Special unitary group

In mathematics, the special unitary group of degree Template:Math, denoted Template:Math, is the Lie group of Template:Math 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 Template:Math, consisting of all Template:Math unitary matrices. As a compact classical group, Template:Math is the group that preserves the standard inner product on <math>\mathbb{C}^n</math>.Template:Efn It is itself a subgroup of the general linear group, <math>\operatorname{SU}(n) \subset \operatorname{U}(n) \subset \operatorname{GL}(n, \mathbb{C} ).</math>

The Template:Math groups find wide application in the Standard Model of particle physics, especially Template:Math in the electroweak interaction and Template:Math in quantum chromodynamics.<ref>Template:Cite book</ref>

The simplest case, Template:Math, is the trivial group, having only a single element. The group Template:Math 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 Template:Math to the [[rotation group SO(3)|rotation group Template:Math]] whose kernel is Template:Math.Template:Efn Since the quaternions can be identified as the even subalgebra of the Clifford Algebra Template:Math, Template:Math is in fact identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations.

Properties

The special unitary group Template:Math is a strictly real Lie group (vs. a more general complex Lie group). Its dimension as a real manifold is Template:Math. Topologically, it is compact and simply connected.<ref>Template:Harvnb, Proposition 13.11</ref> Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below).<ref>Template:Cite book</ref>

The center of Template:Math is isomorphic to the cyclic group <math>\mathbb{Z}/n\mathbb{Z}</math>, and is composed of the diagonal matrices Template:Math for Template:Math an Template:Mathth root of unity and Template:Math the Template:Math identity matrix.

Its outer automorphism group for Template:Math is <math>\mathbb{Z}/2\mathbb{Z},</math> while the outer automorphism group of Template:Math is the trivial group.

A maximal torus of rank Template:Math is given by the set of diagonal matrices with determinant Template:Math. The Weyl group of Template:Math is the symmetric group Template:Math, which is represented by signed permutation matrices (the signs being necessary to ensure that the determinant is Template:Math).

The Lie algebra of Template:Math, denoted by <math>\mathfrak{su}(n)</math>, can be identified with the set of traceless anti‑Hermitian Template:Math complex matrices, with the regular commutator as a Lie bracket. Particle physicists often use a different, equivalent representation: The set of traceless Hermitian Template:Math complex matrices with Lie bracket given by Template:Math times the commutator.

Lie algebra

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

The Lie algebra <math>\mathfrak{su}(n)</math> of <math>\operatorname{SU}(n)</math> consists of Template:Math skew-Hermitian matrices with trace zero.<ref>Template:Harvnb Proposition 3.24</ref> This (real) Lie algebra has dimension Template:Math. More information about the structure of this Lie algebra can be found below in Template:Slink.

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 <math>i</math> from the mathematicians'. With this convention, one can then choose generators Template:Math that are traceless Hermitian complex Template:Math matrices, where:

<math display="block"> 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 </math>

where the Template:Math are the structure constants and are antisymmetric in all indices, while the Template:Math-coefficients are symmetric in all indices.

As a consequence, the commutator is:

<math display="block"> ~ \left[T_a, \, T_b\right] ~ = ~ i \sum_{c=1}^{n^2 -1} \, f_{abc} \, T_c \;,</math>

and the corresponding anticommutator is:

<math display="block"> \left\{T_a, \, T_b\right\} ~ = ~ \tfrac{1}{n} \, \delta_{ab} \, I_n + \sum_{c=1}^{n^2 -1}{d_{abc} \, T_c} ~.</math>

The factor of Template:Math in the commutation relation arises from the physics convention and is not present when using the mathematicians' convention.

The conventional normalization condition is

<math display="block">\sum_{c,e=1}^{n^2 - 1} d_{ace}\,d_{bce} = \frac{\, n^2 - 4 \,}{n} \, \delta_{ab}~ .</math>

The generators satisfy the Jacobi identity:<ref>Template:Cite book</ref>

<math display="block">[T_a,[T_b,T_c]]+[T_b,[T_c,T_a]]+[T_c,[T_a,T_b]]=0.</math>

By convention, in the physics literature the generators <math>T_a</math> are defined as the traceless Hermitian complex matrices with a <math>1/2</math> prefactor: for the <math>SU(2)</math> group, the generators are chosen as <math>\frac{1}{2} \sigma_1, \frac{1}{2} \sigma_2, \frac{1}{2} \sigma_3</math> where <math>\sigma_a</math> are the Pauli matrices, while for the case of <math>SU(3)</math> one defines <math>T_a = \frac{1}{2} \lambda_a</math> where <math>\lambda_a</math> are the Gell-Mann matrices.<ref>Template:Cite book</ref> With these definitions, the generators satisfy the following normalization condition:

