Special unitary group

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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 ${\displaystyle \mathbb {C} ^{n}}$.^[a] It is itself a subgroup of the general linear group, ${\displaystyle \operatorname {SU} (n)\subset \operatorname {U} (n)\subset \operatorname {GL} (n,\mathbb {C} ).}$

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.^[1]

The simplest case, SU(1), is the

(3), that enables a spinor presentation of rotations.

Properties

The special unitary group SU(n) is a strictly real

The

${\displaystyle \mathbb {Z} /n\mathbb {Z} }$, 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 ${\displaystyle \mathbb {Z} /2\mathbb {Z} ,}$ 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 S_n, 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 ${\displaystyle {\mathfrak {su}}(n)}$, can be identified with the set of

n × n complex matrices with Lie bracket given by −i times the commutator.

The Lie algebra ${\displaystyle {\mathfrak {su}}(n)}$ of ${\displaystyle \operatorname {SU} (n)}$ consists of n × n

.

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

The conventional normalization condition is

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

The generators satisfy the Jacobi identity:^[5]

$${\displaystyle [T_{a},[T_{b},T_{c}]]+[T_{b},[T_{c},T_{a}]]+[T_{c},[T_{a},T_{b}]]=0.}$$

By convention, in the physics literature the generators ${\displaystyle T_{a}}$ are defined as the traceless Hermitian complex matrices with a ${\displaystyle 1/2}$ prefactor: for the ${\displaystyle SU(2)}$ group, the generators are chosen as ${\displaystyle {\frac {1}{2}}\sigma _{1},{\frac {1}{2}}\sigma _{2},{\frac {1}{2}}\sigma _{3}}$ where ${\displaystyle \sigma _{a}}$ are the Pauli matrices, while for the case of ${\displaystyle SU(3)}$ one defines ${\displaystyle T_{a}={\frac {1}{2}}\lambda _{a}}$ where ${\displaystyle \lambda _{a}}$ are the Gell-Mann matrices.^[6] With these definitions, the generators satisfy the following normalization condition:

$${\displaystyle Tr(T_{a}T_{b})={\frac {1}{2}}\delta _{ab}.}$$

In the (n² − 1)-dimensional

, the generators are represented by (n² − 1) × (n² − 1) matrices, whose elements are defined by the structure constants themselves:

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

Using matrix multiplication for the binary operation, SU(2) forms a group,^[7]

$${\displaystyle \operatorname {SU} (2)=\left\{{\begin{pmatrix}\alpha &-{\overline {\beta }}\\\beta &{\overline {\alpha }}\end{pmatrix}}:\ \ \alpha ,\beta \in \mathbb {C} ,|\alpha |^{2}+|\beta |^{2}=1\right\}~,}$$

where the overline denotes complex conjugation.

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 SU(2) 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 a group isomorphism. 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 SU(2) is isomorphic to the group of versors.^[8]

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 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

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 matrices.

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

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

.

The Lie algebra serves to work out the representations of SU(2).

The group SU(3) is an 8-dimensional simple Lie group consisting of all 3 × 3 unitary matrices with determinant 1.

The group SU(3) is a simply-connected, compact Lie group.

on the unit sphere ${\displaystyle S^{5}}$ in ${\displaystyle \mathbb {C} ^{3}\cong \mathbb {R} ^{6}}$. The

The SU(2)-bundles over S⁵ 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 ${\displaystyle S_{\text{N}}^{5},S_{\text{S}}^{5}}$ and looking at the transition function on their intersection, which is a copy of S⁴, so

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

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

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

and as ${\displaystyle \pi _{4}(\mathrm {SU} (3))=\{0\}}$ rather than ${\displaystyle \mathbb {Z} /2}$, 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.[12] 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 SU(3) in the defining (particle physics, Hermitian) representation, are

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

where λ_a, 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,

$${\displaystyle {\begin{aligned}\left[\lambda _{a},\lambda _{b}\right]&=2i\sum _{c=1}^{8}f_{abc}\lambda _{c},\\\{\lambda _{a},\lambda _{b}\}&={\frac {4}{3}}\delta _{ab}I_{3}+2\sum _{c=1}^{8}{d_{abc}\lambda _{c}}.\end{aligned}}}$$

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}.^[c]

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:^[13]

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

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

As noted above, the Lie algebra ${\displaystyle {\mathfrak {su}}(n)}$ of SU(n) consists of n × n

The complexification of the Lie algebra ${\displaystyle {\mathfrak {su}}(n)}$ is ${\displaystyle {\mathfrak {sl}}(n;\mathbb {C} )}$, the space of all n × n complex matrices with trace zero.^[15] A Cartan subalgebra then consists of the diagonal matrices with trace zero,^[16] which we identify with vectors in ${\displaystyle \mathbb {C} ^{n}}$ whose entries sum to zero. The roots then consist of all the n(n − 1) permutations of (1, −1, 0, ..., 0).

