Special unitary group

Lie algebra

The Lie algebra ${\displaystyle \mathfrak{s}\mathfrak{u}(n)}$ 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 ${\displaystyle \mathfrak{s}\mathfrak{u}(n)}$ is ⁠${\displaystyle \mathfrak{s}\mathfrak{l}(n;\mathbb{C})}$⁠, 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 ${\displaystyle {\mathbb{C}}^n}$ 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 $${\displaystyle \begin{array}{cc}(& 1,-1,0,\dots ,0,0),\\ (& 0,1,-1,\dots ,0,0),\\ & ⋮\\ (& 0,0,0,\dots ,1,-1).\end{array}}$$

So, SU(n) is of rank n − 1 and its Dynkin diagram is given by A_n−1, a chain of n − 1 nodes: .... Its Cartan matrix is $${\displaystyle \left(\begin{array}{ccccc}2& -1& 0& \dots & 0\\ -1& 2& -1& \dots & 0\\ 0& -1& 2& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & 2\end{array}\right).}$$

Its Weyl group or Coxeter group is the symmetric group S_n, 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 ${\displaystyle i}$ from the mathematicians'. With this convention, one can then choose generators T_a that are traceless Hermitian complex n × n matrices, where: $${\displaystyle T_a{T}_b=\frac{1}{2n}{\delta }_{ab}{I}_n+\frac{1}{2}\sum _{c=1}^{n^2-1}\left(i{f}_{abc}+d_{abc}\right)T_c}$$ 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: $${\displaystyle \left\{T_a,T_b\right\}=\frac{1}{n}{\delta }_{ab}{I}_n+\sum _{c=1}^{n^2-1}{d}_{abc}{T}_c.}$$

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: $${\displaystyle [T_a,[T_b,T_c]]+[T_b,[T_c,T_a]]+[T_c,[T_a,T_b]]=0.}$$

By convention, in the physics literature the generators ${\displaystyle T_a}$ are defined as the traceless Hermitian complex matrices with a 1/2 factor: for SU(2), the generators are chosen as ⁠${\displaystyle \frac{1}{2}{\sigma }_1}$⁠, ⁠${\displaystyle \frac{1}{2}{\sigma }_2}$⁠, ${\displaystyle \frac{1}{2}{\sigma }_3}$ where ${\displaystyle {\sigma }_a}$ are the Pauli matrices, while for the case of SU(3) one defines ${\displaystyle T_a=\frac{1}{2}{\lambda }_a}$ where ${\displaystyle {\lambda }_a}$ are the Gell-Mann matrices. With these definitions, the generators satisfy the following normalization condition: $${\displaystyle Tr(T_a{T}_b)=\frac{1}{2}{\delta }_{ab}.}$$

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: $${\displaystyle {\left(T_a\right)}_{jk}=-i{f}_{ajk}.}$$

SU(2)

Using matrix multiplication for the binary operation, SU(2) forms a group, $${\displaystyle \mathrm{SU}(2)=\left\{\left(\begin{array}{cc}\alpha & -\overline{\beta }\\ \beta & \overline{\alpha }\end{array}\right):\alpha ,\beta \in \mathbb{C},|\alpha {|}^2+|\beta {|}^2=1\right\},}$$ where the overline denotes complex conjugation.

