Special unitary group explained

In mathematics, the special unitary group of degree, denoted, is the Lie group of 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, consisting of all unitary matrices. As a compact classical group, is the group that preserves the standard inner product on

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

The groups find wide application in the Standard Model of particle physics, especially in the electroweak interaction and in quantum chromodynamics.^[1]

The simplest case,, is the trivial group, having only a single element. The group 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 (up to sign), there is a surjective homomorphism from to the rotation group whose kernel is . Since the quaternions can be identified as the even subalgebra of the Clifford Algebra, 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 is a strictly real Lie group (vs. a more general complex Lie group). Its dimension as a real manifold is . Topologically, it is compact and simply connected.^[2] Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below).^[3]

Z /n Z , and is composed of the diagonal matrices for an th root of unity and the identity matrix.

Its outer automorphism group for is

Z / 2 Z , while the outer automorphism group of is the trivial group.

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

The Lie algebra of, denoted by

ak{su}(n) , can be identified with the set of traceless anti‑Hermitian complex matrices, with the regular commutator as a Lie bracket. Particle physicists often use a different, equivalent representation: The set of traceless Hermitian complex matrices with Lie bracket given by times the commutator.

Lie algebra

The Lie algebra

ak{su}(n) of \operatorname{SU}(n) consists of skew-Hermitian matrices with trace zero.^[4] This (real) Lie algebra has dimension . More information about the structure of this Lie algebra can be found below in .

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 from the mathematicians'. With this convention, one can then choose generators that are traceless Hermitian complex matrices, where:T _{a} T _{b} = \tfrac{ 1 }{ 2 n }\delta _{ab} I _{n} + \tfrac{ 1 }{ 2 }\sum ^{n ^{2} -1} _{c =1} \left(if _{abc} + d _{abc} \right) T _{c} where the are the structure constants and are antisymmetric in all indices, while the -coefficients are symmetric in all indices.

As a consequence, the commutator is:

~ \left[T _{a,} T _{b\right]} ~ = ~ i \sum ^{n ^{2} -1} _{c =1} f _{abc} T _{c} , and the corresponding anticommutator is:\left\{T _{a,} T _{b\right\}} ~ = ~ \tfrac{ 1 }{n} \delta _{ab} I _{n} + \sum ^{n ^{2} -1} _{c =1} {d _{abc} T _{c}} ~.

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

The conventional normalization condition is

\sum ^{n ^{2} -1} _{c,e =1} d _{ace} d _{bce} = (n ^{2} -4)/(n) \delta _{ab} ~ . The generators satisfy the Jacobi identity^[5] :[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} are defined as the traceless Hermitian complex matrices with a 1 / 2 prefactor: for the SU( 2 ) group, the generators are chosen as (1)/(2) \sigma _{1,} (1)/(2) \sigma _{2,} (1)/(2) \sigma _{3} where \sigma _{a} are the Pauli matrices, while for the case of SU( 3 ) one defines T _{a} = (1)/(2) λ _{a} where λ _{a} are the Gell-Mann matrices ^[6] . With these definitions, the generators satisfy the following normalization condition:Tr(T _{a} T _{b)} = (1)/(2) \delta _{ab} .

Adjoint representation

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

\left(T _{a\right)} _{jk} = - if _{ajk} .

The group SU(2)

See also: Versor, Pauli matrices and Representation theory of SU(2).

Using matrix multiplication for the binary operation, forms a group,^[7]

\operatorname{SU}( 2 ) = \left\{ \begin{pmatrix} \alpha & - \overline{\beta} \ \beta & \overline{\alpha} \end{pmatrix}: \alpha,\beta \in C , |\alpha| ^{2} + |\beta| ^{2} = 1 \right\}~, where the overline denotes complex conjugation.

Diffeomorphism with the 3-sphere S³

If we consider

\alpha,\beta as a pair in C ^{2} where \alpha = a + bi and \beta = c + di , then the equation |\alpha| ^{2} + |\beta| ^{2} = 1 becomesa ^{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

\begin{align} \varphi \colon C ^{2} \to{} &\operatorname{M}( 2 , C ) \\[ 5 pt] \varphi(\alpha, \beta) = {} &\begin{pmatrix} \alpha & - \overline{\beta}\ \beta & \overline{\alpha}\end{pmatrix}, \end{align} where \operatorname{M}( 2 , C ) denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering C ^{2} diffeomorphic to R ^{4} and \operatorname{M}( 2 , C ) diffeomorphic to R ^{8} ). Hence, the restriction of to the 3-sphere (since modulus is 1), denoted, is an embedding of the 3-sphere onto a compact submanifold of \operatorname{M}( 2 , C ) , namely .