A choice of

is

$${\displaystyle {\begin{aligned}(&1,-1,0,\dots ,0,0),\\(&0,1,-1,\dots ,0,0),\\&\vdots \\(&0,0,0,\dots ,1,-1).\end{aligned}}}$$

So, SU(n) is of rank n − 1 and its Dynkin diagram is given by A_n−1, a chain of n − 1 nodes: ....^[17] Its Cartan matrix is

$${\displaystyle {\begin{pmatrix}2&-1&0&\dots &0\\-1&2&-1&\dots &0\\0&-1&2&\dots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\dots &2\end{pmatrix}}.}$$

Its Weyl group or Coxeter group is the symmetric group S_n, the symmetry group of the (n − 1)-simplex.

For a

.

Specifically, fix a Hermitian matrix A of signature p q in ${\displaystyle \operatorname {GL} (n,\mathbb {R} )}$, then all

$${\displaystyle M\in \operatorname {SU} (p,q,\mathbb {R} )}$$

satisfy

$${\displaystyle {\begin{aligned}M^{*}AM&=A\\\det M&=1.\end{aligned}}}$$

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 ${\displaystyle \mathbb {C} }$ and this gives one of the classical Lie groups. The standard choice for A when ${\displaystyle \operatorname {F} =\mathbb {C} }$ is

$${\displaystyle A={\begin{bmatrix}0&0&i\\0&I_{n-2}&0\\-i&0&0\end{bmatrix}}.}$$

However, there may be better choices for A for certain dimensions which exhibit more behaviour under restriction to subrings of ${\displaystyle \mathbb {C} }$.

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 dimension 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.^[18]

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

In physics the special unitary group is used to represent

are, for p > 1, n − p > 1,

$${\displaystyle \operatorname {SU} (n)\supset \operatorname {SU} (p)\times \operatorname {SU} (n-p)\times \operatorname {U} (1),}$$

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

subgroups,

$${\displaystyle {\begin{aligned}\operatorname {SU} (n)&\supset \operatorname {SO} (n),\\\operatorname {SU} (2n)&\supset \operatorname {Sp} (n).\end{aligned}}}$$

Since the

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,

$${\displaystyle {\begin{aligned}\operatorname {SO} (2n)&\supset \operatorname {SU} (n)\\\operatorname {Sp} (n)&\supset \operatorname {SU} (n)\\\operatorname {Spin} (4)&=\operatorname {SU} (2)\times \operatorname {SU} (2)\\\operatorname {E} _{6}&\supset \operatorname {SU} (6)\\\operatorname {E} _{7}&\supset \operatorname {SU} (8)\\\operatorname {G} _{2}&\supset \operatorname {SU} (3)\end{aligned}}}$$ See Spin group and Simple Lie group for E₆, E₇, and G₂.

There are also the accidental isomorphisms: SU(4) = Spin(6), SU(2) = Spin(3) = Sp(1),^[d] and U(1) = Spin(2) = SO(2).

One may finally mention that SU(2) is the

.

SU(1, 1)

${\displaystyle \mathrm {SU} (1,1)=\left\{{\begin{pmatrix}u&v\\v^{*}&u^{*}\end{pmatrix}}\in M(2,\mathbb {C} ):uu^{*}-vv^{*}=1\right\},}$ where ${\displaystyle ~u^{*}~}$ denotes the complex conjugate of the complex number u.

This group is isomorphic to SL(2,ℝ) and Spin(2,1)

preserved by the group. The expression ${\displaystyle ~uu^{*}-vv^{*}~}$ in the definition of SU(1,1) is an when u and v are expanded with their real components.

