Special unitary group

In mathematics, the special unitary group of degree $n$, denoted $SU(n)$, is the of $n × n$   with  1.

(More may have complex determinants with absolute value 1, rather than real 1 in the special case.)

The group operation is. The special unitary group is a of the  $U(n)$, consisting of all $n×n$ unitary matrices. As a, $U(n)$ is the group that preserves the on $$\mathbb{C}^n$$. It is itself a subgroup of the, $$\operatorname{SU}(n) \subset \operatorname{U}(n) \subset \operatorname{GL}(n, \mathbb{C} )$$.

The $U(n)$ groups find wide application in the of, especially  in the  and  in.

The simplest case, $SU(n)$, is the, having only a single element. The group $SU(n)$ is to the group of s of  1, and is thus  to the. Since s can be used to represent rotations in 3-dimensional space (up to sign), there is a  from $SU(2)$ to the  whose  is $SU(3)$. $SU(1)$ is also identical to one of the symmetry groups of s, (3), that enables a spinor presentation of rotations.

Properties
The special unitary group $SU(2)$ is a real (though not a ). Its dimension as a is $SU(2)$. Topologically, it is and. Algebraically, it is a (meaning its  is simple; see below).

The  of $SO(3)$ is isomorphic to the  $$\mathbb{Z}/n$$, and is composed of the diagonal matrices ${+I, −I}$ for $ζ$ an $n$th root of unity and $I$ the n×n identity matrix.

Its, for $SU(2) → SO(3)$, is $$\mathbb{Z}_2$$, while the outer automorphism group of $SU(2)$ is the.

A maximal torus, of rank $SU(n)$, is given by the set of diagonal matrices with determinant 1. The is the  $n^{2} − 1$, which is represented by  (the signs being necessary to ensure the determinant is 1).

The of $SU(n)$, denoted by $$\mathfrak{su}(n)$$, can be identified with the set of   $ζ I$ complex matrices, with the regular  as Lie bracket. often use a different, equivalent representation: The set of traceless $n ≥ 3$ complex matrices with Lie bracket given by $SU(2)$ times the commutator.

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

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 $n − 1$ that are  complex $S_{n}$ matrices, where:


 * $$T_a T_b = \frac{1}{2n}\delta_{ab}I_n + \frac{1}{2}\sum_{c=1}^{n^2 -1}\left(if_{abc} + d_{abc}\right) T_c$$

where the $SU(n)$ are the and are antisymmetric in all indices, while the $n×n$-coefficients are symmetric in all indices.

As a consequence, the anticommutator and commutator are:


 * $$\begin{align}

\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{align}$$

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

We may also take


 * $$\sum_{c,e=1}^{n^2 - 1} d_{ace}d_{bce} = \frac{n^2 - 4}{n} \delta_{ab}$$

as a normalization convention.

Adjoint representation
In the $n×n$-dimensional, the generators are represented by $−i$× $T_{a}$ matrices, whose elements are defined by the structure constants themselves:


 * $$\left(T_a\right)_{jk} = -if_{ajk}.$$

The group SU(2)
$n×n$ is the following group,


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

where the overline denotes.

with
If we consider $$\alpha,\beta$$ as a pair in $$\mathbb{C}^2$$ where $$\alpha=a+bi$$ and $$\beta=c+di$$, then the equation $$|\alpha|^2 + |\beta|^2 = 1$$ becomes
 * $$a^2 + b^2 + c^2 + d^2 = 1$$

This is the equation of the. This can also be seen using an embedding: the map
 * $$\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}$$

where $$\operatorname{M}(2,\mathbb{C})$$ denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering $$\mathbb{C}^2$$ to $$\mathbb{R}^4$$ and $$\operatorname{M}(2,\mathbb{C})$$ diffeomorphic to $$\mathbb{R}^8$$). Hence, the of $f$ to the  (since modulus is 1), denoted $d$, is an embedding of the 3-sphere onto a compact submanifold of $$\operatorname{M}(2,\mathbb{C})$$, namely $(n^{2} − 1)$.

Therefore, as a manifold, $(n^{2} − 1)$ is diffeomorphic to $(n^{2} − 1)$, which shows that $SU(2)$ is  and that $φ$ can be endowed with the structure of a compact, connected.

with
The complex matrix:
 * $$ \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 :
 * $$a\,\hat{1} + b\,\hat{i} + c\,\hat{j} + d\,\hat{k}$$

This map is in fact an isomorphism. Additionally, the determinant of the matrix is the norm of the corresponding quaternion. Clearly any matrix in $S^{3}$ is of this form and, since it has determinant 1, the corresponding quaternion has norm 1. Thus $φ(S^{3}) = SU(2)$ is isomorphic to the.

