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In mathematics, the special unitary group of degree , denoted , is the
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
of unitary matrices with
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
1. The matrices of the more general unitary group may have
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
determinants with absolute value 1, rather than real 1 in the special case. The group operation is
matrix multiplication In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
. The special unitary group is a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
of the unitary group , 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 n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
, \operatorname(n) \subset \operatorname(n) \subset \operatorname(n, \mathbb ). The groups find wide application in the
Standard Model The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
of
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
, especially in the electroweak interaction and in
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the study of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of ...
. The simplest case, , is the trivial group, having only a single element. The group is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...
to the group of
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (uniquely up to sign), there is a
surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
from to the rotation group whose kernel is . Since the quaternions can be identified as the even subalgebra of the Clifford Algebra , is in fact identical to one of the symmetry groups of
spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
s, Spin(3), that enables a spinor presentation of rotations.


Properties

The special unitary group is a strictly real
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
(vs. a more general complex Lie group). Its dimension as a real manifold 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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
and
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
. Algebraically, it is a simple Lie group (meaning its
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
is simple; see below). The center of is isomorphic to the
cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
\mathbb/n\mathbb, and is composed of the diagonal matrices for an th root of unity and the identity matrix. Its outer automorphism group for is \mathbb/2\mathbb, while the outer automorphism group of is the trivial group. A maximal torus of rank is given by the set of diagonal matrices with determinant . 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 t ...
of 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 grou ...
, which is represented by signed permutation matrices (the signs being necessary to ensure that the determinant is ). The
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
of , denoted by \mathfrak(n), can be identified with the set of traceless anti‑Hermitian complex matrices, with the regular
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. Group theory The commutator of two elements, ...
as a Lie bracket. Particle physicists often use a different, equivalent representation: The set of traceless Hermitian complex matrices with Lie bracket given by times the commutator.


Lie algebra

The Lie algebra \mathfrak(n) of \operatorname(n) consists of skew-Hermitian matrices with trace zero. This (real) Lie algebra has dimension . More information about the structure of this Lie algebra can be found below in '.


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 Hermitian 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 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 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_ = \frac \, \delta_~ . The generators satisfy the Jacobi identity: _a,[T_b,T_c+[T_b,[T_c,T_a">_b,T_c.html" ;"title="_a,[T_b,T_c">_a,[T_b,T_c+[T_b,[T_c,T_a+[T_c,[T_a,T_b=0. By convention, in the physics literature the generators T_a are defined as the traceless Hermitian complex matrices with a 1/2 prefactor: for the SU(2) group, the generators are chosen as \frac \sigma_1, \frac \sigma_2, \frac \sigma_3 where \sigma_a are the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
, while for the case of SU(3) one defines T_a = \frac \lambda_a where \lambda_a are the Gell-Mann matrices. With these definitions, the generators satisfy the following normalization condition: Tr(T_a T_b) = \frac \delta_.


Adjoint representation

In the -dimensional adjoint representation, the generators are represented by matrices, whose elements are defined by the structure constants themselves: \left(T_a\right)_ = -if_.


The group SU(2)

Using
matrix multiplication In mathematics, specifically in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the n ...
for the binary operation, forms a group, \operatorname(2) = \left\~, where the overline denotes
complex conjugation 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 numbers, then the complex conjugate of a + bi is a - ...
.


Diffeomorphism with the 3-sphere ''S''3

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 to \mathbb^4 and \operatorname(2,\mathbb) diffeomorphic to \mathbb^8). Hence, the restriction of to the 3-sphere (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 In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
and that can be endowed with the structure of a compact, connected
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
.


Isomorphism with group of versors

Quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s of norm 1 are called
versor In mathematics, a versor is a quaternion of Quaternion#Norm, norm one, also known as a unit quaternion. Each versor has the form :u = \exp(a\mathbf) = \cos a + \mathbf \sin a, \quad \mathbf^2 = -1, \quad a \in ,\pi where the r2 = −1 conditi ...
s since they generate the
rotation group 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 ...
: The 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 a
group isomorphism In abstract algebra, a group isomorphism is a function between two groups that sets up a bijection between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the ...
. Additionally, the determinant of the matrix is the squared norm of the corresponding quaternion. Clearly any matrix in is of this form and, since it has determinant , the corresponding quaternion has norm . Thus is isomorphic to the group of versors.


Relation to spatial rotations

Every versor is naturally associated to a spatial rotation in 3 dimensions, and the product of versors is associated to the composition of the associated rotations. Furthermore, every rotation arises from exactly two versors in this fashion. In short: there is a 2:1 surjective homomorphism from to ; consequently is isomorphic to the
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For ex ...
, the manifold underlying is obtained by identifying antipodal points of the 3-sphere , and is the
universal cover In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphism ...
of .


