^{5}} and looking at the transition function on their intersection, which is a copy of S4, so
S N 5 ∩ S S 5 ≃ S 4 {\displaystyle S_{\text{N}}^{5}\cap S_{\text{S}}^{5}\simeq S^{4}}
Then, all such transition functions are classified by homotopy classes of maps
[ S 4 , S U ( 2 ) ] ≅ [ S 4 , S 3 ] = π 4 ( S 3 ) ≅ Z / 2 {\displaystyle \left[S^{4},\mathrm {SU} (2)\right]\cong \left[S^{4},S^{3}\right]=\pi _{4}{\mathord {\left(S^{3}\right)}}\cong \mathbb {Z} /2}
and as π 4 ( S U ( 3 ) ) = { 0 } {\displaystyle \pi _{4}(\mathrm {SU} (3))=\{0\}} rather than Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } , SU(3) cannot be the trivial bundle SU(2) × S5 ≅ S3 × S5, and therefore must be the unique nontrivial (twisted) bundle. This can be shown by looking at the induced long exact sequence on homotopy groups.
Representation theory The representation theory of SU(3) is well-understood. Descriptions of these representations, from the point of view of its complexified Lie algebra s l ( 3 ; C ) {\displaystyle {\mathfrak {sl}}(3;\mathbb {C} )} , may be found in the articles on Lie algebra representations or the Clebsch–Gordan coefficients for SU(3).
Lie algebra The generators, T, of the Lie algebra s u ( 3 ) {\displaystyle {\mathfrak {su}}(3)} of SU(3) in the defining (particle physics, Hermitian) representation, are