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In mathematics, the special unitary group of degree , denoted , is the Lie group of
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigrou ...
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
with
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case. The group operation is
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
. The special unitary group is a normal subgroup of the
unitary group In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
, consisting of all unitary matrices. As a compact classical group, is the group that preserves the standard inner product on \mathbb^n. It is itself a subgroup of the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
, \operatorname(n) \subset \operatorname(n) \subset \operatorname(n, \mathbb ). The groups find wide application in the Standard Model of
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, especially in the
electroweak interaction In particle physics, the electroweak interaction or electroweak force is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very differe ...
and in quantum chromodynamics. The groups are important in quantum computing, as they represent the possible quantum logic gate operations in a
quantum circuit In quantum information theory, a quantum circuit is a model for quantum computation, similar to classical circuits, in which a computation is a sequence of quantum gates, measurements, initializations of qubits to known values, and possibly o ...
with n qubits and thus 2^n basis states. (Alternatively, the more general
unitary group In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
U(2^n) can be used, since multiplying by a global
phase factor For any complex number written in polar form (such as ), the phase factor is the complex exponential factor (). As such, the term "phase factor" is related to the more general term phasor, which may have any magnitude (i.e. not necessarily on th ...
e^ does not change the expectation values of a quantum operator.) The simplest case, , is the
trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...
, having only a single element. The group is isomorphic to the group of quaternions of norm 1, and is thus
diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two man ...
to the
3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...
. Since
unit quaternion In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...
s can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
from to the rotation group whose
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...
is . is also identical to one of the symmetry groups of
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
s, Spin(3), that enables a spinor presentation of rotations.


Properties

The special unitary group is a strictly real Lie group (vs. a more general complex Lie group). Its dimension as a
real manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...
is Topologically, it is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
and simply connected. Algebraically, it is a
simple Lie group In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
(meaning its Lie algebra is simple; see below). The
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
of is isomorphic to the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
\mathbb/n\mathbb, and is composed of the diagonal matrices for an ‑th root of unity and the identity matrix. Its
outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...
for is \, \mathbb/2\mathbb \,, while the outer automorphism group of is the
trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...
. A
maximal torus In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefor ...
of
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
is given by the set of diagonal matrices with determinant 1. The
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections ...
of SU(''n'') is the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
, which is represented by
signed permutation matrices In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the non ...
(the signs being necessary to ensure the determinant is 1). The Lie algebra of , denoted by \mathfrak(n), can be identified with the set of
traceless In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace o ...
anti‑Hermitian complex matrices, with the regular commutator as a Lie bracket.
Particle physicists Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) a ...
often use a different, equivalent representation: The set of traceless
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
complex matrices with Lie bracket given by times the commutator.


Lie algebra

The Lie algebra \mathfrak(n) of \operatorname(n) consists of n \times n
skew-Hermitian __NOTOC__ In linear algebra, a square matrix with Complex number, complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix A is skew-Hermitian if it satisf ...
matrices with trace zero. This (real) Lie algebra has dimension n^2 - 1. More information about the structure of this Lie algebra can be found below in the section "Lie algebra structure."


Fundamental representation

In the physics literature, it is common to identify the Lie algebra with the space of trace-zero ''Hermitian'' (rather than the skew-Hermitian) matrices. That is to say, the physicists' Lie algebra differs by a factor of i from the mathematicians'. With this convention, one can then choose generators that are
traceless In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace o ...
Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
complex matrices, where: : T_a \, T_b = \tfrac\,\delta_\,I_n + \tfrac\,\sum_^\left(if_ + d_ \right) \, T_c where the are the
structure constants In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting prod ...
and are antisymmetric in all indices, while the -coefficients are symmetric in all indices. As a consequence, the commutator is: : ~ \left _a, \, T_b\right~ = ~ i \sum_^ \, f_ \, T_c \;, and the corresponding anticommutator is: : \left\ ~ = ~ \tfrac \, \delta_ \, I_n + \sum_^ ~. The factor of i in the commutation relation arises from the physics convention and is not present when using the mathematicians' convention. The conventional normalization condition is :\sum_^ d_\,d_ = \tfrac \, \delta_~ .


Adjoint representation

In the -dimensional
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is G ...
, the generators are represented by × matrices, whose elements are defined by the structure constants themselves: :\left(T_a\right)_ = -if_.


