Special unitary group

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 ${\displaystyle i}$ from the mathematicians'. With this convention, one can then choose generators T_a that are traceless Hermitian complex n×n matrices, where:

${\displaystyle T_{a}\,T_{b}={\tfrac {1}{\,2n\,}}\,\delta _{ab}\,I_{n}+{\tfrac {1}{2}}\,\sum _{c=1}^{n^{2}-1}\left(if_{abc}+d_{abc}\right)\,T_{c}}$

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:

${\displaystyle ~\left[T_{a},\,T_{b}\right]~=~i\sum _{c=1}^{n^{2}-1}\,f_{abc}\,T_{c}\;,}$

and the corresponding anticommutator is:

${\displaystyle \left\{T_{a},\,T_{b}\right\}~=~{\tfrac {1}{n}}\,\delta _{ab}\,I_{n}+\sum _{c=1}^{n^{2}-1}{d_{abc}\,T_{c}}~.}$

The factor of ${\displaystyle i}$ in the commutation relation arises from the physics convention and is not present when using the mathematicians' convention.

The conventional normalization condition is

${\displaystyle \sum _{c,e=1}^{n^{2}-1}d_{ace}\,d_{bce}={\tfrac {\,n^{2}-4\,}{n}}\,\delta _{ab}~.}$

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:

${\displaystyle \left(T_{a}\right)_{jk}=-if_{ajk}.}$

If we consider ${\displaystyle \alpha ,\beta }$ as a pair in ${\displaystyle \mathbb {C} ^{2}}$ where ${\displaystyle \alpha =a+bi}$ and ${\displaystyle \beta =c+di}$, then the equation ${\displaystyle |\alpha |^{2}+|\beta |^{2}=1}$ becomes

${\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=1}$

This is the equation of the 3-sphere S³. This can also be seen using an embedding: the map

${\displaystyle {\begin{aligned}\varphi \colon \mathbb {C} ^{2}\to {}&\operatorname {M} (2,\mathbb {C} )\\[5pt]\varphi (\alpha ,\beta )={}&{\begin{pmatrix}\alpha &-{\overline {\beta }}\\\beta &{\overline {\alpha }}\end{pmatrix}},\end{aligned}}}$

where ${\displaystyle \operatorname {M} (2,\mathbb {C} )}$ denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering ${\displaystyle \mathbb {C} ^{2}}$ diffeomorphic to ${\displaystyle \mathbb {R} ^{4}}$ and ${\displaystyle \operatorname {M} (2,\mathbb {C} )}$ diffeomorphic to ${\displaystyle \mathbb {R} ^{8}}$). 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 ${\displaystyle \operatorname {M} (2,\mathbb {C} )}$, 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.

The complex matrix:

${\displaystyle {\begin{pmatrix}a+bi&c+di\\-c+di&a-bi\end{pmatrix}}\quad (a,b,c,d\in \mathbb {R} )}$

can be mapped to the quaternion:

${\displaystyle a\,{\hat {1}}+b\,{\hat {i}}+c\,{\hat {j}}+d\,{\hat {k}}}$

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]

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 Lie algebra of SU(2) consists of ${\displaystyle 2\times 2}$ skew-Hermitian matrices with trace zero.[7] Explicitly, this means

${\displaystyle {\mathfrak {su}}(2)=\left\{{\begin{pmatrix}i\ a&-{\overline {z}}\\z&-i\ a\end{pmatrix}}:\ a\in \mathbb {R} ,z\in \mathbb {C} \right\}~.}$

The Lie algebra is then generated by the following matrices,

${\displaystyle u_{1}={\begin{pmatrix}0&i\\i&0\end{pmatrix}},\quad u_{2}={\begin{pmatrix}0&-1\\1&0\end{pmatrix}},\quad u_{3}={\begin{pmatrix}i&0\\0&-i\end{pmatrix}}~,}$

which have the form of the general element specified above.

This can also be written as ${\displaystyle {\mathfrak {su}}(2)=\operatorname {span} \left\{i\sigma _{1},i\sigma _{2},i\sigma _{3}\right\}}$ using the pauli matricies.

