Special unitary group

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In mathematics, the special unitary group of degree n, denoted SU(n), is the group of n×n unitary matrices with determinant 1. The group operation is that ofmatrix 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 thegeneral linear group GL(nC).

The SU(n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in QCD.

The simplest case, SU(1), is the 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 surjectivehomomorphism from SU(2) to the rotation group SO(3) whose kernel is { + I, − I}.

Groups

A group expresses the intuitive concept of symmetry.

group (G, •) is a set G closed under a binary operation • satisfying the following 3 axioms:

  • Associativity: For all ab and c in G, (a • b) • ca • (b • c).
  • Identity element: There exists an eG such that for all a in Ge • aa • ea.
  • Inverse element: For each a in G, there is an element b in G such that a • bb • ae, where e is an identity element.

Basic definitions

subset H ⊂ G is a subgroup if the restriction of • to H is a group operation on H. It is called normal, if left and right cosets agree, i.e. gHHg for all g in G.

Given a subset S of a group G, the smallest subgroup of G containing S is called the subgroup generated by S. It is often denoted <S>.

Both subgroups and normal subgroups of a given group form a complete lattice under inclusion of subsets; this property and some related results are described by the lattice theorem.

group homomorphism is a map fG → H between two groups that preserves the structure imposed by the operation, i.e.

f(ab) = f(a) • f(b).

Bijective (in-surjective) maps are isomorphisms of groups (mono-epimorphisms, respectively). The kernel ker(f) is always a normal subgroup of the group. For f as above, the fundamental theorem on homomorphisms relates the structure of G and H, and of the kernel and image of the homomorphism, namely

G / ker(f) ≅ im(f).

One of the fundamental problems of group theory is the classification of groups up to isomorphism.

Groups together with group homomorphisms form a category.

Finiteness conditions

The order |G| (or o(G)) of a group is the cardinality of G. If the order |G| is (in-)finite, then G itself is called (in-)finite. An important class is the group of permutations or symmetric groups of N letters, denoted SNCayley’s theorem exhibits any finite group G as a subgroup of the symmetric group on G.

Abelian groups

The category of groups can be subdivided in several ways. A particularly well-understood class of groups are the so-called abelian (in honor of Niels Abel, or commutative) groups, i.e. the ones satisfying

a • bb • a for all ab in G.

Another way of saying this is that the commutator

[ab] := a−1b−1ab

equals the identity element. A non-abelian group is a group that is not abelian. Even more particular, cyclic groups are the groups generated by a single element. Being either isomorphic to Z or to Zn, the integers modulo n, they are always abelian.

General linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible. The name is because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.

To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over R (the set of real numbers) is the group of n×n invertible matrices of real numbers, and is denoted by GLn(R) or GL(nR).

More generally, the general linear group of degree n over any field F (such as the complex numbers), or a ring R (such as the ring of integers), is the set of n×ninvertible matrices with entries from F (or R), again with matrix multiplication as the group operation.[1] Typical notation is GLn(F) or GL(nF), or simply GL(n) if the field is understood.

More generally still, the general linear group of a vector space GL(V) is the abstract automorphism group, not necessarily written as matrices.

The special linear group, written SL(nF) or SLn(F), is the subgroup of GL(nF) consisting of matrices with a determinant of 1.

The group GL(nF) and its subgroups are often called linear groups or matrix groups (the abstract group GL(V) is a linear group but not a matrix group). These groups are important in the theory of group representations, and also arise in the study of spatial symmetries and symmetries of vector spaces in general, as well as the study of polynomials. The modular group may be realised as a quotient of the special linear group SL(2, Z).

If n ≥ 2, then the group GL(nF) is not abelian.

SU(2)

\operatorname{SU}_2(\mathbb{C}) and \mathfrak{su}_2(\mathbb{C})

A general matrix element of \operatorname{SU}_2(\mathbb{C}) takes the form

U =  \begin{pmatrix} \alpha&-\overline{\beta}\\ \beta&\overline{\alpha} \end{pmatrix}

where \alpha,\beta\in\mathbb{C} such that | α | 2 + | β | 2 = 1. We can consider the following map \varphi : \mathbb{C}^2 \to \operatorname{M}(2,\mathbb{C}), (where \operatorname{M}(2,\mathbb{C}) denotes the set of 2 by 2 complex matrices), defined in the obvious way by

 \varphi(\alpha,\beta) = \begin{pmatrix} \alpha&-\overline{\beta}\\ \beta&\overline{\alpha} \end{pmatrix}.