The group SU(2)

is the following group, :\operatorname(2) = \left\~, where the overline denotes complex conjugation.


Diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
with S3

If we consider \alpha,\beta as a pair in \mathbb^2 where \alpha=a+bi and \beta=c+di, then the equation , \alpha, ^2 + , \beta, ^2 = 1 becomes : a^2 + b^2 + c^2 + d^2 = 1 This is the equation of the 3-sphere S3. This can also be seen using an embedding: the map :\begin \varphi \colon \mathbb^2 \to &\operatorname(2, \mathbb) \\ pt \varphi(\alpha, \beta) = &\begin \alpha & -\overline\\ \beta & \overline\end, \end where \operatorname(2,\mathbb) denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering \mathbb^2
diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two man ...
to \mathbb^4 and \operatorname(2,\mathbb) diffeomorphic to \mathbb^8). Hence, the
restriction Restriction, restrict or restrictor may refer to: Science and technology * restrict, a keyword in the C programming language used in pointer declarations * Restriction enzyme, a type of enzyme that cleaves genetic material Mathematics and logi ...
of to the
3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...
(since modulus is 1), denoted , is an embedding of the 3-sphere onto a compact submanifold of \operatorname(2,\mathbb), namely . Therefore, as a manifold, is diffeomorphic to , which shows that is simply connected and that can be endowed with the structure of a compact, connected Lie group.


Isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
with unit quaternions

The complex matrix: : \begin a + bi & c + di \\ -c + di & a - bi \end \quad (a, b, c, d \in \mathbb) can be mapped to the quaternion: :a\,\hat + b\,\hat + c\,\hat + d\,\hat This map is in fact an isomorphism. Additionally, the determinant of the matrix is the square norm of the corresponding quaternion. Clearly any matrix in is of this form and, since it has determinant 1, the corresponding quaternion has norm 1. Thus is isomorphic to the unit quaternions.


Relation to spatial rotations

Every unit quaternion is naturally associated to a spatial rotation in 3 dimensions, and the product of two quaternions is associated to the composition of the associated rotations. Furthermore, every rotation arises from exactly two unit quaternions in this fashion. In short: there is a 2:1 surjective homomorphism from SU(2) to
SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is a tr ...
; consequently SO(3) is isomorphic to the quotient group SU(2)/, the manifold underlying SO(3) is obtained by identifying antipodal points of the 3-sphere , and SU(2) is the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...
of SO(3).


Lie algebra

The Lie algebra of consists of 2\times 2
skew-Hermitian __NOTOC__ In linear algebra, a square matrix with Complex number, complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix A is skew-Hermitian if it satisf ...
matrices with trace zero. Explicitly, this means :\mathfrak(2) = \left\~. The Lie algebra is then generated by the following matrices, :u_1 = \begin 0 & i \\ i & 0 \end, \quad u_2 = \begin 0 & -1 \\ 1 & 0 \end, \quad u_3 = \begin i & 0 \\ 0 & -i \end~, which have the form of the general element specified above. This can also be written as \mathfrak(2)=\operatorname\left\ using the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
. These satisfy the quaternion relationships u_2\ u_3 = -u_3\ u_2 = u_1~, u_3\ u_1 = -u_1\ u_3 = u_2~, and u_1 u_2 = -u_2\ u_1 = u_3~. The commutator bracket is therefore specified by :\left _3, u_1\right= 2\ u_2, \quad \left _1, u_2\right= 2\ u_3, \quad \left _2, u_3\right= 2\ u_1~. The above generators are related to the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
by u_1 = i\ \sigma_1~, \, u_2 = -i\ \sigma_2 and u_3 = +i\ \sigma_3~. This representation is routinely used in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
to represent the spin of
fundamental particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, antiqu ...
s such as
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
s. They also serve as
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction v ...
s for the description of our 3 spatial dimensions in loop quantum gravity. They also correspond to the Pauli X, Y, and Z gates, which are standard generators for the single qubit gates, corresponding to 3d-rotations about the axes of the
Bloch sphere In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch. Quantum mechanics is mathematically formulated i ...
. The Lie algebra serves to work out the representations of .


The group SU(3)

SU(3) is an 8-dimensional
simple Lie group In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
consisting of all
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigrou ...
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
with
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
1.