<math display="block">Tr(T_a T_b) = \frac{1}{2} \delta_{ab}.</math>

Adjoint representation

In the Template:Math-dimensional adjoint representation, the generators are represented by Template:Math matrices, whose elements are defined by the structure constants themselves:

<math display="block">\left(T_a\right)_{jk} = -if_{ajk}.</math>

The group SU(2)

Using matrix multiplication for the binary operation, Template:Math forms a group,<ref>Template:Harvnb Exercise 1.5</ref>

<math display="block">\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\}~,</math>

where the overline denotes complex conjugation.

Diffeomorphism with the 3-sphere S³

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

<math display="block"> a^2 + b^2 + c^2 + d^2 = 1 </math>

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

<math display="block">\begin{align}

 \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{align}</math>

where <math>\operatorname{M}(2,\mathbb{C})</math> denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering <math>\mathbb{C}^2</math> diffeomorphic to <math>\mathbb{R}^4</math> and <math>\operatorname{M}(2,\mathbb{C})</math> diffeomorphic to <math>\mathbb{R}^8</math>). Hence, the restriction of Template:Math to the 3-sphere (since modulus is 1), denoted Template:Math, is an embedding of the 3-sphere onto a compact submanifold of <math>\operatorname{M}(2,\mathbb{C})</math>, namely Template:Math.

Therefore, as a manifold, Template:Math is diffeomorphic to Template:Math, which shows that Template:Math is simply connected and that Template:Math can be endowed with the structure of a compact, connected Lie group.

Isomorphism with group of versors

Quaternions of norm 1 are called versors since they generate the rotation group SO(3): The Template:Math matrix:

<math display="block"> \begin{pmatrix}

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

</math>

can be mapped to the quaternion

<math display="block">a\,\hat{1} + b\,\hat{i} + c\,\hat{j} + d\,\hat{k}</math>

This map is in fact a group isomorphism. Additionally, the determinant of the matrix is the squared norm of the corresponding quaternion. Clearly any matrix in Template:Math is of this form and, since it has determinant Template:Math, the corresponding quaternion has norm Template:Math. Thus Template:Math is isomorphic to the group of versors.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Relation to spatial rotations

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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 Template:Math to [[3D rotation group|Template:Math]]; consequently Template:Math is isomorphic to the quotient group Template:Math, the manifold underlying Template:Math is obtained by identifying antipodal points of the 3-sphere Template:Math, and Template:Math is the universal cover of Template:Math.

Lie algebra

The Lie algebra of Template:Math consists of Template:Math skew-Hermitian matrices with trace zero.<ref>Template:Harvnb Proposition 3.24</ref> Explicitly, this means

<math display="block">\mathfrak{su}(2) = \left\{ \begin{pmatrix} i\ a & -\overline{z} \\ z & -i\ a \end{pmatrix}:\ a \in \mathbb{R}, z \in \mathbb{C} \right\}~.</math>

The Lie algebra is then generated by the following matrices,

<math display="block">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}~,

</math>

which have the form of the general element specified above.

This can also be written as <math>\mathfrak{s u}(2)=\operatorname{span}\left\{i \sigma_{1}, i \sigma_{2}, i \sigma_{3}\right\}</math> using the Pauli matrices.

These satisfy the quaternion relationships <math>u_2\ u_3 = -u_3\ u_2 = u_1~,</math> <math>u_3\ u_1 = -u_1\ u_3 = u_2~,</math> and <math>u_1 u_2 = -u_2\ u_1 = u_3~.</math> The commutator bracket is therefore specified by

<math display="block">\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~.</math>

The above generators are related to the Pauli matrices by <math>u_1 = i\ \sigma_1~, \, u_2 = -i\ \sigma_2</math> and <math>u_3 = +i\ \sigma_3~.</math> 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 [[representation theory of SU(2)|representations of Template:Math]].

SU(3)

The group Template:Math is an 8-dimensional simple Lie group consisting of all Template:Math unitary matrices with determinant 1.