These satisfy the quaternion relationships ${\displaystyle u_{2}\ u_{3}=-u_{3}\ u_{2}=u_{1}~,}$ ${\displaystyle u_{3}\ u_{1}=-u_{1}\ u_{3}=u_{2}~,}$ and ${\displaystyle u_{1}u_{2}=-u_{2}\ u_{1}=u_{3}~.}$ The commutator bracket is therefore specified by

${\displaystyle \left[u_{3},u_{1}\right]=2\ u_{2},\quad \left[u_{1},u_{2}\right]=2\ u_{3},\quad \left[u_{2},u_{3}\right]=2\ u_{1}~.}$

The above generators are related to the Pauli matrices by ${\displaystyle u_{1}=i\ \sigma _{1}~,\,u_{2}=-i\ \sigma _{2}}$ and ${\displaystyle u_{3}=+i\ \sigma _{3}~.}$ 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).

The group ${\displaystyle SU(3)}$ 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 ${\displaystyle S^{5}}$ in ${\displaystyle \mathbb {C} ^{3}\cong \mathbb {R} ^{6}}$. 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 ${\displaystyle S^{5}}$ with fiber ${\displaystyle S^{3}}$. 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 ${\displaystyle SU(2)}$-bundles over ${\displaystyle S^{5}}$ are classified by ${\displaystyle \pi _{4}{\mathord {\left(S^{3}\right)}}=\mathbb {Z} _{2}}$ since any such bundle can be constructed by looking at trivial bundles on the two hemispheres ${\displaystyle S_{N}^{5},S_{S}^{5}}$ and looking at the transition function on their intersection, which is homotopy equivalent to ${\displaystyle S^{4}}$, so

${\displaystyle S_{N}^{5}\cap S_{S}^{5}\simeq S^{4}}$

Then, all such transition functions are classified by homotopy classes of maps

${\displaystyle \left[S^{4},SU(2)\right]\cong \left[S^{4},S^{3}\right]=\pi _{4}{\mathord {\left(S^{3}\right)}}\cong \mathbb {Z} /2}$

and as ${\displaystyle \pi _{4}(SU(3))=\{0\}}$ rather than ${\displaystyle \mathbb {Z} /2}$, ${\displaystyle SU(3)}$ cannot be the trivial bundle ${\displaystyle SU(2)\times S^{5}\cong S^{3}\times S^{5}}$, and therefore must be the unique nontrivial (twisted) bundle. This can be shown by looking at the induced long exact sequence on homotopy groups.

The representation theory of ${\displaystyle SU(3)}$ is well-understood.[10] Descriptions of these representations, from the point of view of its complexified Lie algebra ${\displaystyle {\mathfrak {sl}}(3;\mathbb {C} )}$, may be found in the articles on Lie algebra representations or the Clebsch–Gordan coefficients for SU(3).

The generators, T, of the Lie algebra ${\displaystyle {\mathfrak {su}}(3)}$ of ${\displaystyle SU(3)}$ in the defining (particle physics, Hermitian) representation, are

${\displaystyle T_{a}={\frac {\lambda _{a}}{2}}~,}$

where λ, the Gell-Mann matrices, are the SU(3) analog of the Pauli matrices for SU(2):

${\displaystyle {\begin{aligned}\lambda _{1}={}&{\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}},&\lambda _{2}={}&{\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}},&\lambda _{3}={}&{\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}},\\[6pt]\lambda _{4}={}&{\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix}},&\lambda _{5}={}&{\begin{pmatrix}0&0&-i\\0&0&0\\i&0&0\end{pmatrix}},\\[6pt]\lambda _{6}={}&{\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}},&\lambda _{7}={}&{\begin{pmatrix}0&0&0\\0&0&-i\\0&i&0\end{pmatrix}},&\lambda _{8}={\frac {1}{\sqrt {3}}}&{\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}}.\end{aligned}}}$

These λ_a span all traceless Hermitian matrices H of the Lie algebra, as required. Note that λ₂, λ₅, λ₇ are antisymmetric.

