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Lie groups andLie algebras
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
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Lie groups inphysics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations
Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Glossary Table of Lie groups
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In mathematics, thespecial unitary group of degreen, denotedSU(n), is theLie group ofn ×nunitary matrices withdeterminant 1.

Thematrices of the more generalunitary group may havecomplex determinants with absolute value 1, rather than real 1 in the special case.

The group operation ismatrix multiplication. The special unitary group is anormal subgroup of theunitary groupU(n), consisting of alln×n unitary matrices. As acompact classical group,U(n) is the group that preserves thestandard inner product on${\displaystyle {\mathbb{C}}^n}$.^[a] It is itself a subgroup of thegeneral linear group,${\displaystyle \mathrm{SU}(n)\subset \mathrm{U}(n)\subset \mathrm{GL}(n,\mathbb{C}).}$

TheSU(n) groups find wide application in theStandard Model ofparticle physics, especiallySU(2) in theelectroweak interaction andSU(3) inquantum chromodynamics.^[1]

The simplest case,SU(1), is thetrivial group, having only a single element. The groupSU(2) isisomorphic to the group ofquaternions ofnorm 1, and is thusdiffeomorphic to the3-sphere. Sinceunit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is asurjectivehomomorphism fromSU(2) to therotation groupSO(3) whosekernel is{+I, −I}.^[b] Since the quaternions can be identified as the even subalgebra of the Clifford AlgebraCl(3),SU(2) is in fact identical to one of the symmetry groups ofspinors,Spin(3), that enables a spinor presentation of rotations.

The special unitary groupSU(n) is a strictly realLie group (vs. a more generalcomplex Lie group). Its dimension as areal manifold isn² − 1. Topologically, it iscompact andsimply connected.^[2] Algebraically, it is asimple Lie group (meaning itsLie algebra is simple; see below).^[3]

Thecenter ofSU(n) is isomorphic to thecyclic group${\displaystyle \mathbb{Z}/n\mathbb{Z}}$, and is composed of the diagonal matricesζI forζ annth root of unity andI then ×n identity matrix.

Itsouter automorphism group forn ≥ 3 is${\displaystyle \mathbb{Z}/2\mathbb{Z},}$ while the outer automorphism group ofSU(2) is thetrivial group.

Amaximal torus ofrankn − 1 is given by the set of diagonal matrices with determinant1. TheWeyl group ofSU(n) is thesymmetric groupS_n, which is represented bysigned permutation matrices (the signs being necessary to ensure that the determinant is1).

TheLie algebra ofSU(n), denoted by${\displaystyle \mathfrak{s}\mathfrak{u}(n)}$, can be identified with the set oftracelessanti‑Hermitiann ×n complex matrices, with the regularcommutator as a Lie bracket.Particle physicists often use a different, equivalent representation: The set of tracelessHermitiann ×n complex matrices with Lie bracket given by−i times the commutator.

The Lie algebra${\displaystyle \mathfrak{s}\mathfrak{u}(n)}$ of${\displaystyle \mathrm{SU}(n)}$ consists ofn ×nskew-Hermitian matrices with trace zero.^[4] This (real) Lie algebra has dimensionn² − 1. More information about the structure of this Lie algebra can be found below in§ Lie algebra structure.

In the physics literature, it is common to identify the Lie algebra with the space of trace-zeroHermitian (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 generatorsT_a that aretracelessHermitian complexn ×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 thef are thestructure constants and are antisymmetric in all indices, while thed-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 ofi 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:^[5]

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

By convention, in the physics literature the generators${\displaystyle T_a}$ are defined as the traceless Hermitian complex matrices with a${\displaystyle 1/2}$ prefactor: for the${\displaystyle SU(2)}$ group, the generators are chosen as${\displaystyle \frac{1}{2}{\sigma }_1,\frac{1}{2}{\sigma }_2,\frac{1}{2}{\sigma }_3}$ where${\displaystyle {\sigma }_a}$ are thePauli matrices, while for the case of${\displaystyle SU(3)}$ one defines${\displaystyle T_a=\frac{1}{2}{\lambda }_a}$ where${\displaystyle {\lambda }_a}$ are theGell-Mann matrices.^[6] With these definitions, the generators satisfy the following normalization condition:

$${\displaystyle Tr(T_a{T}_b)=\frac{1}{2}{\delta }_{ab}.}$$

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

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

Usingmatrix multiplication for the binary operation,SU(2) forms a group,^[7]

where the overline denotescomplex conjugation.