Diffeomorphism with the 3-sphere S3

If we consider ${\displaystyle \alpha ,\beta }$ as a pair in ${\displaystyle {\mathbb{C}}^2}$ where ${\displaystyle \alpha =a+bi}$ and ⁠${\displaystyle \beta =c+di}$⁠ with real numbers ${\displaystyle a,b,c,d}$, 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 $${\displaystyle \begin{array}{cc}\phi :{\mathbb{C}}^2\to & \mathrm{M}(2,\mathbb{C})\\ \phi (\alpha ,\beta )=& \left(\begin{array}{cc}\alpha & -\overline{\beta }\\ \beta & \overline{\alpha }\end{array}\right),\end{array}}$$ where ${\displaystyle \mathrm{M}(2,\mathbb{C})}$ denotes the set of 2-by-2 complex matrices, is an injective real linear map (by considering ${\displaystyle {\mathbb{C}}^2}$ isomorphic to ${\displaystyle {\mathbb{R}}^4}$ and ${\displaystyle \mathrm{M}(2,\mathbb{C})}$ isomorphic to ⁠${\displaystyle {\mathbb{R}}^8}$⁠). Hence, the restriction of φ to the 3-sphere S³ is an embedding of the 3-sphere onto a compact submanifold of ⁠${\displaystyle \mathrm{M}(2,\mathbb{C})}$⁠, 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 $${\displaystyle a\hat{1}+b\hat{i}+c\hat{j}+d\hat{k}}$$ to $${\displaystyle \left(\begin{array}{cc}a+bi& c+di\\ -c+di& a-bi\end{array}\right)(a,b,c,d\in \mathbb{R}).}$$ 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 $${\displaystyle \mathfrak{s}\mathfrak{u}(2)=\left\{\left(\begin{array}{cc}ia& -\overline{z}\\ z& -ia\end{array}\right):a\in \mathbb{R},z\in \mathbb{C}\right\}.}$$

The Lie algebra is then generated by the following matrices, $${\displaystyle u_1=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right),u_2=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),u_3=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right),}$$ which have the form of the general element specified above.

This can also be written as ${\displaystyle \mathfrak{s}\mathfrak{u}(2)=\mathrm{span}\left\{i{\sigma }_1,i{\sigma }_2,i{\sigma }_3\right\}}$ 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 commutator bracket is therefore specified by $${\displaystyle \left[u_3,u_1\right]=2u_2,\left[u_1,u_2\right]=2u_3,\left[u_2,u_3\right]=2u_1.}$$

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).

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 ${\displaystyle S^5}$ 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 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 ${\displaystyle {\pi }_4\left(S^3\right)={\mathbb{Z}}_2}$ 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 $${\displaystyle \left[S^4,\mathrm{S}\mathrm{U}(2)\right]\cong \left[S^4,S^3\right]={\pi }_4\left(S^3\right)\cong \mathbb{Z}/2}$$ and as ${\displaystyle {\pi }_4(\mathrm{S}\mathrm{U}(3))=\{0\}}$ rather than ⁠${\displaystyle \mathbb{Z}/2\mathbb{Z}}$⁠, 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 ⁠${\displaystyle \mathfrak{s}\mathfrak{l}(3;\mathbb{C})}$⁠, 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 ${\displaystyle \mathfrak{s}\mathfrak{u}(3)}$ 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): $${\displaystyle \begin{array}{cccccc}{\lambda }_1=& \left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 0\end{array}\right),& {\lambda }_2=& \left(\begin{array}{ccc}0& -i& 0\\ i& 0& 0\\ 0& 0& 0\end{array}\right),& {\lambda }_3=& \left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 0\end{array}\right),\\ {\lambda }_4=& \left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 1& 0& 0\end{array}\right),& {\lambda }_5=& \left(\begin{array}{ccc}0& 0& -i\\ 0& 0& 0\\ i& 0& 0\end{array}\right),\\ {\lambda }_6=& \left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right),& {\lambda }_7=& \left(\begin{array}{ccc}0& 0& 0\\ 0& 0& -i\\ 0& i& 0\end{array}\right),& {\lambda }_8=\frac{1}{\sqrt{3}}& \left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -2\end{array}\right).\end{array}}$$

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 $${\displaystyle \begin{array}{cc}\left[T_a,T_b\right]& =i\sum _{c=1}^8{f}_{abc}{T}_c,\\ \left\{T_a,T_b\right\}& =\frac{1}{3}{\delta }_{ab}{I}_3+\sum _{c=1}^8{d}_{abc}{T}_c,\end{array}}$$ or, equivalently, $${\displaystyle \begin{array}{cc}\left[{\lambda }_a,{\lambda }_b\right]& =2i\sum _{c=1}^8{f}_{abc}{\lambda }_c,\\ \{{\lambda }_a,{\lambda }_b\}& =\frac{4}{3}{\delta }_{ab}{I}_3+2\sum _{c=1}^8{d}_{abc}{\lambda }_c.\end{array}}$$