Relation to spatial rotations
Every unit quaternion is naturally associated to a spatial rotation in 3 dimensions, and the product of two quaternions is associated to the composition of the associated rotations. Furthermore, every rotation arises from exactly two unit quaternions in this fashion. In short: there is a 2:1 surjective homomorphism from SU(2) to ; consequently SO(3) is isomorphic to the SU(2)/{&plusmn;I}, the manifold underlying SO(3) is obtained by identifying antipodal points of the 3-sphere $S^{3}$, and SU(2) is the  of SO(3).

Lie algebra
The of $SU(2)$ consists of $$2\times 2$$  matrices with trace zero. Explicitly, this means
 * $$\mathfrak{su}(2) = \left\{ \begin{pmatrix} i\ a & -\overline{z} \\ z & -i\ a \end{pmatrix}:\ a \in \mathbb{R}, z \in \mathbb{C} \right\}~.$$

The Lie algebra is then generated by the following matrices,
 * $$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 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  is therefore specified by
 * $$\left[u_3, u_1\right] = 2\ u_2, \quad \left[u_1, u_2\right] = 2\ u_3, \quad \left[u_2, u_3\right] = 2\ u_1~.$$

The above generators are related to the by $$u_1 = i\ \sigma_1~, \, u_2 = -i\ \sigma_2$$  and  $$u_3 = +i\ \sigma_3~.$$ This representation is routinely used in  to represent the  of s such as s. They also serve as s for the description of our 3 spatial dimensions in.

The Lie algebra serves to work out the.

Topology
The group $SU(2)$ is a simply-connected, compact Lie group. Its topological structure can be understood by noting that SU(3) acts transitively on the unit sphere $$S^5$$ in $$\mathbb{C}^3 = \mathbb{R}^6$$. The 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 over the base $$S^5$$ with fiber $$S^3$$. 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 for fiber bundles).

The SU(2)-bundles over $$S^5$$ are classified by $$\pi_4(S^3)=\mathbb{Z}_2$$, and as $$\pi_4(SU(3))=\{0\}$$ rather than $$\mathbb{Z}_2$$, SU(3) cannot be the trivial bundle $$SU(2)\times S^5\cong S^3\times S^5$$, and therefore must be the unique nontrivial (twisted) bundle.

Representation theory
The representation theory of $S^{3}$ is well understood. Descriptions of these representations, from the point of view of its complexified Lie algebra $$\operatorname{sl}(3; \mathbb{C})$$, may be found in the articles on or.

Lie algebra
The generators, $T$, of the Lie algebra $$\mathfrak{su}(3)$$ of $SU(2)$ in the defining (particle physics, Hermitian) representation, are
 * $$T_a = \frac{\lambda_a}{2}~, $$

where $SU(2)$, the, are the $S^{3}$ analog of the for $SU(2)$:
 * $$\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}$$

These $SU(2)$ span all  $H$ of the, as required. Note that $SU(3)$ are antisymmetric.

They obey the relations
 * $$\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}$$ or, equivalently,
 * $$\{\lambda_a, \lambda_b\} = \frac{4}{3}\delta_{ab} I_3 + 2\sum_{c=1}^8{d_{abc} \lambda_c}$$.

The $f$ are the of the Lie algebra, given by
 * $$f_{123} = 1$$,
 * $$f_{147} = -f_{156} = f_{246} = f_{257} = f_{345} = -f_{367} = \frac{1}{2} $$,
 * $$f_{458} = f_{678} = \frac{\sqrt{3}}{2}$$,

while all other $SU(3)$ 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 $SU(3)$ take the values
 * $$d_{118} = d_{228} = d_{338} = -d_{888} =  \frac{1}{\sqrt{3}}$$
 * $$d_{448} = d_{558} = d_{668} =  d_{778} = -\frac{1}{2\sqrt{3}}$$
 * $$d_{344} = d_{355} = -d_{366} = -d_{377}                                         = \frac{1}{2}  ~.$$

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

A generic $λ$ group element generated by a traceless 3×3 Hermitian matrix $1/6$, normalized as $SU(3)$, can be expressed as a second order matrix polynomial in $H$:
 * $$\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}$$

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

Lie algebra structure
As noted above, the Lie algebra $$\mathfrak{su}(n)$$ of $$\operatorname{SU}(n)$$ consists of $$n\times n$$ matrices with trace zero.

The of the Lie algebra $$\mathfrak{su}(n)$$ is $$\mathfrak{sl}(n; \mathbb{C})$$, the space of all $$n\times n$$ complex matrices with trace zero. A Cartan subalgebra then consists of the diagonal matrices with trace zero, which we identify with vectors in $$\mathbb C^n$$ whose entries sum to zero. The then consist of all the $SU(2)$ permutations of $λ_{a}$.

A choice of s is
 * $$\begin{align}

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

So, $λ_{2}, λ_{5}, λ_{7}$ is of $f_{abc}$ and its  is given by $f_{abc}$, a chain of $d$ nodes: .... Its is
 * $$\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 or  is the  $SU(3)$, the  of the $tr(H^{2}) = 2$-.