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 the3-sphere S³. This can also be seen using an embedding: the map

where${\displaystyle \mathrm{M}(2,\mathbb{C})}$ denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering${\displaystyle {\mathbb{C}}^2}$diffeomorphic to${\displaystyle {\mathbb{R}}^4}$ and${\displaystyle \mathrm{M}(2,\mathbb{C})}$ diffeomorphic to${\displaystyle {\mathbb{R}}^8}$). Hence, therestriction ofφ to the3-sphere (since modulus is 1), denotedS³, is an embedding of the 3-sphere onto a compact submanifold of${\displaystyle \mathrm{M}(2,\mathbb{C})}$, namelyφ(S³) = SU(2).

Therefore, as a manifold,S³ is diffeomorphic toSU(2), which shows thatSU(2) issimply connected and thatS³ can be endowed with the structure of a compact, connectedLie group.

Quaternions of norm 1 are calledversors since they generate therotation group SO(3):TheSU(2) matrix:

can be mapped to the quaternion

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

This map is in fact agroup isomorphism. Additionally, the determinant of the matrix is the squared norm of the corresponding quaternion. Clearly any matrix inSU(2) is of this form and, since it has determinant 1, the corresponding quaternion has norm1. ThusSU(2) is isomorphic to the group of versors.^[8]

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 fromSU(2) toSO(3); consequentlySO(3) is isomorphic to thequotient groupSU(2)/{±I}, the manifold underlyingSO(3) is obtained by identifying antipodal points of the 3-sphereS³, andSU(2) is theuniversal cover ofSO(3).

TheLie algebra ofSU(2) consists of2 × 2skew-Hermitian matrices with trace zero.^[9] Explicitly, this means

The Lie algebra is then generated by the following matrices,

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 thePauli matrices.

These satisfy thequaternion relationships${\displaystyle u_2{u}_3=-u_3{u}_2=u_1,}$${\displaystyle u_3{u}_1=-u_1{u}_3=u_2,}$ and${\displaystyle u_1{u}_2=-u_2{u}_1=u_3.}$ Thecommutator 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 thePauli 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 inquantum mechanics to represent thespin offundamental particles such aselectrons. They also serve asunit vectors for the description of our 3 spatial dimensions inloop quantum gravity. They also correspond to thePauli X, Y, and Z gates, which are standard generators for the single qubit gates, corresponding to 3d rotations about the axes of theBloch sphere.

The Lie algebra serves to work out therepresentations ofSU(2).

The groupSU(3) is an 8-dimensionalsimple Lie group consisting of all3 × 3unitarymatrices withdeterminant 1.

The groupSU(3) is a simply-connected, compact Lie group.^[10] Its topological structure can be understood by noting thatSU(3) actstransitively on the unit sphere${\displaystyle S^5}$ in${\displaystyle {\mathbb{C}}^3\cong {\mathbb{R}}^6}$. Thestabilizer of an arbitrary point in the sphere is isomorphic toSU(2), which topologically is a 3-sphere. It then follows thatSU(3) is afiber bundle over the baseS⁵ with fiberS³. Since the fibers and the base are simply connected, the simple connectedness ofSU(3) then follows by means of a standard topological result (thelong exact sequence of homotopy groups for fiber bundles).^[11]

TheSU(2)-bundles overS⁵ 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 ofS⁴, 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${\displaystyle \mathbb{Z}/2}$,SU(3) cannot be the trivial bundleSU(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.

The representation theory ofSU(3) is well-understood.^[12] 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 onLie algebra representations orthe Clebsch–Gordan coefficients forSU(3).

The generators,T, of the Lie algebra${\displaystyle \mathfrak{s}\mathfrak{u}(3)}$ ofSU(3) in the defining (particle physics, Hermitian) representation, are

$${\displaystyle T_a=\frac{{\lambda }_a}{2},}$$

whereλ_a, theGell-Mann matrices, are theSU(3) analog of thePauli matrices forSU(2):

Theseλ_a span alltracelessHermitian matricesH of theLie algebra, as required. Note thatλ₂,λ₅,λ₇ are antisymmetric.

They obey the relations

or, equivalently,

Thef are thestructure constants of the Lie algebra, given by

while all otherf_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}.^[c]

The symmetric coefficientsd take the values

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

A genericSU(3) group element generated by a traceless 3×3 Hermitian matrixH, normalized astr(H²) = 2, can be expressed as asecond order matrix polynomial inH:^[13]

LPwhere

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

As noted above, the Lie algebra${\displaystyle \mathfrak{s}\mathfrak{u}(n)}$ ofSU(n) consists ofn ×nskew-Hermitian matrices with trace zero.^[14]

Thecomplexification of the Lie algebra${\displaystyle \mathfrak{s}\mathfrak{u}(n)}$ is${\displaystyle \mathfrak{s}\mathfrak{l}(n;\mathbb{C})}$, the space of alln ×n complex matrices with trace zero.^[15] ACartan subalgebra then consists of the diagonal matrices with trace zero,^[16] which we identify with vectors in${\displaystyle {\mathbb{C}}^n}$ whose entries sum to zero. Theroots then consist of all then(n − 1) permutations of(1, −1, 0, ..., 0).