Topology

The group SU(3) is a simply-connected, compact Lie group. Its topological structure can be understood by noting that SU(3) acts transitively on the unit sphere S^5 in \mathbb^3 \cong \mathbb^6. The stabilizer of an arbitrary point in the sphere is isomorphic to SU(2), which topologically is a 3-sphere. It then follows that SU(3) is a fiber bundle over the base S^5 with fiber S^3. Since the fibers and the base are simply connected, the simple connectedness of SU(3) then follows by means of a standard topological result (the long exact sequence of homotopy groups for fiber bundles). The SU(2)-bundles over S^5 are classified by \pi_4\mathord\left(S^3\right) = \mathbb_2 since any such bundle can be constructed by looking at trivial bundles on the two hemispheres S^5_N, S^5_S and looking at the transition function on their intersection, which is homotopy equivalent to S^4, so : S^5_N \cap S^5_S \simeq S^4 Then, all such transition functions are classified by homotopy classes of maps : \left ^4, SU(2)\right\cong \left ^4, S^3\right= \pi_4\mathord\left(S^3\right) \cong \mathbb/2 and as \pi_4(SU(3)) = \ rather than \mathbb/2, SU(3) cannot be the trivial bundle SU(2) \times S^5 \cong S^3 \times S^5, and therefore must be the unique nontrivial (twisted) bundle. This can be shown by looking at the induced long exact sequence on homotopy groups.


Representation theory

The representation theory of SU(3) is well-understood. Descriptions of these representations, from the point of view of its complexified Lie algebra \mathfrak(3; \mathbb), may be found in the articles on Lie algebra representations or the Clebsch–Gordan coefficients for SU(3).


Lie algebra

The generators, , of the Lie algebra \mathfrak(3) of SU(3) in the defining (particle physics, Hermitian) representation, are :T_a = \frac~, where , the
Gell-Mann matrices The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics. They span the Lie algebra of the SU(3) group in t ...
, are the analog of the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used ...
for : :\begin \lambda_1 = &\begin 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end, & \lambda_2 = &\begin 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end, & \lambda_3 = &\begin 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end, \\ pt \lambda_4 = &\begin 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end, & \lambda_5 = &\begin 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end, \\ pt \lambda_6 = &\begin 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end, & \lambda_7 = &\begin 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end, & \lambda_8 = \frac &\begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end. \end These span all
traceless In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace o ...
Hermitian matrices In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
of the Lie algebra, as required. Note that are antisymmetric. They obey the relations :\begin \left _a, T_b\right&= i \sum_^8 f_ T_c, \\ \left\ &= \frac \delta_ I_3 + \sum_^8 d_ T_c, \end or, equivalently, :\ = \frac\delta_ I_3 + 2\sum_^8. The are the
structure constants In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting prod ...
of the Lie algebra, given by : \begin f_ &= 1, \\ f_ = -f_ = f_ = f_ = f_ = -f_ &= \frac, \\ f_ = f_ &= \frac, \end while all other not related to these by permutation are zero. In general, they vanish unless they contain an odd number of indices from the set . The symmetric coefficients take the values : \begin d_ = d_ = d_ = -d_ &= \frac \\ d_ = d_ = d_ = d_ &= -\frac \\ d_ = d_ = -d_ = -d_ = -d_ = d_ = d_ = d_ &= \frac ~. \end They vanish if the number of indices from the set is odd. A generic group element generated by a traceless 3×3 Hermitian matrix , normalized as , can be expressed as a ''second order'' matrix polynomial in : : \begin \exp(i\theta H) = &\left \frac I\sin\left(\varphi + \frac\right) \sin\left(\varphi - \frac\right) - \frac~H\sin(\varphi) - \frac~H^2\right \frac \\ pt & + \left \frac~I\sin(\varphi) \sin\left(\varphi - \frac\right) - \frac~H\sin\left(\varphi + \frac\right) - \frac~H^\right \frac \\ pt & + \left \frac~I\sin(\varphi) \sin\left(\varphi + \frac\right) - \frac~H \sin\left(\varphi - \frac\right) - \frac~H^2\right \frac \end where :\varphi \equiv \frac\left arccos\left(\frac\det H\right) - \frac\right


