|
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1 (i.e., real-valued determinant, not complex as for general unitary matrices). The group operation is that of matrix multiplication. The special unitary group is a subgroup of the unitary group , consisting of all unitary matrices. As a compact classical group, is the group that preserves the standard inner product on .〔For a characterization of and hence in terms of preservation of the standard inner product on , see Classical group.〕 It is itself a subgroup of the general linear group, . The groups find wide application in the Standard Model of particle physics, especially in the electroweak interaction and in quantum chromodynamics. The simplest case, , is the trivial group, having only a single element. The group is isomorphic to the group of quaternions of norm 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), there is a surjective homomorphism from to the whose kernel is }.〔For an explicit description of the homomorphism , see Connection between SO(3) and SU(2).〕 is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations. ==Properties== The special unitary group is a real Lie group (though not a complex Lie group). Its dimension as a real manifold is . Topologically, it is compact and simply connected. Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below). 〔Wybourne, B G (1974). ''Classical Groups for Physicists'', Wiley-Interscience. ISBN 0471965057 .〕 The center of is isomorphic to the cyclic group , and is composed of the diagonal matrices for an th root of unity and the ''n''×''n'' identity matrix. Its outer automorphism group, for , is , 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 1. The Weyl group is the symmetric group , which is represented by signed permutation matrices (the signs being necessary to ensure the determinant is 1). The Lie algebra of , denoted by , can be identified with the set of traceless antihermitian complex matrices, with the regular commutator as 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「special unitary group」の詳細全文を読む スポンサード リンク
|