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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the special unitary group of degree n, denoted SU(n), is the group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 of n×n unitary
Unitary matrix

In mathematics, a unitary matrix is an n by n complex number matrix U satisfying the condition where is the identity matrix and is the conjugate transpose of U....
 matrices
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
 with determinant
Determinant

In algebra, a determinant is a function depending on n that associates a scalar , det, to an n?n square matrix A. The fundamental geometric meaning of a determinant is a scale factor for measure when A is regarded as a linear transformation....
 1. The group operation is that of matrix multiplication
Matrix multiplication

In mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix. This article gives an overview of the various ways to perform matrix multiplication....
. The special unitary group is a subgroup
Subgroup

In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *....
 of the unitary group
Unitary group

In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrix, with the group operation that of matrix multiplication....
 U(n), consisting of all n×n unitary matrices, which is itself a subgroup of the general linear group
General linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrix, together with the operation of ordinary matrix multiplication....
 GL(n, C).

The SU(n) groups find wide application in the standard model
Standard Model

The Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions....
 of physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, especially SU(2) in the electroweak interaction
Electroweak interaction

In particle physics, the electroweak interaction is the unified description of two of the four fundamental interactions of nature: electromagnetism and the weak interaction....
 and SU(3) in QCD
Quantum chromodynamics

Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons ....
.

The simplest case, SU(1), is the trivial group
Trivial group

In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group....
, having only a single element.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the special unitary group of degree n, denoted SU(n), is the group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 of n×n unitary
Unitary matrix

In mathematics, a unitary matrix is an n by n complex number matrix U satisfying the condition where is the identity matrix and is the conjugate transpose of U....
 matrices
Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
 with determinant
Determinant

In algebra, a determinant is a function depending on n that associates a scalar , det, to an n?n square matrix A. The fundamental geometric meaning of a determinant is a scale factor for measure when A is regarded as a linear transformation....
 1. The group operation is that of matrix multiplication
Matrix multiplication

In mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix. This article gives an overview of the various ways to perform matrix multiplication....
. The special unitary group is a subgroup
Subgroup

In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *....
 of the unitary group
Unitary group

In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrix, with the group operation that of matrix multiplication....
 U(n), consisting of all n×n unitary matrices, which is itself a subgroup of the general linear group
General linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrix, together with the operation of ordinary matrix multiplication....
 GL(n, C).

The SU(n) groups find wide application in the standard model
Standard Model

The Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions....
 of physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, especially SU(2) in the electroweak interaction
Electroweak interaction

In particle physics, the electroweak interaction is the unified description of two of the four fundamental interactions of nature: electromagnetism and the weak interaction....
 and SU(3) in QCD
Quantum chromodynamics

Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons ....
.

The simplest case, SU(1), is the trivial group
Trivial group

In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group....
, having only a single element. The group SU(2) is isomorphic to the group of quaternion
Quaternion

Quaternions, in mathematics, are a non-commutative number system that extends the complex numbers. The quaternions were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space....
s of absolute value
Absolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3....
 1, and is thus diffeomorphic to the 3-sphere
3-sphere

In mathematics, a '3-sphere' is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space....
. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), we have a surjective homomorphism
Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ???? meaning "same" and ???f? meaning "shape"....
 from SU(2) to the rotation group
Rotation group

In classical mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of functional composition....
 SO(3) whose kernel
Kernel (mathematics)

In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the Additive identity , as in kernel and kernel ....
 is .

Properties


The special unitary group SU(n) is a real matrix Lie group of dimension n2 − 1. Topologically, it is compact and simply connected. Algebraically, it is a simple Lie group
Simple Lie group

In mathematics, a simple Lie group is a connected space nonabelian group Lie group G which does not have nontrivial connected normal subgroups....
 (meaning its Lie algebra
Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds....
 is simple; see below). The center of SU(n) is isomorphic to the cyclic group
Cyclic group

In group theory, a cyclic group or monogenous group is a group that can be generating set of a group by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g ....
 Zn. Its outer automorphism group
Outer automorphism group

In mathematics, the outer automorphism group of a group Gis the quotient group Aut / Inn, where Aut is the automorphism group of G and Inn is the subgroup consisting of inner automorphisms....
, for n = 3, is Z2, while the outer automorphism group of SU(2) is the trivial group
Trivial group

In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group....
.

The SU(n) algebra is generated by n2 operators, which satisfy the commutator relationship (for i,j,k,l = 1, 2, ..., n)

Additionally, the operator

satisfies

which implies that the number of independent generators of SU(n) is n2-1.

Generators

In general the infinitesimal generators
Infinitesimal generator

In mathematics, the term infinitesimal generator may refer to:* an element of the Lie algebra associated to a Lie group;* the Infinitesimal generator of a stochastic processes;...
 of SU(n), T, are represented
Group representation

In the mathematics field of representation theory, group representations describe abstract group in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrix so that the group operation can be represented by matrix multiplication....
 as traceless
Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
 Hermitian matrices
Hermitian matrix

A Hermitian matrix is a square matrix with complex number entries which is equal to its own conjugate transpose — that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for all indices i and j:...
. I.e:



and

Fundamental representation

In the defining or fundamental representation the generators are represented by n×n matrices where:
where the f are the structure constants and are antisymmetric in all indices, whilst the d are symmetric in all indices.
As a consequence:


We also have


as a normalization convention.