The f are the structure constants of the Lie algebra, given by $${\displaystyle \begin{array}{cc}{f}_{123}& =1,\\ f_{147}=-f_{156}=f_{246}=f_{257}=f_{345}=-f_{367}& =\frac{1}{2},\\ f_{458}=f_{678}& =\frac{\sqrt{3}}{2},\end{array}}$$ 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}.

The symmetric coefficients d take the values $${\displaystyle \begin{array}{cc}{d}_{118}=d_{228}=d_{338}=-d_{888}& =\frac{1}{\sqrt{3}}\\ d_{448}=d_{558}=d_{668}=d_{778}& =-\frac{1}{2\sqrt{3}}\\ d_{344}=d_{355}=-d_{366}=-d_{377}=-d_{247}=d_{146}=d_{157}=d_{256}& =\frac{1}{2}.\end{array}}$$

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: $${\displaystyle \begin{array}{cc}\exp (i\theta H)=& \left[-\frac{1}{3}I\sin \left(\phi +\frac{2\pi }{3}\right)\sin \left(\phi -\frac{2\pi }{3}\right)-\frac{1}{2\sqrt{3}}H\sin (\phi )-\frac{1}{4}{H}^2\right]\frac{\exp \left(\frac{2}{\sqrt{3}}i\theta \sin (\phi )\right)}{\cos \left(\phi +\frac{2\pi }{3}\right)\cos \left(\phi -\frac{2\pi }{3}\right)}\\ & +\left[-\frac{1}{3}I\sin (\phi )\sin \left(\phi -\frac{2\pi }{3}\right)-\frac{1}{2\sqrt{3}}H\sin \left(\phi +\frac{2\pi }{3}\right)-\frac{1}{4}{H}^2\right]\frac{\exp \left(\frac{2}{\sqrt{3}}i\theta \sin \left(\phi +\frac{2\pi }{3}\right)\right)}{\cos (\phi )\cos \left(\phi -\frac{2\pi }{3}\right)}\\ & +\left[-\frac{1}{3}I\sin (\phi )\sin \left(\phi +\frac{2\pi }{3}\right)-\frac{1}{2\sqrt{3}}H\sin \left(\phi -\frac{2\pi }{3}\right)-\frac{1}{4}{H}^2\right]\frac{\exp \left(\frac{2}{\sqrt{3}}i\theta \sin \left(\phi -\frac{2\pi }{3}\right)\right)}{\cos (\phi )\cos \left(\phi +\frac{2\pi }{3}\right)}\end{array}}$$ where $${\displaystyle \phi \equiv \frac{1}{3}\left[\arccos \left(\frac{3\sqrt{3}}{2}detH\right)-\frac{\pi }{2}\right].}$$

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 SU(n) that are important in GUT physics are, for p > 1, n − p > 1, $${\displaystyle \mathrm{SU}(n)\supset \mathrm{SU}(p)\times \mathrm{SU}(n-p)\times \mathrm{U}(1),}$$ 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 orthogonal and symplectic subgroups, $${\displaystyle \begin{array}{cc}\mathrm{SU}(n)& \supset \mathrm{SO}(n),\\ \mathrm{SU}(2n)& \supset \mathrm{Sp}(n).\end{array}}$$

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, $${\displaystyle \begin{array}{cc}\mathrm{SO}(2n)& \supset \mathrm{SU}(n)\\ \mathrm{Sp}(n)& \supset \mathrm{SU}(n)\\ \mathrm{Spin}(4)& =\mathrm{SU}(2)\times \mathrm{SU}(2)\\ {\mathrm{E}}_6& \supset \mathrm{SU}(6)\\ {\mathrm{E}}_7& \supset \mathrm{SU}(8)\\ {\mathrm{G}}_2& \supset \mathrm{SU}(3)\end{array}}$$ 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), 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.