Therefore, as a manifold, is diffeomorphic to, which shows that is simply connected and that 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 matrix:

\begin{pmatrix} a + bi & c + di \ - c + di & a - bi \end{pmatrix} (a, b, c, d \in R ) can be mapped to the quaterniona\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 is of this form and, since it has determinant , the corresponding quaternion has norm . Thus is isomorphic to the group of versors.^[8]

Relation to spatial rotations

See main article: Quaternions and spatial rotation. 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 to ; consequently is isomorphic to the quotient group, the manifold underlying is obtained by identifying antipodal points of the 3-sphere, and is the universal cover of .

Lie algebra

The Lie algebra of consists of skew-Hermitian matrices with trace zero.^[9] Explicitly, this means

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

The Lie algebra is then generated by the following matrices,

u _{1} = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}, u _{2} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, 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

ak{s u}( 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~,} u _{3 u} _{1} = - u _{1 u} _{3} = u _{2~,} and u _{1} u _{2} = - u _{2 u} _{1} = u _{3~.} The commutator bracket is therefore specified by\left[u _{3,} u _{1\right]} = 2 u _{2,} \left[u _{1,} u _{2\right]} = 2 u _{3,} \left[u _{2,} u _{3\right]} = 2 u _{1~.}

The above generators are related to the Pauli matrices by

u _{1} = i \sigma _{1~,} u _{2} = - i \sigma _{2} and 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(3)

See also: Clebsch–Gordan coefficients for SU(3).

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

Topology

The group is a simply-connected, compact Lie group.^[10] Its topological structure can be understood by noting that acts transitively on the unit sphere

S ^{5} in C ^{3} \cong R ^{6} . The stabilizer of an arbitrary point in the sphere is isomorphic to, which topologically is a 3-sphere. It then follows that is a fiber bundle over the base with fiber . Since the fibers and the base are simply connected, the simple connectedness of then follows by means of a standard topological result (the long exact sequence of homotopy groups for fiber bundles).^[11]

The -bundles over are classified by

\pi ^{3 \right)} _{4 d\left(S} = Z _{2} since any such bundle can be constructed by looking at trivial bundles on the two hemispheres S ^{5} _{N ,} S ^{5} _{S} and looking at the transition function on their intersection, which is a copy of, soS ^{5} _{N} \cap S ^{5} _{S} \simeq S ^{4}

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

\left[S ^{4,} SU ( 2 )\right] \cong \left[S ^{4,} S ^{3\right]} = \pi ^{3 \right)} _{4 d\left(S} \cong Z / 2 and as \pi _{4(SU(3))} = \{ 0 \} rather than Z / 2 , cannot be the trivial bundle, 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 is well-understood.^[12] Descriptions of these representations, from the point of view of its complexified Lie algebra

ak{sl}(3; C ) , may be found in the articles on Lie algebra representations or the Clebsch–Gordan coefficients for .

Lie algebra

The generators,, of the Lie algebra

ak{su}(3) of in the defining (particle physics, Hermitian) representation, areT _{a} = (λ _{a})/(2) ~, where, the Gell-Mann matrices, are the analog of the Pauli matrices for :\begin{align} λ _{1} = {} &\begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix}, & λ _{2} = {} &\begin{pmatrix} 0 & - i & 0 \ i & 0 & 0 \ 0 & 0 & 0 \end{pmatrix}, & λ _{3} = {} &\begin{pmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 0 \end{pmatrix}, \\[ 6 pt] λ _{4} = {} &\begin{pmatrix} 0 & 0 & 1 \ 0 & 0 & 0 \ 1 & 0 & 0 \end{pmatrix}, & λ _{5} = {} &\begin{pmatrix} 0 & 0 & - i \ 0 & 0 & 0 \ i & 0 & 0 \end{pmatrix}, \\[ 6 pt] λ _{6} = {} &\begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{pmatrix}, & λ _{7} = {} &\begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & - i \ 0 & i & 0 \end{pmatrix}, & λ _{8} = (1)/(\sqrt{ 3) } &\begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end.\end

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

They obey the relations

\begin{align} \left[T _{a,} T _{b\right]} & = i \sum ^{8} _{c =1} f _{abc} T _{c,} \\ \left\{T _{a,} T _{b\right\}} & = (1)/(3) \delta _{ab} I _{3} + \sum ^{8} _{c =1} d _{abc} T _{c, \end{align}} or, equivalently,\begin{align} \left[ λ _{a,} λ _{b\right]} & = 2 i \sum ^{8} _{c =1} f _{abc} λ _{c,} \\ \{ λ _{a,} λ _{b\}} & = (4)/(3) \delta _{ab} I _{3} + 2 \sum ^{8 {d} _{abc} λ _{c}. \end{align}}

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

\begin{align} f _{123} & = 1 , \\ f _{147} = - f _{156} = f _{246} = f _{257} = f _{345} = - f _{367} & = (1)/(2) , \\ f _{458} = f _{678} & = (\sqrt{ 3)/() },\endwhile all other not related to these by permutation are zero. In general, they vanish unless they contain an odd number of indices from the set .

