# Special unitary group

In

mathematics , the**special unitary group**of degree "n", denoted SU("n"), is the group of "n"×"n" unitary matrices withdeterminant 1. The group operation is that ofmatrix multiplication . The special unitary group is asubgroup of theunitary group U("n"), consisting of all "n"×"n" unitary matrices, which is itself a subgroup of thegeneral linear group GL("n",**C**).The SU(n) groups find wide application in the

standard model ofphysics , especially SU(2) in theelectroweak interaction and SU(3) in QCD.The simplest case, SU(1), is the

trivial group , having only a single element. The group SU(2) isisomorphic to the group ofquaternion s ofabsolute value 1, and is thusdiffeomorphic to the3-sphere . Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), we have asurjective homomorphism from SU(2) to therotation group SO(3) whose kernel is $\{+I,\; -I\}$.**Properties**The special unitary group SU("n") is a real

matrix Lie group of dimension "n"^{2}− 1. Topologically, it iscompact andsimply connected . Algebraically, it is asimple Lie group (meaning itsLie algebra is simple; see below). The center of SU("n") is isomorphic to thecyclic group **Z**_{"n"}. Itsouter automorphism group , for "n" ≥ 3, is**Z**_{2}, while the outer automorphism group of SU(2) is thetrivial group .The SU(n) algebra is generated by "n"

^{2}operators, which satisfy the commutator relationship (for i,j,k,l = 1, 2, ..., n):$left\; [\; hat\{O\}\_\{ij\}\; ,\; hat\{O\}\_\{kl\}\; ight\; ]\; =\; delta\_\{jk\}\; hat\{O\}\_\{il\}\; -\; delta\_\{il\}\; hat\{O\}\_\{kj\}$

Additionally, the operator

:$hat\{N\}\; =\; sum\_\{i=1\}^n\; hat\{O\}\_\{ii\}$

satisfies

:$left\; [\; hat\{N\},\; hat\{O\}\_\{ij\}\; ight\; ]\; =\; 0$

which implies that the number of "independent" generators of SU(n) is n

^{2}-1. [*R.R. Puri, "Mathematical Methods of Quantum Optics", Springer, 2001.*]**Generators**In general the generators of SU(n), "T", are represented as traceless Hermitian matrices. I.e:

:*$operatorname\{tr\}(T\_a)\; =\; 0\; ,$and:*$T\_a\; =\; T\_a^dagger\; ,$

**Fundamental representation**In the defining or fundamental representation the generators are represented by "n"×"n" matrices where: :*$T\_a\; T\_b\; =\; frac\{1\}\{2n\}delta\_\{ab\}I\_n\; +\; frac\{1\}\{2\}sum\_\{c=1\}^\{n^2\; -1\}\{(if\_\{abc\}\; +\; d\_\{abc\})\; T\_c\}\; ,$:where the "f" are the "

structure constant s" and are antisymmetric in all indices, whilst the "d" are symmetric in all indices. As a consequence: :*$left\; [T\_a,\; T\_b\; ight]\; \_+\; =\; frac\{1\}\{n\}delta\_\{ab\}\; +\; sum\_\{c=1\}^\{n^2\; -1\}\{d\_\{abc\}\; T\_c\}\; ,$:*$left\; [T\_a,\; T\_b\; ight]\; \_-\; =\; i\; sum\_\{c=1\}^\{n^2\; -1\}\{f\_\{abc\}\; T\_c\}\; ,$We also have:*$sum\_\{c,e=1\}^\{n^2\; -1\}d\_\{ace\}d\_\{bce\}=\; frac\{n^2-4\}\{n\}delta\_\{ab\}\; ,$

as a normalization convention.

**Adjoint representation**In the

adjoint representation the generators are represented by $(n^2-1)$×$(n^2-1)$ matrices whose elements are defined by the structure constants:::*$(T\_a)\_\{jk\}\; =\; -if\_\{ajk\}\; ,$**U(2)**For SU(2), the generators T, in the defining representation, are proportional to the

Pauli matrices , $sigma\; ,$ via:::$T\_a\; =\; frac\{sigma\_a\}\{2\}\; .,$where::$sigma\_1\; =\; egin\{pmatrix\}01\backslash 10end\{pmatrix\},\; quad\; sigma\_2\; =\; egin\{pmatrix\}0-i\backslash i0end\{pmatrix\},quad\; sigma\_3\; =\; egin\{pmatrix\}10\backslash 0-1end\{pmatrix\}.$

Note that all the generators are traceless Hermitian matrices as required.