An early appearance of this group was as the "unit sphere" of

in 1852. Let

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

Then ${\displaystyle ~j\,k={\begin{bmatrix}0&-1\\1&\;~0\end{bmatrix}}=-i~,~}$ ${\displaystyle ~i\,j\,k=I_{2}\equiv {\begin{bmatrix}1&0\\0&1\end{bmatrix}}~,~}$ the 2×2 identity matrix, ${\displaystyle ~k\,i=j~,}$ and ${\displaystyle \;i\,j=k\;,}$ and the elements i, j, and k all anticommute, as in quaternions. Also ${\displaystyle i}$ is still a square root of −I₂ (negative of the identity matrix), whereas ${\displaystyle ~j^{2}=k^{2}=+I_{2}~}$ are not, unlike in quaternions. For both quaternions and

, all scalar quantities are treated as implicit multiples of I₂ and notated as 1.

The coquaternion ${\displaystyle ~q=w+x\,i+y\,j+z\,k~}$ with scalar w, has conjugate ${\displaystyle ~q=w-x\,i-y\,j-z\,k~}$ similar to Hamilton's quaternions. The quadratic form is ${\displaystyle ~q\,q^{*}=w^{2}+x^{2}-y^{2}-z^{2}.}$

Note that the 2-sheet hyperboloid ${\displaystyle \left\{xi+yj+zk:x^{2}-y^{2}-z^{2}=1\right\}}$ 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 Spin(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 ${\displaystyle ~p\neq \pm i~}$. The

has been introduced.

When an element of SU(1, 1) is interpreted as a

complex projective line

, the action of SU(1,1) is given by

$${\displaystyle {\begin{pmatrix}u&v\\v^{*}&u^{*}\end{pmatrix}}\,{\bigl [}\;z,\;1\;{\bigr ]}=[\;u\,z+v,\,v^{*}\,z+u^{*}\;]\,=\,\left[\;{\frac {uz+v}{v^{*}z+u^{*}}},\,1\;\right]}$$

since in

projective coordinates

${\displaystyle (\;u\,z+v,\;v^{*}\,z+u^{*}\;)\thicksim \left(\;{\frac {\,u\,z+v\,}{v^{*}\,z+u^{*}}},\;1\;\right).}$

Writing ${\displaystyle \;suv+{\overline {suv}}=2\,\Re {\mathord {\bigl (}}\,suv\,{\bigr )}\;,}$ complex number arithmetic shows

$${\displaystyle {\bigl |}u\,z+v{\bigr |}^{2}=S+z\,z^{*}\quad {\text{ and }}\quad {\bigl |}v^{*}\,z+u^{*}{\bigr |}^{2}=S+1~,}$$ where ${\displaystyle S=v\,v^{*}\left(z\,z^{*}+1\right)+2\,\Re {\mathord {\bigl (}}\,uvz\,{\bigr )}.}$

Therefore, ${\displaystyle z\,z^{*}<1\implies {\bigl |}uz+v{\bigr |}<{\bigl |}\,v^{*}\,z+u^{*}\,{\bigr |}}$ so that their ratio lies in the open disk.[21]

• Unitary group
• Projective special unitary group
  , PSU(n)
• Orthogonal group
• Generalizations of Pauli matrices
• Representation theory of SU(2)

Citations

1. ISBN 0-471-88741-2
   .
2. ^ Hall 2015, Proposition 13.11
3. ISBN 0471965057
   .
4. ^ Hall 2015 Proposition 3.24
5. ISBN 978-0-429-49921-0
   .
6. ISBN 978-0-429-49921-0
   .
7. ^ Hall 2015 Exercise 1.5
8. ^ Savage, Alistair. "LieGroups" (PDF). MATH 4144 notes.
9. ^ Hall 2015 Proposition 3.24
10. ^ Hall 2015 Proposition 13.11
11. ^ Hall 2015 Section 13.2
12. ^ Hall 2015 Chapter 6
13. S2CID 119679825
    .
14. ^ Hall 2015 Proposition 3.24
15. ^ Hall 2015 Section 3.6
16. ^ Hall 2015 Section 7.7.1
17. ^ Hall 2015 Section 8.10.1
18. arXiv:math/0509708
    .
19. MR 1275599
    .
20. S2CID 119146980
    .
21. ISBN 0-471-79080 X
    .

References

• Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,
  ISBN 978-3319134666
• Iachello, Francesco (2006), Lie Algebras and Applications, Lecture Notes in Physics, vol. 708, Springer,
  ISBN 3540362363