Topology

The group Template:Math is a simply-connected, compact Lie group.<ref>Template:Harvnb Proposition 13.11</ref> Its topological structure can be understood by noting that Template:Math acts transitively on the unit sphere <math>S^5</math> in <math>\mathbb{C}^3 \cong \mathbb{R}^6</math>. The stabilizer of an arbitrary point in the sphere is isomorphic to Template:Math, which topologically is a 3-sphere. It then follows that Template:Math is a fiber bundle over the base Template:Math with fiber Template:Math. Since the fibers and the base are simply connected, the simple connectedness of Template:Math then follows by means of a standard topological result (the long exact sequence of homotopy groups for fiber bundles).<ref>Template:Harvnb Section 13.2</ref>

The Template:Math-bundles over Template:Math are classified by <math>\pi_4\mathord\left(S^3\right) = \mathbb{Z}_2</math> since any such bundle can be constructed by looking at trivial bundles on the two hemispheres <math>S^5_\text{N}, S^5_\text{S}</math> and looking at the transition function on their intersection, which is a copy of Template:Math, so

<math display="block">S^5_\text{N} \cap S^5_\text{S} \simeq S^4</math>

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

<math display="block">\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</math>

and as <math>\pi_4(\mathrm{SU}(3)) = \{0\}</math> rather than <math>\mathbb{Z}/2</math>, Template:Math cannot be the trivial bundle Template:Math, 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 Template:Math is well-understood.<ref>Template:Harvnb Chapter 6</ref> Descriptions of these representations, from the point of view of its complexified Lie algebra <math>\mathfrak{sl}(3; \mathbb{C})</math>, may be found in the articles on Lie algebra representations or [[Clebsch–Gordan coefficients for SU(3)#Representations of the SU(3) group|the Clebsch–Gordan coefficients for Template:Math]].

Lie algebra

The generators, Template:Mvar, of the Lie algebra <math>\mathfrak{su}(3)</math> of Template:Math in the defining (particle physics, Hermitian) representation, are

<math display="block">T_a = \frac{\lambda_a}{2}~, </math>

where Template:Math, the Gell-Mann matrices, are the Template:Math analog of the Pauli matrices for Template:Math:

<math display="block">\begin{align}

 \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{align}</math>

These Template:Math span all traceless Hermitian matrices Template:Mvar of the Lie algebra, as required. Note that Template:Math are antisymmetric.

They obey the relations

<math display="block">\begin{align}

   \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{align}</math>

or, equivalently,

<math display="block">\begin{align}

           \left[\lambda_a, \lambda_b\right] &= 2i \sum_{c=1}^8 f_{abc} \lambda_c, \\
            \{\lambda_a, \lambda_b\} &= \frac{4}{3}\delta_{ab} I_3 + 2\sum_{c=1}^8{d_{abc} \lambda_c}.

\end{align}</math>

The Template:Mvar are the structure constants of the Lie algebra, given by

<math display="block">\begin{align}

                                                     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{align}</math>

while all other Template:Math not related to these by permutation are zero. In general, they vanish unless they contain an odd number of indices from the set Template:Math.Template:Efn

The symmetric coefficients Template:Math take the values

<math display="block">\begin{align}

                                            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{align}</math>

They vanish if the number of indices from the set Template:Math is odd.

A generic Template:Math group element generated by a traceless 3×3 Hermitian matrix Template:Mvar, normalized as Template:Math, can be expressed as a second order matrix polynomial in Template:Mvar:<ref>Template:Cite journal; Template:Cite journal</ref>

<math display="block">\begin{align}

 \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{align}</math> LP where

<math display="block">\varphi \equiv \frac{1}{3}\left[\arccos\left(\frac{3\sqrt{3}}{2}\det H\right) - \frac{\pi}{2}\right].</math>

Lie algebra structure

As noted above, the Lie algebra <math>\mathfrak{su}(n)</math> of Template:Math consists of Template:Math skew-Hermitian matrices with trace zero.<ref>Template:Harvnb Proposition 3.24</ref>

The complexification of the Lie algebra <math>\mathfrak{su}(n)</math> is <math>\mathfrak{sl}(n; \mathbb{C})</math>, the space of all Template:Math complex matrices with trace zero.<ref>Template:Harvnb Section 3.6</ref> A Cartan subalgebra then consists of the diagonal matrices with trace zero,<ref>Template:Harvnb Section 7.7.1</ref> which we identify with vectors in <math>\mathbb C^n</math> whose entries sum to zero. The roots then consist of all the Template:Math permutations of Template:Math.