The symmetric coefficients take the values

\begin{align} d _{118} = d _{228} = d _{338} = - d _{888} & = (1)/(\sqrt{ 3) } \\ d_ = d_ = d_ = d_ &= -\frac \\ d_ = d_ = -d_ = -d_ = -d_ = d_ = d_ = d_ &= \frac ~.\end

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

A generic group element generated by a traceless 3×3 Hermitian matrix, normalized as, can be expressed as a second order matrix polynomial in :^[13]

\begin{align} \exp(i\theta H) = {} &\left[ - (1)/(3) I\sin\left(\varphi + (2 \pi)/(3) \right) \sin\left(\varphi - (2 \pi)/(3) \right) - (1)/(2 \sqrt{ 3) }~H\sin(\varphi) - \frac~H^2\right] \frac \\[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 \\[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 \endwhere\varphi \equiv (1)/(\left[\arccos\left( 3) (3 \sqrt{ 3)/() }\det H\right) - \frac\right].

Lie algebra structure

As noted above, the Lie algebra

ak{su}(n) of consists of skew-Hermitian matrices with trace zero.^[14]

The complexification of the Lie algebra

ak{su}(n) is ak{sl}(n; C ) , the space of all 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 C ^{n} whose entries sum to zero. The roots then consist of all the permutations of .

A choice of simple roots is

\begin{align} (& 1 , -1 , 0 , ... , 0 , 0 ), \\ (& 0 , 1 , -1 , ... , 0 , 0 ), \\ &\vdots \\ (& 0 , 0 , 0 , ... , 1 , -1 ). \end{align}

So, is of rank and its Dynkin diagram is given by, a chain of nodes: ....^[17] Its Cartan matrix is

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

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

Generalized special unitary group

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

Specifically, fix a Hermitian matrix of signature in

\operatorname{GL}(n, R ) , then allM \in \operatorname{SU}(p, q, R ) satisfy\begin{align} M ^{*} A M & = A \\ \det M & = 1 . \end{align}

Often one will see the notation without reference to a ring or field; in this case, the ring or field being referred to is

C and this gives one of the classical Lie groups. The standard choice for when \operatorname{F} = C isA = \begin{bmatrix} 0 & 0 & i \\ 0 & I _{n-2} & 0 \\ - i & 0 & 0 \end{bmatrix}.

However, there may be better choices for for certain dimensions which exhibit more behaviour under restriction to subrings of

C .

Example

\operatorname{SU}( 2 , 1 ; Z [i]) which acts (projectively) on complex hyperbolic space of degree two, in the same way that \operatorname{SL}( 2 , 9 ; 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 .^[18]

A further example is

\operatorname{SU}( 1 , 1 ; C ) , which is isomorphic to \operatorname{SL}( 2 , R ) .

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 that are important in GUT physics are, for,

\operatorname{SU}(n) \supset \operatorname{SU}(p) x \operatorname{SU}(n - p) x \operatorname{U}( 1 ), where × denotes the direct product and, 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,

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

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

\begin{align} \operatorname{SO}( 2 n) &\supset \operatorname{SU}(n) \\ \operatorname{Sp}(n) &\supset \operatorname{SU}(n) \\ \operatorname{Spin}( 4 ) & = \operatorname{SU}( 2 ) x \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} See Spin group and Simple Lie group for,, and .

There are also the accidental isomorphisms:,, and .

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

SU(1, 1)

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

This group is isomorphic to and ^[19] where the numbers separated by a comma refer to the signature of the quadratic form preserved by the group. The expression

~u u ^{*} - v v ^{*~} in the definition of is an Hermitian form which becomes an isotropic quadratic form when and 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

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

Then

~jk = \begin{bmatrix} 0 & -1 \ 1 & ~ 0 \end{bmatrix} = - i ~,~ ~ ijk = I _{2} \equiv \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}~,~ the 2×2 identity matrix, ~ki = j ~, and ij = k , and the elements and all anticommute, as in quaternions. Also i is still a square root of (negative of the identity matrix), whereas ~j ^{2} = k ^{2} = + I _{2~} are not, unlike in quaternions. For both quaternions and coquaternions, all scalar quantities are treated as implicit multiples of and notated as .