A choice ofsimple roots is

So,SU(n) is ofrankn − 1 and itsDynkin diagram is given byA_n−1, a chain ofn − 1 nodes:....^[17] ItsCartan matrix is

ItsWeyl group orCoxeter group is thesymmetric groupS_n, thesymmetry group of the(n − 1)-simplex.

For afieldF, thegeneralized special unitary group overF,SU(p,q;F), is thegroup of alllinear transformations ofdeterminant 1 of avector space of rankn =p +q overF which leave invariant anondegenerate,Hermitian form ofsignature(p,q). This group is often referred to as thespecial unitary group of signaturepq overF. The fieldF can be replaced by acommutative ring, in which case the vector space is replaced by afree module.

Specifically, fix aHermitian matrixA of signaturepq in${\displaystyle \mathrm{GL}(n,\mathbb{R})}$, then all

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

satisfy

Often one will see the notationSU(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 classicalLie groups. The standard choice forA when${\displaystyle \mathrm{F}=\mathbb{C}}$ is

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

An important example of this type of group is thePicard 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 realhyperbolic space of dimension two. In 2005 Gábor Francsics andPeter Lax computed an explicit fundamental domain for the action of this group onHC².^[18]

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

In physics the special unitary group is used to representfermionic symmetries. In theories ofsymmetry breaking it is important to be able to find the subgroups of the special unitary group. Subgroups ofSU(n) that are important inGUT physics are, forp > 1,n −p > 1,

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

where × denotes thedirect product andU(1), known as thecircle group, is the multiplicative group of allcomplex numbers withabsolute value 1.

For completeness, there are also theorthogonal andsymplectic subgroups,

Since therank ofSU(n) isn − 1 and ofU(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,

SeeSpin group andSimple Lie group forE₆,E₇, andG₂.

There are also theaccidental isomorphisms:SU(4) = Spin(6),SU(2) = Spin(3) = Sp(1),^[d] andU(1) = Spin(2) = SO(2).

One may finally mention thatSU(2) is thedouble covering group ofSO(3), a relation that plays an important role in the theory of rotations of 2-spinors in non-relativisticquantum mechanics.

where${\displaystyle u^{\ast }}$ denotes thecomplex conjugate of the complex numberu.

This group is isomorphic toSL(2,ℝ) andSpin(2,1)^[19] where the numbers separated by a comma refer to thesignature of thequadratic form preserved by the group. The expression${\displaystyle u{u}^{\ast }-v{v}^{\ast }}$ in the definition ofSU(1,1) is anHermitian form which becomes anisotropic quadratic form whenu andv are expanded with their real components.

An early appearance of this group was as the "unit sphere" ofcoquaternions, introduced byJames Cockle in 1852. Let

Then the 2×2 identity matrix,${\displaystyle ki=j,}$ and${\displaystyle ij=k,}$ and the elementsi, j, andk allanticommute, as inquaternions. Also${\displaystyle i}$ is still a square root of−I₂ (negative of the identity matrix), whereas${\displaystyle j^2=k^2=+I_2}$ are not, unlike in quaternions. For both quaternions andcoquaternions, all scalar quantities are treated as implicit multiples ofI₂ and notated as1.

The coquaternion${\displaystyle q=w+xi+yj+zk}$ with scalarw, 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-sheethyperboloid${\displaystyle \left\{xi+yj+zk:x^2-y^2-z^2=1\right\}}$ corresponds to theimaginary units in the algebra so that any pointp on this hyperboloid can be used as apole of a sinusoidal wave according toEuler's formula.

The hyperboloid is stable underSU(1, 1), illustrating the isomorphism withSpin(2, 1). The variability of the pole of a wave, as noted in studies ofpolarization, might viewelliptical polarization as an exhibit of the elliptical shape of a wave withpole${\displaystyle p\ne \pm i}$. ThePoincaré sphere model used since 1892 has been compared to a 2-sheet hyperboloid model,^[20] and the practice ofSU(1, 1) interferometry has been introduced.

When an element ofSU(1, 1) is interpreted as aMöbius transformation, it leaves theunit disk stable, so this group represents themotions of thePoincaré disk model of hyperbolic plane geometry. Indeed, for a point[z, 1] in thecomplex projective line, the action ofSU(1,1) is given by

since inprojective 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, so that their ratio lies in the open disk.^[21]

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

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

• Lie groups
• Mathematical physics

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