A choice of simple roots is

<math display="block">\begin{align}

 (&1, -1,  0, \dots, 0,  0), \\
 (&0,  1, -1, \dots, 0,  0), \\
  &\vdots                    \\
 (&0,  0,  0, \dots, 1, -1).

\end{align}</math>

So, Template:Math is of rank Template:Math and its Dynkin diagram is given by Template:Math, a chain of Template:Math nodes: Template:Dynkin...Template:Dynkin.<ref>Template:Harvnb Section 8.10.1</ref> Its Cartan matrix is

<math display="block">\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}.

</math>

Its Weyl group or Coxeter group is the symmetric group Template:Math, the symmetry group of the Template:Math-simplex.

Generalized special unitary group

For a field Template:Math, the generalized special unitary group over F, Template:Math, is the group of all linear transformations of determinant 1 of a vector space of rank Template:Math over Template:Math which leave invariant a nondegenerate, Hermitian form of signature Template:Math. This group is often referred to as the special unitary group of signature Template:Math over Template:Math. The field Template:Math can be replaced by a commutative ring, in which case the vector space is replaced by a free module.

Specifically, fix a Hermitian matrix Template:Math of signature Template:Math in <math>\operatorname{GL}(n, \mathbb{R})</math>, then all

<math display="block">M \in \operatorname{SU}(p, q, \mathbb{R})</math>

satisfy

<math display="block">\begin{align}

 M^{*} A M &= A \\
    \det M &= 1.

\end{align}</math>

Often one will see the notation Template:Math without reference to a ring or field; in this case, the ring or field being referred to is <math>\mathbb C</math> and this gives one of the classical Lie groups. The standard choice for Template:Math when <math>\operatorname{F} = \mathbb{C}</math> is

<math display="block">A = \begin{bmatrix}

    0 & 0       & i \\
    0 & I_{n-2} & 0 \\
   -i & 0       & 0
 \end{bmatrix}.

</math>

However, there may be better choices for Template:Math for certain dimensions which exhibit more behaviour under restriction to subrings of <math>\mathbb C</math>.

Example

An important example of this type of group is the Picard modular group <math>\operatorname{SU}(2, 1; \mathbb{Z}[i])</math> which acts (projectively) on complex hyperbolic space of dimension two, in the same way that <math>\operatorname{SL}(2,9;\mathbb{Z})</math> 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 Template:Math.<ref>Template:Cite arXiv</ref>

A further example is <math>\operatorname{SU}(1, 1; \mathbb{C})</math>, which is isomorphic to <math>\operatorname{SL}(2, \mathbb{R})</math>.

Important subgroups

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. Subgroups of Template:Math that are important in GUT physics are, for Template:Math,

<math display="block">\operatorname{SU}(n) \supset \operatorname{SU}(p) \times \operatorname{SU}(n - p) \times \operatorname{U}(1),</math>

where × denotes the direct product and Template:Math, 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,

<math display="block">\begin{align}

  \operatorname{SU}(n) &\supset \operatorname{SO}(n), \\
 \operatorname{SU}(2n) &\supset \operatorname{Sp}(n).

\end{align}</math>

Since the rank of Template:Math is Template:Math and of Template:Math 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. Template:Math is a subgroup of various other Lie groups,

<math display="block">\begin{align}

  \operatorname{SO}(2n) &\supset \operatorname{SU}(n) \\
   \operatorname{Sp}(n) &\supset \operatorname{SU}(n) \\
 \operatorname{Spin}(4) &= \operatorname{SU}(2) \times \operatorname{SU}(2) \\
     \operatorname{E}_6 &\supset \operatorname{SU}(6) \\
     \operatorname{E}_7 &\supset \operatorname{SU}(8) \\
     \operatorname{G}_2 &\supset \operatorname{SU}(3)

\end{align}</math> See Spin group and Simple Lie group for Template:Math, Template:Math, and Template:Math.

There are also the accidental isomorphisms: Template:Math, Template:Math,Template:Efn and Template:Math.

