Representations of SU(2) describe non-relativistic spin, due to being a double covering of the rotation group of Euclidean 3-space. Relativistic spin is described by the representation theory of SL 2 (C), a supergroup of SU(2), which in a similar way covers SO + …

李群里面有两个很重要的群，分别是 SU (2) 群和 SO (3) 群，这两个是最简单的非平庸非阿贝尔李群。 这两个群在经典力学和量子力学中都有着至关重要的意义，量子力学中的电子自旋和经典力学中的刚体转动都可以用这两个群描述。 而且 SU (2) 群和 SO (3) 群之间还有着密切的关系。 这里总结一下如何从 SU (2) 群导出 SO (3) 群。 首先给出群同态的定义：给定两个群 (G, *), (H, \cdot) ，群同态是一个映射 h: (G, *) \rightarrow (H, \cdot) ，使得对于所有属于群 G 的元素 u, v ，都 …

In mathematics, the special unitary group of degree n, denoted SU (n), is the Lie group of n × n unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 in the special case. The group operation is matrix multiplication.

2024年1月1日 · 下面我们求SU(2)的表示，第一步先选择一个基，我们选\psi^j_m(x_1,x_2)=\frac{(-1)^{j-m}}{\sqrt{(j+m)!(j-m)!}}x_1^{j-m}x_2^{j+m},一个二维特殊酉矩阵的作用下，它变成\psi^j_m(U^{-1}\vec{x}),这个东西可以重新在\psi^j_m(x_1,x_2)下展开，得到展开矩阵，感兴趣可以算算，但是计算过于冗长。

2020年12月25日 · SU(2)群是所有行列式为1的二阶幺正矩阵构成的群，S表示行列式为1，U表示幺正，2表示二阶。任意二阶幺正矩阵可以表示为 \left[ \begin{array}{c c} a&b\\ -b^* &a^* \end{array} \right] ，令a=x1+ix2, b=x3+ix4，那么行列式为1等价于 x_1^2+x_2^2+x_3^2+x_4^2 = 1 ，即四维空间中的一个球面 ...

There are two examples of such Gthat we have seen: SO(3) and its connected double cover SU(2). These Lie groups are not homeomorphic, as their fundamental groups are distinct. Also, by inspecting the adjoint action of a maximal torus, SU(2) has center f 1gof order

2017年4月25日 · We will now work out in detail the properties of SU(2) and its representations. We have already seen that the generators may be chosen to be. with σi = the Pauli matrices. βij = −ǫaibǫbja = 2δij. Thus our generators are not quite canonically normalized, but are all nor-malized equally, and β is positive definite.

This definition immediately implies that SU(n) ⊂ U(n), but actually SU(n) is also a Subgroup of U(n), something we will prove for n=2 below. SU(2) a Worked Example. Using our definition above when n=2, we can see that SU(2) is the set of 2 × 2 unitary matrices which also have a determinant of 1. We can write this as follows:

In this article we want to list a few examples of isometric SU(2); SO(3)-actions on all known examples of positively curved 6-manifolds, namely S6; CP 3; SU(3)=T 2; SU(3)==T 2. We list the isotropy groups and give geometric description of the orbit spaces.

1. R(Γ,SU(2)) is a topological space (as a subspace of SU(2)n). 2. R(Γ,SU(2)) is a real algebraic variety as SU(2) is algebraic and re-lations are algebraic maps. More precisely, SU(2) may be defined as {(x,y,z,t) ∈ R4,x2+y2+z2+t2 = 1} putting α= x+iyand β= z+it. It is easy to verify that these structures do not depend on the choices of