They obey the relations

${\displaystyle {\begin{aligned}\left[T_{a},T_{b}\right]&=i\sum _{c=1}^{8}f_{abc}T_{c},\\\left\{T_{a},T_{b}\right\}&={\frac {1}{3}}\delta _{ab}I_{3}+\sum _{c=1}^{8}d_{abc}T_{c},\end{aligned}}}$

or, equivalently,

${\displaystyle \{\lambda _{a},\lambda _{b}\}={\frac {4}{3}}\delta _{ab}I_{3}+2\sum _{c=1}^{8}{d_{abc}\lambda _{c}}}$.

The f are the structure constants of the Lie algebra, given by

${\displaystyle {\begin{aligned}f_{123}&=1,\\f_{147}=-f_{156}=f_{246}=f_{257}=f_{345}=-f_{367}&={\frac {1}{2}},\\f_{458}=f_{678}&={\frac {\sqrt {3}}{2}},\end{aligned}}}$

while all other f_abc 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}.[lower-alpha 3]

The symmetric coefficients d take the values

${\displaystyle {\begin{aligned}d_{118}=d_{228}=d_{338}=-d_{888}&={\frac {1}{\sqrt {3}}}\\d_{448}=d_{558}=d_{668}=d_{778}&=-{\frac {1}{2{\sqrt {3}}}}\\d_{344}=d_{355}=-d_{366}=-d_{377}=-d_{247}=d_{146}=d_{157}=d_{256}&={\frac {1}{2}}~.\end{aligned}}}$

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]

${\displaystyle {\begin{aligned}\exp(i\theta H)={}&\left[-{\frac {1}{3}}I\sin \left(\varphi +{\frac {2\pi }{3}}\right)\sin \left(\varphi -{\frac {2\pi }{3}}\right)-{\frac {1}{2{\sqrt {3}}}}~H\sin(\varphi )-{\frac {1}{4}}~H^{2}\right]{\frac {\exp \left({\frac {2}{\sqrt {3}}}~i\theta \sin(\varphi )\right)}{\cos \left(\varphi +{\frac {2\pi }{3}}\right)\cos \left(\varphi -{\frac {2\pi }{3}}\right)}}\\[6pt]&{}+\left[-{\frac {1}{3}}~I\sin(\varphi )\sin \left(\varphi -{\frac {2\pi }{3}}\right)-{\frac {1}{2{\sqrt {3}}}}~H\sin \left(\varphi +{\frac {2\pi }{3}}\right)-{\frac {1}{4}}~H^{2}\right]{\frac {\exp \left({\frac {2}{\sqrt {3}}}~i\theta \sin \left(\varphi +{\frac {2\pi }{3}}\right)\right)}{\cos(\varphi )\cos \left(\varphi -{\frac {2\pi }{3}}\right)}}\\[6pt]&{}+\left[-{\frac {1}{3}}~I\sin(\varphi )\sin \left(\varphi +{\frac {2\pi }{3}}\right)-{\frac {1}{2{\sqrt {3}}}}~H\sin \left(\varphi -{\frac {2\pi }{3}}\right)-{\frac {1}{4}}~H^{2}\right]{\frac {\exp \left({\frac {2}{\sqrt {3}}}~i\theta \sin \left(\varphi -{\frac {2\pi }{3}}\right)\right)}{\cos(\varphi )\cos \left(\varphi +{\frac {2\pi }{3}}\right)}}\end{aligned}}}$

where

${\displaystyle \varphi \equiv {\frac {1}{3}}\left[\arccos \left({\frac {3{\sqrt {3}}}{2}}\det H\right)-{\frac {\pi }{2}}\right].}$

An important example of this type of group is the Picard modular group ${\displaystyle \operatorname {SU} (2,1;\mathbb {Z} [i])}$ which acts (projectively) on complex hyperbolic space of degree two, in the same way that ${\displaystyle \operatorname {SL} (2,9;\mathbb {Z} )}$ 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.[16]

A further example is ${\displaystyle \operatorname {SU} (1,1;\mathbb {C} )}$, which is isomorphic to ${\displaystyle \operatorname {SL} (2,\mathbb {R} )}$.