One may finally mention that Template:Math is the double covering group of Template:Math, a relation that plays an important role in the theory of rotations of 2-spinors in non-relativistic quantum mechanics.

SU(1, 1)

<math>\mathrm{SU}(1,1) = \left \{ \begin{pmatrix}u & v \\ v^* & u^* \end{pmatrix} \in M(2,\mathbb{C}): u u^* - v v^* = 1 \right \},</math> where <math>~u^*~</math> denotes the complex conjugate of the complex number Template:Mvar.

This group is isomorphic to Template:Math and Template:Math<ref>Template:Cite book</ref> where the numbers separated by a comma refer to the signature of the quadratic form preserved by the group. The expression <math>~u u^* - v v^*~</math> in the definition of Template:Math is an Hermitian form which becomes an isotropic quadratic form when Template:Mvar and Template:Math 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

<math display="block">

 j = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\,, \quad
 k = \begin{bmatrix} 1 & \;~0 \\ 0 & -1 \end{bmatrix}\,, \quad
 i = \begin{bmatrix} \;~0 & 1 \\ -1 & 0 \end{bmatrix}~.

</math>

Then <math>~j\,k = \begin{bmatrix} 0 & -1 \\ 1 & \;~0 \end{bmatrix} = -i ~,~</math> <math>~ i\,j\,k = I_2 \equiv \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}~,~</math> the 2×2 identity matrix, <math>~k\,i = j ~,</math> and <math>\;i\,j = k \;,</math> and the elements Template:Mvar and Template:Mvar all anticommute, as in quaternions. Also <math>i</math> is still a square root of Template:Math (negative of the identity matrix), whereas <math>~j^2 = k^2 = +I_2~</math> are not, unlike in quaternions. For both quaternions and coquaternions, all scalar quantities are treated as implicit multiples of Template:Mvar₂ and notated as Template:Math.

The coquaternion <math>~q = w + x\,i + y\,j + z\,k~</math> with scalar Template:Mvar, has conjugate <math>~q = w - x\,i - y\,j - z\,k~</math> similar to Hamilton's quaternions. The quadratic form is <math>~q\,q^* = w^2 + x^2 - y^2 - z^2.</math>

Note that the 2-sheet hyperboloid <math>\left\{ x i + y j + z k : x^2 - y^2 - z^2 = 1 \right\}</math> corresponds to the imaginary units in the algebra so that any point Template:Mvar on this hyperboloid can be used as a pole of a sinusoidal wave according to Euler's formula.

The hyperboloid is stable under Template:Math, illustrating the isomorphism with Template:Math. 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 Template:Nowrap The Poincaré sphere model used since 1892 has been compared to a 2-sheet hyperboloid model,<ref>Template:Cite journal</ref> and the practice of [[SU(1,1) interferometry|Template:Math interferometry]] has been introduced.

When an element of Template:Math 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 Template:Math in the complex projective line, the action of Template:Math is given by

<math display="block">\begin{pmatrix}u & v \\ v^* & u^* \end{pmatrix}\,\bigl[\;z,\;1\;\bigr] = [\;u\,z + v, \, v^*\,z +u^*\;] \, = \, \left[\;\frac{uz + v}{v^*z +u^*}, \, 1 \;\right]</math>

since in projective coordinates <math>(\;u\,z + v, \; v^*\,z +u^*\;) \thicksim \left(\;\frac{\,u\,z + v\,}{v^*\,z +u^*}, \; 1 \;\right).</math>

Writing <math>\;suv + \overline{suv} = 2\,\Re\mathord\bigl(\,suv\,\bigr)\;,</math> complex number arithmetic shows

<math display="block">\bigl|u\,z + v\bigr|^2 = S + z\,z^* \quad \text{ and } \quad \bigl|v^*\,z + u^*\bigr|^2 = S + 1~,</math> where <math>S = v\,v^* \left(z\,z^* + 1\right) + 2\,\Re\mathord\bigl(\,uvz\,\bigr).</math>

Therefore, <math>z\,z^* < 1 \implies \bigl|uz + v\bigr| < \bigl|\,v^*\,z + u^*\,\bigr|</math> so that their ratio lies in the open disk.<ref>Template:Cite book</ref>

See also

• Unitary group
• Projective special unitary group, Template:Math
• Orthogonal group
• Generalizations of Pauli matrices
• Representation theory of SU(2)

References

• Template:Citation
• Template:Citation