Generalized special unitary group
For a $n(n − 1)$, the generalized special unitary group over F, $(1, −1, 0, ..., 0)$, is the  of all s of  1 of a  of rank $SU(n)$ over $n − 1$ which leave invariant a,  of  $A_{n−1}$. This group is often referred to as the special unitary group of signature $n − 1$ over $S_{n}$. The field $(n − 1)$ can be replaced by a, in which case the vector space is replaced by a.

Specifically, fix a $F$ of signature $SU(p, q; F)$ in $$\operatorname{GL}(n, \mathbb{R})$$, then all
 * $$M \in \operatorname{SU}(p, q, \mathbb{R})$$

satisfy
 * $$\begin{align}

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

Often one will see the notation $n = p + q$ without reference to a ring or field; in this case, the ring or field being referred to is $$\mathbb C$$ and this gives one of the classical. The standard choice for $F$ when $\operatorname{F} = \mathbb{C}$ is
 * $$A = \begin{bmatrix}

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

However, there may be better choices for $(p, q)$ for certain dimensions which exhibit more behaviour under restriction to subrings of $$\mathbb C$$.

Example
An important example of this type of group is the $$\operatorname{SU}(2, 1; \mathbb{Z}[i])$$ which acts (projectively) on complex hyperbolic space of degree two, in the same way that $$\operatorname{SL}(2,9;\mathbb{Z})$$ acts (projectively) on real  of dimension two. In 2005 Gábor Francsics and computed an explicit fundamental domain for the action of this group on $p q$.

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

Important subgroups
In physics the special unitary group is used to represent symmetries. In theories of it is important to be able to find the subgroups of the special unitary group. Subgroups of $F$ that are important in are, for $F$,
 * $$\operatorname{SU}(n) \supset \operatorname{SU}(p) \times \operatorname{SU}(n - p) \times \operatorname{U}(1),$$

where × denotes the and $A$, known as the, is the multiplicative group of all s with  1.

For completeness, there are also the and  subgroups,
 * $$\begin{align}

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

Since the of $p q$ is $SU(p, q)$ and of $A$ 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. $A$ is a subgroup of various other Lie groups,
 * $$\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}$$ See, and for E6, E7, and G2.

There are also the : $HC^{2}$, $SU(n)$, and $p > 1, n − p > 1$.

One may finally mention that $U(1)$ is the of $SU(n)$, a relation that plays an important role in the theory of rotations of 2-s in non-relativistic.

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

This group is locally isomorphic to $n − 1$ and $U(1)$ where the numbers separated by a comma refer to the of the  preserved by the group. The expression $$u u^* - v v^* $$ in the definition of $SU(n)$ is an which becomes an  when u and v are expanded with their real components. An early appearance of this group was as the "unit sphere" of s, introduced by in 1852. Let
 * $$j = \begin{pmatrix}0 & 1\\ 1 & 0 \end{pmatrix}, \quad k = \begin{pmatrix}1 & 0\\ 0 & -1 \end{pmatrix} , \quad i = \begin{pmatrix}0 & 1\\ -1 & 0 \end{pmatrix} .$$

Then $$j k = \begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix} = -i .$$ 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 property so that the quadratic form is   $$ q q^* \ = \ w^2 + x^2 - y^2 - z^2. $$

Note that the 2-sheet $$\{x i + y j + z k : x^2 - y^2 - z^2 = 1 \} $$ corresponds to the s in the algebra so that any point p on this hyperbola can be used as a pole of a sinusoidal wave according to.

The hyperboloid is stable under $SU(4) = Spin(6)$, illustrating the isomorphism with $SU(2) = Spin(3) = Sp(1)$. The variability of the pole of a wave, as noted in studies of, might view as an exhibit of the elliptical shape of a wave with pole $p \ne \pm i$. The model used since 1892 has been compared to a 2-sheet hyperboloid model.

When an element of $Sp(n)$ is interpreted as a, it leaves the stable, so this group represents the s of the  of hyperbolic plane geometry. Indeed, for a point [z, 1] in the, the action of $USp(2n)$ is given by
 * $$[z,1]\begin{pmatrix}u & v \\ v^* & u^* \end{pmatrix} \ = \ [uz + v^*, \ vz +u^*] \ = \ \left[\frac{uz + v^*}{vz +u^*}, \ 1 \right]$$

since in $$(uz + v^*, \ vz +u^*) \ \thicksim \ \left(\frac{uz + v^*}{vz +u^*}, \ 1 \right).$$

Writing $$suv + \overline{suv} \ = \ 2 \Re(suv),$$ complex number arithmetic shows
 * $$|uz + v^*|^2 \ = \ S + zz^* \quad \text{ and } \quad |vz +u^*|^2 \ = \ S + 1,$$

where $$S \ = \ vv^*(z z^* + 1) + 2 \Re(uvz).$$ Therefore, $$zz^* < 1 \implies |uz + v^*| < |vz + u^*|$$ so that their ratio lies in the open disk.