In mathematics, the special unitary group of degree n, denoted SU(n), is the group of n×n unitary matrices with unit determinant. The group operation is that of matrix multiplication. The special unitary group is a subgroup of the unitary group U(n), consisting of all n×n unitary matrices, which is itself a subgroup of the general linear group GL(n, C). Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... This article gives an overview of the various ways to multiply matrices. ... 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 *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices with complex entries, with the group operation that of matrix multiplication. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ...

The simplest case, SU(1), is a trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of absolute value 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), we have a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is { + I, − I}. The following list in mathematics contains the finite groups of small order up to group isomorphism. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In abstract algebra, a homomorphism is a structure-preserving map. ... In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ... The word kernel has several meanings in mathematics, some related to each other and some not. ...

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 (meaning its Lie algebra is simple; see below). The center of SU(n) is isomorphic to the cyclic group Z_n. Its outer automorphism group, for n ≥ 3, is Z₂, while the outer automorphism group of SU(2) is the trivial group. In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In mathematics, a simple Lie group is a Lie group which is also a simple group. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of G — if... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... In mathematics, the outer automorphism group of a group G is the quotient of the automorphism group Aut(G) by its inner automorphism group Inn(G). ... The following list in mathematics contains the finite groups of small order up to group isomorphism. ...

Lie algebra

The Lie algebra corresponding to SU(n) is denoted by . It consists of the traceless antihermitian complex matrices, with the regular commutator as Lie bracket. Note that this is a real and not a complex Lie algebra, in the convention used by mathematicians. A factor i is often inserted by particle physicists who find the different, complex Lie algebra convenient. In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... 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 (the diagonal from the upper left to the lower right) of A, i. ... In linear algebra, a square matrix (or more generally, a linear transformation from a complex vector space with a sesquilinear norm to itself) A is said to be skew-Hermitian or antihermitian if its conjugate transpose A* is also its negative. ... In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ... A lie bracket can refer to: Lie algebra Lie derivative This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Leonhard Euler is considered by many people to be one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is mathematics. ... Particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... A physicist is a scientist trained in physics. ...

For example, the following matrices used in quantum mechanics form a basis for over : For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...

(where i is the imaginary unit.) In mathematics, the imaginary unit (sometimes also represented by the Latin or the Greek iota) allows the real number system to be extended to the complex number system . ...

This representation is often used in quantum mechanics (see Pauli matrices and Gell-Mann matrices), to represent the spin of fundamental particles such as electrons. They also serve as unit vectors for the description of our 3 spatial dimensions in quantum relativity. For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ... The Pauli matrices are a set of 2 × 2 complex Hermitian matrices developed by Wolfgang Pauli. ... The Gell-Mann matrices, named after Murray Gell-Mann, are the infinitesimal generators of su(3). ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is generated by the motion of its center of mass about an external point. ... In particle physics, an elementary particle is a particle of which other, larger particles are composed. ... Properties The electron is a lightweight fundamental subatomic particle that carries a negative electric charge. ... In mathematics, a unit vector in a normed vector space is a vector (most commonly a spatial vector) whose length is 1. ...

Note that the product of any two different generators is another generator, and that the generators anticommute. Together with the identity matrix (times i), For an electrical switch that periodically reverses the current see commutator (electric) In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ... In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...

these are also generators of the Lie algebra .

Note: make clearer the fact that under matrix multiplication (which is anticommutative in this case), we generate the Clifford algebra Cl₃, whereas you generate the Lie algebra with commutator brackets instead.

Back to general SU(n):

If we choose an (arbitrary) particular basis, then the subspace of traceless diagonal matrices with imaginary entries forms an n − 1 dimensional Cartan subalgebra. Screenshot (from SSCX Star Warzone). ... In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ... In mathematics, a Cartan subalgebra is a certain kind of subalgebra of a Lie algebra. ...

Complexify the Lie algebra, so that any traceless matrix is now allowed. The weight eigenvectors are the Cartan subalgebra itself and the matrices with only one nonzero entry which is off diagonal. Even though the Cartan subalgebra 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 ith basis vector is the matrix with 1 on the ith 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). In mathematics, the complexification of a vector space V over the real number field is the corresponding vector space VC over the complex number field. ... Given a set S of complex matrices, each of which is diagonalizable and any two of which commute under multiplication, it is always possible to diagonalize all the elements of S simultaneously. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

So, has a rank of n − 1 and its Dynkin diagram is given by A_{n − 1}, a chain of n − 1 vertices. In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ... See also Simple Lie group. ...

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 . The construction given two paragraphs ago explains why. A choice of simple roots is In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... See also Simple Lie group. ...

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,

…,

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Its Cartan matrix is In mathematics, the term Cartan matrix has two meanings. ...

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Its Weyl group or Coxeter group is the symmetric group S_n, the symmetry group of the (n − 1)-simplex. In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is the subgroup of the isometry group of the root system generated by reflections through the hyperplanes orthogonal to the roots. ... In mathematics, a Coxeter group is a group with a presentation of the form where mi,j ≥ 2; the condition mi,j = ∞ means no relation of the form (xixj)m should be imposed. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... The symmetry group of an object (e. ... In geometry, a simplex (plural: simplices) 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 group of; all linear transformations of determinant 1 of a free module 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. ==GROUP== CONTENTS : 1. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, a free module is a module having a free basis. ... In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some non-zero x in V for all y ∈ V. A nondegenerate form is one that is not degenerate. ... In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. ... In mathematics, signature can refer to The signature of a permutation is ±1 according to whether it is an even/odd permutation. ...

Specifically, fix a hermitian matrix A of signature p q in GL(n,R), then all A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex 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...

satisfy

M ^* AM = A

det M = 1

Often one will see the notation SU_p,q without reference to a ring, in this case the ring being reffered to is C which is one of the classical Lie groups. The standard choice for A when R = C is In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...

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 of dimension two. In 2003 Gábor Francsics and Peter D. Lax computed a fundamental domain for the action of this group on HC², see [1] In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. ...

External links

• Physics 558 - Lecture 1, Winter 2003