Lie algebra

The
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
of consists of skew-Hermitian 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 that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
. These satisfy the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
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 that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
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 the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
to represent the spin of fundamental particles such as
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s. They also serve as
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
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. Mathematically each quantum mechanical syst ...
. The Lie algebra serves to work out the representations of .


SU(3)

The group is an 8-dimensional simple Lie group consisting of all unitary matrices with
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
1.


Topology

The group is a simply-connected, compact Lie group. Its topological structure can be understood by noting that 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 , which topologically is a 3-sphere. It then follows that is a fiber bundle over the base with fiber . Since the fibers and the base are simply connected, the simple connectedness of then follows by means of a standard topological result (the long exact sequence of homotopy groups for fiber bundles). The -bundles over 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_\text, S^5_\text and looking at the transition function on their intersection, which is a copy of , so S^5_\text \cap S^5_\text \simeq S^4 Then, all such transition functions are classified by homotopy classes of maps \left ^4, \mathrm(2)\right\cong \left ^4, S^3\right= \pi_4\mathord\left(S^3\right) \cong \mathbb/2 and as \pi_4(\mathrm(3)) = \ rather than \mathbb/2, cannot be the trivial bundle , 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 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 .


Lie algebra

The generators, , of the Lie algebra \mathfrak(3) of in the defining (particle physics, Hermitian) representation, are T_a = \frac~, where , the Gell-Mann matrices, are the analog of the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
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 Hermitian matrices of the
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
, 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, \begin \left lambda_a, \lambda_b\right&= 2i \sum_^8 f_ \lambda_c, \\ \ &= \frac\delta_ I_3 + 2\sum_^8. \end The are the structure constants 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 LP 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 consists of skew-Hermitian matrices with trace zero. The complexification of the Lie algebra \mathfrak(n) is \mathfrak(n; \mathbb), the space of all 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 focusin ...
then consist of all the permutations of . A choice of simple roots 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 and its Dynkin diagram is given by , a chain of nodes: .... Its Cartan matrix 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 t ...
or Coxeter group 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 grou ...
, the
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
of the - simplex.


Generalized special unitary group

For a field , the generalized special unitary group over ''F'', , is the group 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 pr ...
s of
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
1 of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
of rank over which leave invariant a nondegenerate,
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 map, linear in each of its arguments, but a sesquilinear f ...
of
signature A signature (; from , "to sign") is a 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. Signatures are often, but not always, Handwriting, handwritt ...
. This group is often referred to as the special unitary group of signature over . The field can be replaced by a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
, in which case the vector space is replaced by a
free module In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commu ...
. 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 ...
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 group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
s. The standard choice for when \operatorname = \mathbb 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 \operatorname(2, 1; \mathbb which acts (projectively) on complex hyperbolic space of dimension two, in the same way that \operatorname(2,9;\mathbb) acts (projectively) on real hyperbolic space of dimension two. In 2005 Gábor Francsics and Peter Lax 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 fermionic symmetries. In theories of symmetry breaking 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 for ...
s with
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
 1. For completeness, there are also the
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
and symplectic subgroups, \begin \operatorname(n) &\supset \operatorname(n), \\ \operatorname(2n) &\supset \operatorname(n). \end Since the rank 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'' and '' Simple Lie group'' for , , and . There are also the accidental isomorphisms: , , and . One may finally mention that is the double covering group of , a relation that plays an important role in the theory of rotations of 2-
spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
s in non-relativistic
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
.


SU(1, 1)

\mathrm(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 numbers, then the complex conjugate of a + bi is 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 , "to sign") is a 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. Signatures are often, but not always, Handwriting, handwritt ...
of the
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
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 map, linear in each of its arguments, but a sesquilinear f ...
which becomes an isotropic quadratic form 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 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
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s. 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 mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
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, for ...
. 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 as an exhibit of the elliptical shape of a wave with The Poincaré sphere model used since 1892 has been compared to a 2-sheet hyperboloid model, and the practice of interferometry has been introduced. When an element of is interpreted as a
Möbius transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying . Geometrically ...
, it leaves the
unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose d ...
stable, so this group represents the
motion In physics, motion is when an object changes its position with respect to a reference point in a given time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and frame of reference to an o ...
s of the
Poincaré disk model In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk t ...
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 (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex ...
, the action of is given by \beginu & v \\ v^* & u^* \end\,\bigl ;z,\;1\;\bigr= ;u\,z + v, \, v^*\,z +u^*\;\, = \, \left ;\frac, \, 1 \;\right/math> since in
projective coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
(\;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 * Projective special unitary group, *
Orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
* Generalizations of Pauli matrices * Representation theory of SU(2)


Footnotes


Citations


References

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