Lie algebra structure

As noted above, the Lie algebra \mathfrak(n) of \operatorname(n) consists of n\times n
skew-Hermitian __NOTOC__ In linear algebra, a square matrix with Complex number, complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix A is skew-Hermitian if it satisf ...
matrices with trace zero. The
complexification In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include ...
of the Lie algebra \mathfrak(n) is \mathfrak(n; \mathbb), the space of all n\times n complex matrices with trace zero. A Cartan subalgebra then consists of the diagonal matrices with trace zero, which we identify with vectors in \mathbb C^n whose entries sum to zero. The
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
then consist of all the permutations of . A choice of
simple root Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
s is :\begin (&1, -1, 0, \dots, 0, 0), \\ (&0, 1, -1, \dots, 0, 0), \\ &\vdots \\ (&0, 0, 0, \dots, 1, -1). \end So, is of
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
and its
Dynkin diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
is given by , a chain of nodes: .... Its
Cartan matrix In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Ki ...
is :\begin 2 & -1 & 0 & \dots & 0 \\ -1 & 2 & -1 & \dots & 0 \\ 0 & -1 & 2 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 2 \end. Its
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections ...
or
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
is the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
, the symmetry group of the - simplex.


Generalized special unitary group

For a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
, the generalized special unitary group over ''F'', , is the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of all
linear transformation In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s of
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
1 of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of rank over which leave invariant a
nondegenerate In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case. T ...
,
Hermitian form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...
of
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
. This group is often referred to as the special unitary group of signature over . The field can be replaced by a commutative ring, in which case the vector space is replaced by a free module. Specifically, fix a
Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -t ...
of signature in \operatorname(n, \mathbb), then all :M \in \operatorname(p, q, \mathbb) satisfy :\begin M^ A M &= A \\ \det M &= 1. \end Often one will see the notation without reference to a ring or field; in this case, the ring or field being referred to is \mathbb C and this gives one of the classical
Lie groups In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
. The standard choice for when is : A = \begin 0 & 0 & i \\ 0 & I_ & 0 \\ -i & 0 & 0 \end. However, there may be better choices for for certain dimensions which exhibit more behaviour under restriction to subrings of \mathbb C.


Example

An important example of this type of group is the
Picard modular group In mathematics, a Picard modular group, studied by , is a group of the form SU(''J'',''L''), where ''L'' is a 3-dimensional lattice over the ring of integers of an imaginary quadratic field and ''J'' is a hermitian form on ''L'' of signature  ...
\operatorname(2, 1; \mathbb which acts (projectively) on complex hyperbolic space of degree two, in the same way that \operatorname(2,9;\mathbb) acts (projectively) on real
hyperbolic space In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
of dimension two. In 2005 Gábor Francsics and
Peter Lax Peter David Lax (born Lax Péter Dávid; 1 May 1926) is a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics. Lax has made important contributions to integrable systems, fluid ...
computed an explicit fundamental domain for the action of this group on . A further example is \operatorname(1, 1; \mathbb), which is isomorphic to \operatorname(2, \mathbb).


Important subgroups

In physics the special unitary group is used to represent
bosonic In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
symmetries. In theories of
symmetry breaking In physics, symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observe ...
it is important to be able to find the subgroups of the special unitary group. Subgroups of that are important in GUT physics are, for , :\operatorname(n) \supset \operatorname(p) \times \operatorname(n - p) \times \operatorname(1), where × denotes the direct product and , known as the circle group, is the multiplicative group of all
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s with absolute value 1. For completeness, there are also the orthogonal and symplectic subgroups, :\begin \operatorname(n) &\supset \operatorname(n), \\ \operatorname(2n) &\supset \operatorname(n). \end Since the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
of is and of 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. is a subgroup of various other Lie groups, :\begin \operatorname(2n) &\supset \operatorname(n) \\ \operatorname(n) &\supset \operatorname(n) \\ \operatorname(4) &= \operatorname(2) \times \operatorname(2) \\ \operatorname_6 &\supset \operatorname(6) \\ \operatorname_7 &\supset \operatorname(8) \\ \operatorname_2 &\supset \operatorname(3) \end See
spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a ...
, and simple Lie groups for E6, E7, and G2. There are also the accidental isomorphisms: , , and . One may finally mention that is the
double covering group In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous group homomorphism. The map ''p'' is called the covering homomorphism. ...
of , a relation that plays an important role in the theory of rotations of 2-
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
s in non-relativistic
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
.


