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.

(More general unitary matrices 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 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 Cⁿ.^{[nb 1]} It is itself a subgroup of the general linear group, SU(n) ⊂ U(n) ⊂ GL(n, 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 trivial group, having only a single element. The group SU(2) 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 SU(2) to the rotation group SO(3) whose kernel is {+I, −I}.^{[nb 2]} SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations.

Properties[edit]

The special unitary group SU(n) is a real Lie group (though not a complex Lie group). Its dimension as a real manifold is n² − 1. Topologically, it is compact and simply connected.^[2] Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below).^[3]

The center of SU(n) is isomorphic to the cyclic group Z_n, and is composed of the diagonal matrices ζ I for ζ an n^th root of unity and I the n×n identity matrix.

Its outer automorphism group, for n ≥ 3, is Z₂, 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 is the symmetric group S_n, which is represented by signed permutation matrices (the signs being necessary to ensure the determinant is 1).

The Lie algebra of SU(n), denoted by ${\displaystyle {\mathfrak {su}}(n)}$, can be identified with the set of traceless antiHermitian n×n complex matrices, with the regular commutator as Lie bracket. Particle physicists often use a different, equivalent representation: the set of traceless Hermitian n×n complex matrices with Lie bracket given by −i times the commutator.

Lie algebra[edit]

The Lie algebra ${\displaystyle \operatorname {su} (n)}$ of ${\displaystyle \operatorname {SU} (n)}$ consists of ${\displaystyle n\times n}$ skew-Hermitian matrices with trace zero.^[4] This (real) Lie algebra has dimension ${\displaystyle n^{2}-1}$. More information about the structure of this Lie algebra can be found below in the section "Lie algebra structure."

Fundamental representation[edit]

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:

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

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 anticommutator and commutator are:

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

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

We may also take

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

as a normalization convention.

Adjoint representation[edit]

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

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

The group SU(2)[edit]

SU(2) is the following group,^[5]

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

where the overline denotes complex conjugation.

There is a 2:1 homomorphism from SU(2) to SO(3).

Diffeomorphism with S³[edit]

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

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

where M(2, C) denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering C² diffeomorphic to R⁴ and M(2, C) diffeomorphic to R⁸). 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 M(2, C), namely φ(S³) = SU(2).

Therefore, as a manifold, S³ is diffeomorphic to SU(2), which shows that S³ can be endowed with the structure of a compact, connected Lie group.

Isomorphism with unit quaternions[edit]

The complex matrix:

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

can be mapped to the quaternion:

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

This map is in fact an isomorphism. Additionally, the determinant of the matrix is the norm of the corresponding quaternion. Clearly any matrix in SU(2) is of this form and, since it has determinant 1, the corresponding quaternion has norm 1. Thus SU(2) is isomorphic to the unit quaternions.^[6]

Lie algebra[edit]

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

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

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.

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

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

The above generators are related to the Pauli matrices by ${\displaystyle u_{1}=i\ \sigma _{1}~,\,u_{2}=-i\ \sigma _{2}}$ and ${\displaystyle u_{3}=+i\ \sigma _{3}~.}$ This representation is routinely used in quantum mechanics to represent the spin of fundamental particles such as electrons. They also serve as unit vectors for the description of our 3 spatial dimensions in loop quantum gravity.

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

The group SU(3)[edit]

Topology[edit]

The group SU(3) is a simply-connected, compact Lie group,^[8] its topological structure can be understood by noting that SU(3) acts transitively on the unit sphere ${\displaystyle S^{5}}$ in ${\displaystyle \mathbb {C} ^{3}=\mathbb {R} ^{6}}$. The stabilizer of an arbitrary point in the sphere is isomorphic to SU(2), which topologically is a 3-sphere, it then follows that SU(3) is a fiber bundle over the base ${\displaystyle S^{5}}$ with fiber ${\displaystyle S^{3}}$. Since the fibers and the base are simply connected, the simple connectedness of SU(3) then follows by means of a standard topological result (the long exact sequence of homotopy groups for fiber bundles).^[9]

The SU(2)-bundles over ${\displaystyle S^{5}}$ are classified by ${\displaystyle \pi _{4}(S^{3})=\mathbb {Z} _{2}}$, and as ${\displaystyle \pi _{4}(SU(3))=\{0\}}$ rather than ${\displaystyle \mathbb {Z} _{2}}$, SU(3) cannot be the trivial bundle ${\displaystyle SU(2)\times S^{5}\cong S^{3}\times S^{5}}$, and therefore must be the unique nontrivial (twisted) bundle.