The coquaternion

~q = w + xi + yj + zk~ with scalar, has conjugate ~q = w - xi - yj - zk~ similar to Hamilton's quaternions. The quadratic form is ~qq ^{*} = w ^{2} + x ^{2} - y ^{2} - z ^{2.} \left\{ x i + y j + z k : x ^{2} - y ^{2} - z ^{2} = 1 \right\} corresponds to the imaginary units in the algebra so that any point on this hyperboloid can be used as a pole of a sinusoidal wave according to Euler's formula.

The hyperboloid is stable under, illustrating the isomorphism with . 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 The Poincaré sphere model used since 1892 has been compared to a 2-sheet hyperboloid model,^[20] and the practice of interferometry has been introduced.

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

l [ z, 1 r ]\begin{pmatrix}u & v \ v ^{*} & u ^{*} \end{pmatrix} = [ uz + v ^{*,} vz + u ^{* ]} = \left[ (uz + v ^{*})/(vz + u ^{*}) , 1 \right] since in projective coordinates ( uz + v ^{*,} vz + u ^{* )} \thicksim \left( (uz + v ^{*})/(vz + u ^{*}) , 1 \right)~.

Writing

suv + \overline{suv} = 2 \Re d l (suv r ) , complex number arithmetic showsl |uz + v ^{*r|} ^{2} = S + zz ^{*} and l |vz + u ^{*r|} ^{2} = S + 1 ~, where S = vv ^{*} \left(zz ^{*} + 1 \right) + 2 \Re d l (uvz r ). Therefore, zz ^{*} < 1 \implies l |uz + v ^{*r|} < l |vz + u ^{*r|} so that their ratio lies in the open disk.^[21]

See also

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

Notes and References

1. Book: Halzen, Francis . Francis Halzen . Martin, Alan . Alan Martin (physicist). Quarks & Leptons: An Introductory Course in Modern Particle Physics . registration . John Wiley & Sons . 1984 . 0-471-88741-2.
2. , Proposition 13.11
3. Book: Brian Garner Wybourne . Wybourne, B.G. . 1974 . Classical Groups for Physicists . Wiley-Interscience . 0471965057.
4. Proposition 3.24
5. Book: Georgi, Howard . Lie Algebras in Particle Physics: From Isospin to Unified Theories . 2018-05-04 . CRC Press . 978-0-429-49921-0 . 1 . Boca Raton . en . 10.1201/9780429499210.
6. Book: Georgi, Howard . Lie Algebras in Particle Physics: From Isospin to Unified Theories . 2018-05-04 . CRC Press . 978-0-429-49921-0 . 1 . Boca Raton . en . 10.1201/9780429499210.
7. Exercise 1.5
8. Web site: LieGroups . MATH 4144 notes . Savage, Alistair.
9. Proposition 3.24
10. Proposition 13.11
11. Section 13.2
12. Chapter 6
13. Rosen. S P. Finite Transformations in Various Representations of SU(3). Journal of Mathematical Physics. 12. 4. 1971. 673–681 . 10.1063/1.1665634. 1971JMP....12..673R.

    10.1016/S0034-4877(15)30040-9. Elementary results for the fundamental representation of SU(3). Curtright, T L. Zachos, C K. 2015. Reports on Mathematical Physics. 76. 3. 401–404. 2015RpMP...76..401C. 1508.00868. 119679825.

14. Proposition 3.24
15. Section 3.6
16. Section 7.7.1
17. Section 8.10.1
18. math/0509708 . September 2005 . An explicit fundamental domain for the Picard modular group in two complex dimensions . Francsics . Gabor . Lax . Peter D..
19. Book: Gilmore, Robert . 1974 . Lie Groups, Lie Algebras and some of their Applications . 52, 201−205 . . 1275599.
20. R.D. . Mota . D. . Ojeda-Guillén . M. . Salazar-Ramírez . V.D. . Granados . 2016 . SU(1,1) approach to Stokes parameters and the theory of light polarization . Journal of the Optical Society of America B . 33 . 8 . 1696–1701 . 1602.03223 . 10.1364/JOSAB.33.001696. 2016JOSAB..33.1696M . 119146980 .
21. Book: C. L. Siegel . Siegel, C.L. . 1971 . Topics in Complex Function Theory . 2 . 13–15 . Shenitzer, A. . Tretkoff, M. . Wiley-Interscience . 0-471-79080 X.