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Special unitary group

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	Topics Special unitary group Topic Home In mathematicsMathematics Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... , the special unitary group of degree n, denoted SU(n), is the groupGroup (mathematics) In mathematics, a group is a set together with a binary operation satisfying certain axioms, detailed below.... of n×n unitary matricesUnitary matrix Summary In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition... with determinantDeterminant Overview In algebra, a determinant is a function depending on n that associates a scalar, det, to every n'n square matrix,... 1. The group operation is that of matrix multiplicationMatrix multiplication This article gives an overview of the various ways to multiply matrices. ... . The special unitary group is a subgroupSubgroup In group theory, given a group G under a binary operation , we say that some subset H of G is a subgroup* of G... of the unitary groupUnitary group In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the g... U(n), consisting of all n×n unitary matrices, which is itself a subgroup of the general linear groupGeneral linear group In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with ... GL(n, C). The SU(n) groups find wide application in the standard modelStandard Model The Standard Model of particle physics is a theory which describes the strong, weak, and electromagnetic fundamental forces,... of physicsPhysics Physics , the most fundamental physical science, is concerned with the underlying principles of the natural world.... , especially SU(2) in the electroweak interactionElectroweak interaction In particle physics, the electroweak interaction is the unified description of two of the four fundamental interactions of n... and SU(3) in QCDQuantum chromodynamics Quantum chromodynamics is the theory of the strong interaction , a fundamental force describing the interactions of the qua... . The simplest case, SU(1), is the trivial groupTrivial group In mathematics, the trivial group is the group consisting of a single element, e.... , having only a single element. The group SU(2) is isomorphic to the group of quaternionQuaternion In mathematics, quaternions are a non-commutative extension of complex numbers.... s of absolute valueAbsolute value In mathematics, the absolute value of a real number is its numerical value without regard to its sign.... 1, and is thus diffeomorphic to the 3-sphere3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere.... . Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), we have a surjective homomorphismHomomorphism In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures .... from SU(2) to the rotation groupRotation group In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space,... SO(3) whose kernelKernel (mathematics) In mathematics, especially algebra, kernels are general constructions which measure the failure of a homomorphism or function to b... 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 groupFacts About Simple Lie group In mathematics, a simple Lie group is a Lie group which is... (meaning its Lie algebraLie algebra In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups a... is simple; see below). The center of SU(n) is isomorphic to the cyclic groupFacts About Cyclic group In group theory, a cyclic group or monogenous group is a group that can be generated by a single element, in the sense... Z_n. Its outer automorphism groupOuter automorphism group In mathematics, the outer automorphism group of a group G... , for n = 3, is Z₂, while the outer automorphism group of SU(2) is the trivial groupTrivial group In mathematics, the trivial group is the group consisting of a single element, e.... . 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 generators of SU(n), T, are representedGroup representation Group representation theory is the branch of mathematics that studies properties of abstract groups via their representation... as tracelessTrace (linear algebra) Overview 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 d... Hermitian matricesHermitian matrix A Hermitian matrix is a square matrix with complex entries which is equal to its own conjugate transpose — that is, th... . 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 representationAdjoint representation In mathematics, the adjoint representation of a Lie group G is the natural representation of G on its own Lie algebr... the generators are represented by (n^2-1)×(n^2-1) matrices whose elements are defined by the structure constants: SU(2) For SU(2), the generators T, in the defining representation, are proportional to the Pauli matricesPauli matrices Summary The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices.... , via: where: Note that all the generators are tracelessTrace (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 d... Hermitian matricesHermitian matrix A Hermitian matrix is a square matrix with complex entries which is equal to its own conjugate transpose — that is, th... as required. The structure constants for SU(2) are defined by the Levi-Civita symbolLevi-Civita symbol The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol use... ; the rest can be determined by antisymmetry. All the d values vanish. SU(3) The generators of SU(3), T, in the defining representation, are: where , the Gell-Mann matricesGell-Mann matrices Summary The Gell-Mann matrices, named for Murray Gell-Mann, are one possible representation of the infinitesimal generators of the ... , are the SU(3) analog of the Pauli matrices for SU(2): Note that they are all tracelessTrace (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 d... Hermitian matricesHermitian matrix A Hermitian matrix is a square matrix with complex entries which is equal to its own conjugate transpose — that is, th... 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 algebraLie algebra Summary In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups a... corresponding to is denoted by . Its standard mathematical representation consists of the traceless antihermitian complex matrices, with the regular commutatorCommutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative... as Lie bracketLie bracket A lie bracket can refer to:Lie algebra... . A factor is often inserted by particleFacts About Particle physics Particle physics* is a branch of physics that studies the elementary constituents of matter and radiation, and the interactio... physicistPhysicist A physicist is a scientist who studies or practices physics.... 