The structure constants for SU(2) are defined by the

Levi-Civita symbol ::$f^\{123\}\; =\; 1\; ,$; the rest can be determined by antisymmetry.All the "d" values vanish.**U(3)**The generators of SU(3), "T", in the defining representation, are:::$T\_a\; =\; frac\{lambda\_a\}\{2\}\; .,$where $lambda\; ,$, the

Gell-Mann matrices , are the SU(3) analog of the Pauli matrices for SU(2): :Note that they are all traceless Hermitian matrices as required.

These obey the relations:*$left\; [T\_a,\; T\_b\; ight]\; =\; i\; sum\_\{c=1\}^8\{f\_\{abc\}\; T\_c\}\; ,$:where the "f" are the "structure constants", as previously defined, and have values given by::$f^\{123\}\; =\; 1\; ,$::$f^\{147\}\; =\; -f^\{156\}\; =\; f^\{246\}\; =\; f^\{257\}\; =\; f^\{345\}\; =\; -f^\{367\}\; =\; frac\{1\}\{2\}\; ,$::$f^\{458\}\; =\; f^\{678\}\; =\; frac\{sqrt\{3\{2\}\; ,$

The "d" take the values:::$d^\{118\}\; =\; d^\{228\}\; =\; d^\{338\}\; =\; -d^\{888\}\; =\; frac\{1\}\{sqrt\{3\; ,$::$d^\{448\}\; =\; d^\{558\}\; =\; d^\{668\}\; =\; d^\{778\}\; =\; -frac\{1\}\{2sqrt\{3\; ,$::$d^\{146\}\; =\; d^\{157\}\; =\; -d^\{247\}\; =\; d^\{256\}\; =\; d^\{344\}\; =\; d^\{355\}\; =\; -d^\{366\}\; =\; -d^\{377\}\; =\; frac\{1\}\{2\}\; ,$

**Lie algebra**The

Lie algebra corresponding to $mathrm\{SU\}(n)$ is denoted by $mathfrak\{su\}(n)$. Its standard mathematical representation consists of thetraceless antihermitian $n\; imes\; n$ complex matrices, with the regularcommutator asLie bracket . A factor $i$ is often inserted by particlephysicist s, so that all matrices become hermitian. This is simply a different, more convenient, representation of the same real Lie algebra. Note that $mathfrak\{su\}(n)$ is a Lie algebra over $mathbb\{R\}$.For example, the following antihermitian matrices used in

quantum mechanics form a basis for $mathfrak\{su\}(2)$ over $mathbb\{R\}$::$isigma\_x\; =\; egin\{bmatrix\}\; 0\; i\; \backslash \; i\; 0\; end\{bmatrix\}$:$isigma\_y\; =\; egin\{bmatrix\}\; 0\; 1\; \backslash \; -1\; 0\; end\{bmatrix\}$:$isigma\_z\; =\; egin\{bmatrix\}\; i\; 0\; \backslash \; 0\; -i\; end\{bmatrix\}$ (where $i$ is theimaginary unit .)This representation is often used in

quantum mechanics (see "Pauli matrices " and "Gell-Mann matrices "), to represent the spin offundamental particle s such aselectron s. They also serve asunit vector s for the description of our 3 spatial dimensions inquantum relativity .Note that the product of any two different generators is another generator, and that the generators

anticommute . Together with theidentity matrix (times $i$),:$i\; I\_2\; =\; egin\{bmatrix\}\; i\; 0\; \backslash \; 0\; i\; end\{bmatrix\}$these are also generators of the Lie algebra $mathfrak\{u\}(2)$.Here it depends of course on the problem whether one works finally, as in non-relativistic quantummechanics, with 2-

spinors ; or, as in the relativistic Dirac theory, one needs an extension to 4-spinors; or in mathematics even toClifford algebra s."Note: make clearer the fact that under matrix multiplication (which is anticommutative in this case), we generate the Clifford algebra $mathrm\{Cl\}\_3$, whereas you generate the Lie algebra $mathfrak\{u\}(2)$ with commutator brackets instead."

Back to general $mathrm\{SU\}(n)$:

If we choose an (arbitrary) particular basis, then the

subspace of traceless diagonal $n\; imes\; n$ matrices with imaginary entries forms an $n\; -\; 1$ dimensionalCartan subalgebra .Complexify the Lie algebra, so that any traceless $n\; imes\; n$ matrix is now allowed. The weighteigenvector s are the Cartan subalgebra itself and the matrices with only one nonzero entry which is off diagonal. Even though the Cartan subalgebra $mathrm\{h\}$ is only $n\; -\; 1$ dimensional, to simplify calculations, it is often convenient to introduce an auxiliary element, the unit matrix which commutes with everything else (which should not be thought of as an element of the Lie algebra!) for the purpose of computing weights and that only. So, we have a basis where the $i$th basis vector is the matrix with $1$ on the $i$th diagonal entry and zero elsewhere. Weights would then be given by $n$ coordinates and the sum over all $n$ coordinates has to be zero (because the unit matrix is only auxiliary).So, $mathfrak\{su\}(n)$ has a rank of $n\; -\; 1$ and its