The group SU(1,1)

SU(1,1) = \left \~,~ where ~u^*~ denotes the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of the complex number . This group is isomorphic to and where the numbers separated by a comma refer to the
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
of the quadratic form preserved by the group. The expression ~u u^* - v v^*~ in the definition of is an
Hermitian form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...
which becomes an
isotropic quadratic form In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector s ...
when and are expanded with their real components. An early appearance of this group was as the "unit sphere" of coquaternions, introduced by
James Cockle Sir James Cockle FRS FRAS FCPS (14 January 1819 – 27 January 1895) was an English lawyer and mathematician. Cockle was born on 14 January 1819. He was the second son of James Cockle, a surgeon, of Great Oakley, Essex. Educated at Charte ...
in 1852. Let : j = \begin 0 & 1 \\ 1 & 0 \end\,, \quad k = \begin 1 & \;~0 \\ 0 & -1 \end\,, \quad i = \begin \;~0 & 1 \\ -1 & 0 \end~. Then ~j\,k = \begin 0 & -1 \\ 1 & \;~0 \end = -i ~,~ ~ i\,j\,k = I_2 \equiv \begin 1 & 0 \\ 0 & 1 \end~,~ the 2×2 identity matrix, ~k\,i = j ~, and \;i\,j = k \;, and the elements and all anticommute, as in quaternions. Also i is still a square root of (negative of the identity matrix), whereas ~j^2 = k^2 = +I_2~ are not, unlike in quaternions. For both quaternions and coquaternions, all scalar quantities are treated as implicit multiples of   and notated as  . The coquaternion ~q = w + x\,i + y\,j + z\,k~ with scalar , has conjugate ~q = w - x\,i - y\,j - z\,k~ similar to Hamilton's quaternions. The quadratic form is ~q\,q^* = w^2 + x^2 - y^2 - z^2~. Note that the 2-sheet hyperboloid \left\ corresponds to the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
s in the algebra so that any point on this hyperboloid can be used as a pole of a sinusoidal wave according to
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that fo ...
. The hyperboloid is stable under , illustrating the isomorphism with . The variability of the pole of a wave, as noted in studies of polarization, might view
elliptical polarization In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An el ...
as an exhibit of the elliptical shape of a wave with The
Poincaré sphere Poincaré sphere may refer to: * Poincaré sphere (optics), a graphical tool for visualizing different types of polarized light ** Bloch sphere, a related tool for representing states of a two-level quantum mechanical system * Poincaré homology s ...
model used since 1892 has been compared to a 2-sheet hyperboloid model. When an element of is interpreted as a Möbius transformation, it leaves the unit disk stable, so this group represents the
motion In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and m ...
s of the Poincaré disk model of hyperbolic plane geometry. Indeed, for a point in the
complex projective line In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers p ...
, the action of is given by :\bigl ;z,\;1\;\bigr,\beginu & v \\ v^* & u^* \end = ;u\,z + v^*, \, v\,z +u^*\;\, = \, \left ;\frac, \, 1 \;\right/math> since in projective coordinates (\;u\,z + v^*, \; v\,z +u^*\;) \thicksim \left(\;\frac, \; 1 \;\right)~. Writing \;suv + \overline = 2\,\Re\mathord\bigl(\,suv\,\bigr)\;, complex number arithmetic shows :\bigl, u\,z + v^*\bigr, ^2 = S + z\,z^* \quad \text \quad \bigl, v\,z + u^*\bigr, ^2 = S + 1~, where ~S = v\,v^* \left(z\,z^* + 1\right) + 2\,\Re\mathord\bigl(\,uvz\,\bigr)~. Therefore, ~z\,z^* < 1 \implies \bigl, uz + v^*\bigr, < \bigl, \,v\,z + u^*\,\bigr, ~ so that their ratio lies in the open disk.


See also

*
Unitary group In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
*
Projective special unitary group In mathematics, the projective unitary group is the quotient of the unitary group by the right multiplication of its center, , embedded as scalars. Abstractly, it is the holomorphic isometry group of complex projective space, just as the projectiv ...
, * Orthogonal group * Generalizations of Pauli matrices *
Representation theory of SU(2) In the study of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abel ...


Footnotes


Citations


References

* * {{DEFAULTSORT:Special Unitary Group Lie groups Mathematical physics