By considering \mathbb{C}^2 diffeomorphic to \mathbb{R}^4 and \operatorname{M}(2,\mathbb{C}) diffeomorphic to \mathbb{R}^8 we can see that \varphi is an injective real linear map and hence an embedding. Now considering the restriction of \varphi to the 3-sphere, denoted S3, we can see that this is an embedding of the 3-sphere onto a compact submanifold of \operatorname{M}(2,\mathbb{C}). However it is also clear that \varphi(S^3) = \operatorname{SU}_2(\mathbb{C}), which as a manifold is diffeomorphic to \operatorname{SU}_2(\mathbb{C}), making \operatorname{SU}_2(\mathbb{C}) a compact, connected Lie group.

Now considering the Lie algebra \mathfrak{su}_2(\mathbb{C}), a general element takes the form

 U' =  \begin{pmatrix} ix & -\overline{\beta}\\ \beta & -ix \end{pmatrix}

where x \in \mathbb{R} and \beta \in \mathbb{C}. It is easily verified that matrices of this form have trace zero and are antihermitian. The Lie algebra is then generated by the following matrices

 u_1 = \begin{pmatrix} 0 & i\\ i & 0 \end{pmatrix} \qquad u_2 = \begin{pmatrix} 0 & -1\\ 1 & 0 \end{pmatrix} \qquad u_3 = \begin{pmatrix} i & 0\\ 0 & -i \end{pmatrix}

which are easily seen to have the form of the general element specified above. These satisfy the relations u3u2 = − u2u3 = − u1 and u2u1 = − u1u2 = − u3. The commutator bracket is therefore specified by

 [u_3,u_1]=2u_2, \qquad [u_1,u_2] = 2u_3, \qquad [u_2,u_3] = 2u_1.

The above generators are related to the Pauli matrices by u1iσ1u2 = − iσ2 and u3iσ3.

SU(3)

The generators of \mathfrak{su}(3), T, in the defining representation, are:

T_a = \frac{\lambda_a }{2}.\,

where \lambda \,, the Gell-Mann matrices, are the SU(3) analog of the Pauli matrices for SU(2):

\lambda_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \lambda_2 = \begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \lambda_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{pmatrix}
\lambda_4 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix} \lambda_5 = \begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{pmatrix} \lambda_6 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}
\lambda_7 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix} \lambda_8 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -2 \end{pmatrix}.

Note that they are all traceless Hermitian matrices as required.

These obey the relations

\left[T_a, T_b \right] = i \sum_{c=1}^8{f_{abc} T_c} \,

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

f_{123} = 1 \,
f_{147} = -f_{156} = f_{246} = f_{257} = f_{345} = -f_{367} = \frac{1}{2} \,
f_{458} = f_{678} = \frac{\sqrt{3}}{2}, \,

and all other fabc not related to these by permutation are zero.

The d take the values:

d_{118} = d_{228} = d_{338} = -d_{888} = \frac{1}{\sqrt{3}} \,
d_{448} = d_{558} = d_{668} = d_{778} = -\frac{1}{2\sqrt{3}} \,
d_{146} = d_{157} = -d_{247} = d_{256} = d_{344} = d_{355} = -d_{366} = -d_{377} = \frac{1}{2}. \,

References

http://en.wikipedia.org/wiki/Special_unitary_group

Quarks and Leptons: An Introductory Course in Modern Particle Physics

http://ebookee.org/Quarks-and-Leptons-An-Introductory-Course-in-Modern-Particle-Physics_542136.html

http://www.ima.umn.edu/~miller/symmetrygroups.html

http://en.wikipedia.org/wiki/Group_theory

http://en.wikipedia.org/wiki/Glossary_of_group_theory

http://en.wikipedia.org/wiki/General_linear_group

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