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 mechanicsFacts About Quantum mechanics Quantum mechanics is a first quantized quantum theory that supersedes classical mechanics at the atomic and subatomic levels... form a basisFacts About 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 s... for over : (where is the imaginary unitImaginary unit In mathematics, the imaginary unit allows the real number system to be extended to the complex number system .... .) This representation is often used in quantum mechanicsQuantum mechanics Overview Quantum mechanics is a first quantized quantum theory that supersedes classical mechanics at the atomic and subatomic levels... (see Pauli matricesPauli matrices The Pauli matrices are a set of 2 × 2 complex Hermitian and unitary matrices.... and Gell-Mann matricesGell-Mann matrices The Gell-Mann matrices, named for Murray Gell-Mann, are one possible representation of the infinitesimal generators of the ... ), to represent the spinSpin (physics) In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the mo... of fundamental particles such as electronElectron The electron is a fundamental subatomic particle that carries an electric charge.... s. They also serve as unit vectorUnit vector In mathematics, a unit vector in a normed vector space is a vector whose length is 1.... 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 matrixIdentity matrix In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and ... (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 theoryDirac equation In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac... , one needs an extension to 4-spinors; or in mathematics even to Clifford algebraClifford algebra Clifford algebras are a type of associative algebra in mathematics.... 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 subspaceSubspace Subspace may refer to:* Linear subspace or vector subspace, in linear algebra and related fields of mathematics... of traceless diagonalDiagonal matrix In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero.... matrices with imaginary entries forms an dimensional Cartan subalgebraCartan subalgebra In mathematics, a Cartan subalgebra is a nilpotent subalgebra of a Lie algebra that is self-normalising .... . Complexify the Lie algebra, so that any traceless matrix is now allowed. The weightWeight (representation theory) Weight is a concept arising often in representation theory of Lie groups and Lie algebras, a branch of mathematics.... 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 rankRank (linear algebra) In linear algebra, the column rank of a matrix A with entries in some field is defined to be the maximal number of colu... of and its Dynkin diagram is given by , a chain of vertices. Its root systemRoot system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties.... consists of roots spanning a Euclidean spaceEuclidean space Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", wh... . 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 matrixCartan matrix In mathematics, the term Cartan matrix has two meanings. Both of these are named after Elie Cartan. ... is . Its Weyl groupWeyl group In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is the subgroup of the isome... or Coxeter groupCoxeter group In mathematics, a Coxeter group, named after H.S.M.... is the symmetric groupSymmetric group In mathematics, the symmetric group on a set X, denoted by SX or Sym, is the group whose underlying set is the set o... , the symmetry groupSymmetry group The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operatio... of the -simplexSimplex Overview In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle.... . Generalized special unitary group The generalized special unitary group over R, SU(p,q;F), is the groupGroup (mathematics) In mathematics, a group is a set together with a binary operation satisfying certain axioms, detailed below.... of all linear transformationFacts About Linear transformation In mathematics, a linear transformation is a function between two vector spaces that preserves the operations of vector add... s of determinantDeterminant In algebra, a determinant is a function depending on n that associates a scalar, det, to every n'n square matrix,... 1 of a free moduleFree module In mathematics, a free module is a module having a free basis.... of rank n = p + q over a commutative ring R 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 R. Specifically, fix a hermitian matrixHermitian matrix A Hermitian matrix is a square matrix with complex entries which is equal to its own conjugate transpose — that is, th... A of signature p q in GL(n,R), then all satisfy Often one will see the notation without reference to a ring, in this case the ring being referred to is C which is one of the classical Lie groups. The standard choice for A when R = 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 spaceHyperbolic space In mathematics, hyperbolic n-space, denoted H'n, is the maximally symmetric, simply connected, n-dimensional ... of dimension two. In 2003 Gábor Francsics and Peter D. Lax 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 breakingSymmetry breaking Symmetry breaking in physics describes a phenomenon where small fluctuations acting on a system crossing a critical point de... 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 physicsGrand unification theory Grand unification, grand unified theory, or GUT is one of several very similar theories or models in physics tha... are, for p>1, n-p>1: For completeness there are also the orthogonal and symplectic subgroups: Since the rankRank Rank is a very broad term with several meanings.... of SU(n) is n-1 and 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: There are also the identities SU(4)=O(6), SU(2)=O(3)=USp(2) and U(1)=O(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-spinors in non-relativistic quantum mechanics. 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 repres... External links