Dynkin diagram is given by $A\_\{n\; -\; 1\}$, a chain of $n\; -\; 1$ vertices.Its

root system consists of $n(n\; -\; 1)$ roots spanning a $n\; -\; 1$Euclidean space . Here, we use $n$ redundant coordinates instead of $n\; -\; 1$ to emphasize the symmetries of the root system (the $n$ coordinates have to add up to zero). In other words, we are embedding this $n\; -\; 1$ dimensional vector space in an $n$-dimensional one. Then, the roots consists of all the $n(n\; -\; 1)$ permutations of $(1,\; -1,\; 0,\; dots,\; 0)$. The construction given two paragraphs ago explains why. A choice ofsimple root s is:$(1,\; -1,\; 0,\; dots,\; 0)$,:$(0,\; 1,\; -1,\; dots,\; 0)$,:…,:$(0,\; 0,\; 0,\; dots,\; 1,\; -1)$.Its

Cartan matrix is :$egin\{pmatrix\}\; 2\; -1\; 0\; dots\; 0\; \backslash -1\; 2\; -1\; dots\; 0\; \backslash \; 0\; -1\; 2\; dots\; 0\; \backslash \; vdots\; vdots\; vdots\; ddots\; vdots\; \backslash \; 0\; 0\; 0\; dots\; 2\; end\{pmatrix\}$.Its

Weyl group orCoxeter group is thesymmetric group $S\_n$, thesymmetry group of the $(n\; -\; 1)$-simplex .**Generalized special unitary group**For a field "F", the

**generalized special unitary group over F**, SU("p","q";"F"), is the group of alllinear transformation s ofdeterminant 1 of avector space of rank "n" = "p" + "q" over "F" which leave invariant anondegenerate ,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 acommutative ring , in which case the vector space is replaced by afree module .Specifically, fix a

hermitian matrix "A" of signature "p" "q" in GL("n","R"), then all:$M\; in\; SU(p,q,R)$

satisfy

:$M^\{*\}\; A\; M\; =\; A\; ,$

:$det\; M\; =\; 1.\; ,$

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 classicalLie groups . The standard choice for "A" when "F" =**C**is: $A\; =\; egin\{bmatrix\}\; 0\; 0\; i\; \backslash \; 0\; I\_\{n-2\}\; 0\; \backslash \; -i\; 0\; 0\; end\{bmatrix\}.$However there may be better choices for "A" for certain dimensions which exhibit more behaviour under restriction to subrings of**C**.**Example**A very important example of this type of group is the

Picard modular group SU(2,1;**Z**["i"] ) which acts (projectively) oncomplex hyperbolic space of degree two, in the same way that SL(2,**Z**) acts (projectively) on realhyperbolic space of dimension two. In 2003Gábor Francsics andPeter Lax computed a fundamental domain for the action of this group on $HC^2$, see [*http://www.esi.ac.at/Preprint-shadows/esi1273.html*] .Another example is SU(2,1;**C**) which is isomorphic to SL(2,**R**).**Important Subgroups**In physics the special unitary group is used to represent

bosonic symmetries. In theories ofsymmetry breaking it is important to be able to find the subgroups of the special unitary group. Important subgroups of SU(n) that are important in GUT physics are, for p>1, n-p>1::$SU(n)\; supset\; SU(p)\; imes\; SU(n-p)\; imes\; U(1)$For completeness there are also theorthogonal and symplectic subgroups::$SU(n)\; supset\; O(n)$:$SU(2n)\; supset\; USp(2n)$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::$SO(2n)\; supset\; SU(n)$:$USp(2n)\; supset\; SU(n)$:$Spin(4)\; =\; SU(2)\; imes\; SU(2)$ (seeSpin group ):$E\_6\; supset\; SU(6)$:$E\_7\; supset\; SU(8)$:$G\_2\; supset\; SU(3)$ (seeSimple Lie groups for E_{6}, E_{7}, and G_{2})There are also the identities**SU(4)=Spin(6)**,**SU(2)=Spin(3)=USp(2)**and**U(1)=Spin(2)=SO(2)**.One should 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-spinor s in non-relativisticquantum mechanics .**See also***

Representation theory of SU(2)

*Projective special unitary group , "PSU(n)"**References***

* [

*http://arxiv.org/abs/math/0605784v1 Maximal Subgroups of Compact Lie Groups*]**External links*** [

*http://courses.washington.edu/phys55x/Physics%20558_lec1_03.htm Physics 558 - Lecture 1, Winter 2003*]

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