Representation theory[edit]

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

Lie algebra[edit]

The generators, T, of the Lie algebra su(3) of SU(3) in the defining (particle physics, Hermitian) representation, are

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

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

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

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

They obey the relations

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

or, equivalently,

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

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

${\displaystyle f_{123}=1}$,

${\displaystyle f_{147}=-f_{156}=f_{246}=f_{257}=f_{345}=-f_{367}={\frac {1}{2}}}$,

${\displaystyle f_{458}=f_{678}={\frac {\sqrt {3}}{2}}}$,

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}.^{[nb 3]}

The symmetric coefficients d take the values

${\displaystyle d_{118}=d_{228}=d_{338}=-d_{888}={\frac {1}{\sqrt {3}}}}$

${\displaystyle d_{448}=d_{558}=d_{668}=d_{778}=-{\frac {1}{2{\sqrt {3}}}}}$

${\displaystyle d_{344}=d_{355}=-d_{366}=-d_{377}={\frac {1}{2}}~.}$

They vanish if the number of indices from the set {2, 5, 7} is odd.

A generic SU(3) group element generated by a traceless 3×3 Hermitian matrix H, normalized as tr(H²) = 2, can be expressed as a second order matrix polynomial in H^[11]:

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

where

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

Lie algebra structure[edit]

As noted above, the Lie algebra ${\displaystyle \operatorname {su} (n)}$ of ${\displaystyle \operatorname {SU} (n)}$ consists of ${\displaystyle n\times n}$ skew-Hermitian matrices with trace zero.^[12]

The complexification of the Lie algebra ${\displaystyle \operatorname {su} (n)}$ is ${\displaystyle \operatorname {sl} (n;\mathbb {C} )}$, the space of all ${\displaystyle n\times n}$ complex matrices with trace zero.^[13] A Cartan subalgebra then consists of the diagonal matrices with trace zero,^[14] 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

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

So, SU(n) is of rank n − 1 and its Dynkin diagram is given by A_n−1, a chain of n − 1 nodes: ....^[15] 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.

Generalized special unitary group[edit]

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 GL(n, R), then all

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

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 C and this gives one of the classical Lie groups. The standard choice for A when F = 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 C.

Example[edit]

An important example of this type of group is the Picard modular group SU(2, 1; Z[i]) which acts (projectively) on complex hyperbolic space of degree two, in the same way that 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 HC².^[16]

A further example is SU(1, 1; C), which is isomorphic to SL(2,R).

Important subgroups[edit]

In physics the special unitary group is used to represent bosonic 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 ,

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

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 orthogonal and symplectic subgroups,

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

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,

${\displaystyle {\begin{aligned}\operatorname {SO} (2n)&\supset \operatorname {SU} (n)\\\operatorname {Sp} (n)&\supset \operatorname {SU} (n)\\\operatorname {Spin} (4)&=\operatorname {SU} (2)\times \operatorname {SU} (2)\\\operatorname {E} _{6}&\supset \operatorname {SU} (6)\\\operatorname {E} _{7}&\supset \operatorname {SU} (8)\\\operatorname {G} _{2}&\supset \operatorname {SU} (3)\end{aligned}}}$

See spin group, and simple Lie groups for E₆, E₇, and G₂.

There are also the accidental isomorphisms: SU(4) = Spin(6) , SU(2) = Spin(3) = Sp(1) ,^{[nb 4]} 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.

The group SU(1,1)[edit]

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

This group is isomorphic to SO(2,1) and SL(2,ℝ)^[17] where the numbers separated by a comma refer to the signature of the quadratic form preserved by the group; the expression ${\displaystyle uu^{*}-vv^{*}}$ 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, introduced by James Cockle in 1852. Let

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

Then ${\displaystyle jk={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}=-i.}$ Also i is a square root of −1 (negative of the identity matrix), while j² = k² = identity matrix. Similar to Hamilton's quaternions, here q = w + x i + y j + z k is a coquaternion with conjugate q * = w – x i – y j – z k; the elements i, j, and k have the anticommutativity property so that the quadratic form is ${\displaystyle qq^{*}\ =\ w^{2}+x^{2}-y^{2}-z^{2}.}$

Note that the 2-sheet hyperboloid ${\displaystyle \{xi+yj+zk:x^{2}-y^{2}-z^{2}=1\}}$ corresponds to the imaginary units in the algebra so that any point on this hyperbola 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 SO(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 p ≠ ± i. The Poincaré sphere model used since 1892 has been compared to a 2-sheet hyperboloid model.^[18]

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.

See also[edit]

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

Remarks[edit]

Notes[edit]

References[edit]

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