Generalized special unitary group

For a field F, the generalized special unitary group over 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 special unitary group of signature p q over F. 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 ⁠${\displaystyle \mathrm{GL}(n,\mathbb{R})}$⁠, then all $${\displaystyle M\in \mathrm{SU}(p,q,\mathbb{R})}$$ satisfy $${\displaystyle \begin{array}{cc}{M}^{\ast }AM& =A\\ detM& =1.\end{array}}$$

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 ${\displaystyle \mathrm{F}=\mathbb{C}}$ is $${\displaystyle A=\left[\begin{array}{ccc}0& 0& i\\ 0& I_{n-2}& 0\\ -i& 0& 0\end{array}\right].}$$

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

Example

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

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

SU(1, 1)

${\displaystyle \mathrm{S}\mathrm{U}(1,1)=\left\{\left(\begin{array}{cc}u& v\\ v^{\ast }& u^{\ast }\end{array}\right)\in M(2,\mathbb{C}):u{u}^{\ast }-v{v}^{\ast }=1\right\},}$ where ${\displaystyle u^{\ast }}$ denotes the complex conjugate of the complex number u.

This group is isomorphic to SL(2,ℝ) and Spin(2,1) where the numbers separated by a comma refer to the signature of the quadratic form preserved by the group. The expression ${\displaystyle u{u}^{\ast }-v{v}^{\ast }}$ in the definition of SU(1,1) is an Hermitian form which becomes an isotropic quadratic form when u and v are expanded with their real components.

An early appearance of this group was as the "unit sphere" of coquaternions (split-quaternions), introduced by James Cockle in 1852. Let $${\displaystyle j=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],k=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],i=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right].}$$

Then ⁠${\displaystyle jk=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]=-i}$⁠, ⁠${\displaystyle ijk=I_2\equiv \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]}$⁠, the 2 × 2 identity matrix, ⁠${\displaystyle ki=j}$⁠, and ⁠${\displaystyle ij=k}$⁠, 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 j² = k² = I₂ are not, unlike in quaternions. For both quaternions and coquaternions, all scalar quantities are treated as implicit multiples of I₂ and notated as 1.

The coquaternion ${\displaystyle q=w+xi+yj+zk}$ with scalar w, has conjugate ${\displaystyle q=w-xi-yj-zk}$ similar to Hamilton's quaternions. The quadratic form is ${\displaystyle q{q}^{\ast }=w^2+x^2-y^2-z^2}$.

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 ⁠${\displaystyle p\ne \pm i}$⁠. The Poincaré sphere model used since 1892 has been compared to a 2-sheet hyperboloid model, and the practice of SU(1, 1) interferometry has been introduced.

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 $${\displaystyle [z,1]\left(\begin{array}{cc}u& v^{\ast }\\ v& u^{\ast }\end{array}\right)=[uz+v,v^{\ast }z+u^{\ast }]=\left[\frac{uz+v}{v^{\ast }z+u^{\ast }},1\right]}$$ since in projective coordinates ⁠${\displaystyle (uz+v,v^{\ast }z+u^{\ast })\sim \left(\frac{uz+v}{v^{\ast }z+u^{\ast }},1\right)}$⁠.

Writing ⁠${\displaystyle suv+\overline{suv}=2\mathrm{\Re }(suv)}$⁠, complex number arithmetic shows $${\displaystyle |uz+v{|}^2=S+z{z}^{\ast }\text{ and }|v^{\ast }z+u^{\ast }{|}^2=S+1,}$$ where ⁠${\displaystyle S=v{v}^{\ast }\left(z{z}^{\ast }+1\right)+2\mathrm{\Re }(uvz)}$⁠.

Therefore, ${\displaystyle z{z}^{\ast }<1⟹\left|uz+v\right|<\left|v^{\ast }z+u^{\ast }\right|}$ so that their ratio lies in the open disk.