Adjoint representation

In the adjoint representation
Adjoint representation

In mathematics, the adjoint representation of a Lie group G is the natural group representation of G on its own Lie algebra. This representation is the linearized version of the group action of G on itself by conjugation ....
 the generators are represented by × matrices whose elements are defined by the structure constants:

SU(2)


A general SU(2) matrix takes the form

where a and b are complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s satisfying . Here denotes complex conjugation.

In the defining representation the generators are proportional to the Pauli matrices
Pauli matrices

The Pauli matrices are a set of 2 × 2 complex number Hermitian matrix and Unitary matrix matrix Usually indicated by the Greek letter 'sigma' , they are occasionally denoted with a 'tau' when used in connection with isospin symmetries....
  via:
where:

Note that all the generators are traceless
Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
 Hermitian matrices
Hermitian matrix

A Hermitian matrix is a square matrix with complex number entries which is equal to its own conjugate transpose — that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for all indices i and j:...
 as required.

The structure constants for SU(2) are defined by the Levi-Civita symbol
Levi-Civita symbol

The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematics symbol used in particular in tensor calculus....
; the rest can be determined by antisymmetry.
For example,


All the d values vanish.

SU(3)

The generators of SU(3), T, in the defining representation, are:
where , the Gell-Mann matrices
Gell-Mann matrices

The Gell-Mann matrices, named for Murray Gell-Mann, are one possible representation of the Lie group#The Lie algebra associated to a Lie groups of the special unitary group called SU....
, are the SU(3) analog of the Pauli matrices for SU(2):



Note that they are all traceless
Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
 Hermitian matrices
Hermitian matrix

A Hermitian matrix is a square matrix with complex number entries which is equal to its own conjugate transpose — that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for all indices i and j:...
 as required.

These obey the relations
where the f are the structure constants, as previously defined, and have values given by


The d take the values:


Lie algebra


The Lie algebra
Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds....
 corresponding to is denoted by . Its standard mathematical representation consists of the traceless antihermitian complex matrices, with the regular commutator
Commutator

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory....
 as Lie bracket
Lie bracket

Lie bracket can refer to:*Lie algebra*Lie bracket of vector fields...
. A factor is often inserted by particle
Particle physics

Particle physics is a branch of physics that studies the elementary particle constituents of matter and radiation, and the interactions between them....
 physicist
Physicist

A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many Physics#Major fields of physics spanning all length scales: from atom particles of which all ordinary matter is made to the behavior of the material Universe as a whole ....
s, so that all matrices become hermitian. This is simply a different, more convenient, representation of the same real Lie algebra. Note that is a Lie algebra over .

For example, the following antihermitian matrices used in quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
 form a basis
Basis (linear algebra)

In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space or free module, and such that no element of the set can be represented as a linear combination of the others....
 for over : (where is the imaginary unit
Imaginary unit

In mathematics, physics, and engineering, the imaginary unit is denoted by  or the Latin   or the Greek iota . It allows the real number system, to be extended to the complex number system,   Its precise definition is dependent upon the particular method of extension....
.)

This representation is often used in quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
 (see Pauli matrices
Pauli matrices

The Pauli matrices are a set of 2 × 2 complex number Hermitian matrix and Unitary matrix matrix Usually indicated by the Greek letter 'sigma' , they are occasionally denoted with a 'tau' when used in connection with isospin symmetries....
 and Gell-Mann matrices
Gell-Mann matrices

The Gell-Mann matrices, named for Murray Gell-Mann, are one possible representation of the Lie group#The Lie algebra associated to a Lie groups of the special unitary group called SU....
), to represent the spin
Spin (physics)

In quantum mechanics, spin is a fundamental property of atomic nucleus, hadrons, and elementary particles. For particles with non-zero spin, spin direction is an important intrinsic degrees of freedom ....
 of fundamental particles such as electron
Electron

The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
s. They also serve as unit vector
Unit vector

In mathematics, a unit vector in a normed vector space is a Vector space whose Norm is 1 . A unit vector is often denoted by a lowercase letter with a superscribed caret or ?hat?, like this: ....
s for the description of our 3 spatial dimensions in quantum relativity.

Note that the product of any two different generators is another generator, and that the generators anticommute. Together with the identity matrix
Identity matrix

In linear algebra, the identity matrix or unit matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere....
 (times ), these are also generators of the Lie algebra .

Here it depends of course on the problem whether one works finally, as in non-relativistic quantum mechanics, with 2-spinors; or, as in the relativistic Dirac theory
Dirac equation

In physics, the Dirac equation is a theory of relativity quantum mechanics wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary particle spin-? particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity....
, one needs an extension to 4-spinors; or in mathematics even to Clifford algebra
Clifford algebra

In mathematics, Clifford algebras are a type of associative algebra. They can be thought of as one of the possible generalizations of the complex numbers and quaternions....
s.

Note: make clearer the fact that under matrix multiplication (which is anticommutative in this case), we generate the Clifford algebra , whereas you generate the Lie algebra with commutator brackets instead.

Back to general :

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

Subspace may refer to:Mathematics* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication....
 of traceless diagonal
Diagonal matrix

In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero....
  matrices with imaginary entries forms an dimensional Cartan subalgebra
Cartan subalgebra

In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent Lie algebra subalgebra of a Lie algebra that is self-normalising ....
.

Complexify the Lie algebra, so that any traceless matrix is now allowed. The weight
Weight (representation theory)

In the mathematics field of representation theory, a weight of an algebra over a field A over a field F is an algebra homomorphism from A to F, or equivalently, a one dimensional representation of A over F....
 eigenvectors are the Cartan subalgebra itself and the matrices with only one nonzero entry which is off diagonal. Even though the Cartan subalgebra is only 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 th basis vector is the matrix with on the th diagonal entry and zero elsewhere. Weights would then be given by coordinates and the sum over all coordinates has to be zero (because the unit matrix is only auxiliary).

So, has a rank
Rank (linear algebra)

The column rank of a matrix_ A is the maximal number of linear independence columns of A. Likewise, the row rank is the maximal number of linearly independent rows of A....
 of and its Dynkin diagram is given by , a chain of vertices.

Its root system
Root system

In mathematics, a root system is a configuration of vector spaces in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras....
 consists of roots spanning a Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
. Here, we use redundant coordinates instead of to emphasize the symmetries of the root system (the coordinates have to add up to zero). In other words, we are embedding this dimensional vector space in an -dimensional one. Then, the roots consists of all the permutations of . The construction given two paragraphs ago explains why. A choice of simple roots is , ,
…,
.

Its Cartan matrix
Cartan matrix

In mathematics, the term Cartan matrix has three meanings. All of these are named after the France mathematician ?lie Cartan. In an example of Stigler's law of eponymy, Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan....
 is .

Its Weyl group
Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system....
 or Coxeter group
Coxeter group

In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....
 is the symmetric group
Symmetric group

In mathematics, the symmetric group on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
 , the symmetry group
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
 of the -simplex
Simplex

In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of affine transformation Point s in some Euclidean space of dimension n or higher ....
.

Generalized special unitary group

For a field
Field (mathematics)

In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication and division , satisfying certain axioms....
 F, the generalized special unitary group over F, SU(p,q;F), is the group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 of all linear transformation
Linear transformation

In mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication....
s of determinant
Determinant

In algebra, a determinant is a function depending on n that associates a scalar , det, to an n?n square matrix A. The fundamental geometric meaning of a determinant is a scale factor for measure when A is regarded as a linear transformation....
 1 of a vector space
Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
 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
Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a Ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
, in which case the vector space is replaced by a free module
Free module

In mathematics, a free module is a free object in the category of module s. Given a set S, a free module on S is a free module with basis S....
.

Specifically, fix a hermitian matrix
Hermitian matrix

A Hermitian matrix is a square matrix with complex number entries which is equal to its own conjugate transpose — that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for all indices i and j:...
 A of signature p q in GL(n,R), then all

satisfy

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 A when F = C is
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) on complex hyperbolic space of degree two, in the same way that SL(2,Z) acts (projectively) on real hyperbolic space
Hyperbolic space

In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1....
 of dimension two. In 2003 Gábor Francsics and Peter Lax
Peter Lax

Peter David Lax is a mathematician working in the areas of pure and applied mathematics. He has made important contributions to integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other fields....
 computed a fundamental domain for the action of this group on , see . 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 of symmetry breaking
Symmetry breaking

Symmetry breaking in physics describes a phenomenon where small fluctuations acting on a system crossing a Critical point decide a system's fate, by determining which branch of a Bifurcation theory is taken....
 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
Grand unification theory

Grand Unification, grand unified theory, or GUT refers to any of several very similar unified field theory or models in physics that predicts that at extremely high energies , the electromagnetic, weak nuclear, and strong nuclear forces are fused into a single unified field....
 are, for p>1, n-p>1:

For completeness there are also the orthogonal and symplectic
Symplectic group

In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical group . In this article, we shall denote these two groups Sp and Sp....
 subgroups:

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:

(see Spin group
Spin group

In mathematics the spin group Spin is the covering space of the special orthogonal group SO, such that there exists a short exact sequence of Lie groups...
)

(see Simple Lie groups for E6, E7, and G2) 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
Spinor

In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector and tensor....
s in non-relativistic quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
.

See also

  • Representation theory of SU(2)
    Representation theory of SU(2)

    In the study of the representation theory of Lie groups, the study of representations of SU is fundamental to the study of representations of semisimple Lie groups....
  • Projective special unitary group, PSU(n)


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