Generalized Lie Algebras
Generalized Lie Algebras
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DOI：
10.1007/978-1-4899-1219-0_21
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发表时间：
1993
期刊：
影响因子：
--
通讯作者：
U. Franz;B. Gruber
中科院分区：
文献类型：
--
作者：
U. Franz;B. Gruber
将李代数L用于物理应用时，会遇到这样一个事实：Cartan子代数H的本征值是以一个常数的整数倍，即根的整数倍来相互联系的。一维谐振子具有哈密顿算符H=a+a+1/2和[a，a+]=i，a，a+玻色子算符，就是这种情况的标准例子。同样，角动量J3的第三个分量的本征值与根1的整数倍相关。然而，角动量C=J2是二次型，是L=su(2)的包络代数U的一个元素。它的特征值是不可约表示中的常数，是非线性的，形式为j(j+1)。
Using Lie algebras L in physical applications one is faced with the fact that the eigenvalues of the Cartan subalgebras H are related to each other by integer multiples of a constant, i. e. integer multiples of the roots. A standard example for this situation is the one-dimensional harmonic oscillator with its HamiltonianH=a+a+ 1/2 and [a,a+] =I, a, a+boson operators. Similarly, the eigenvalues of the third component of angular momentumJ3are related to each other by integer multiples of the root 1. The angular momentumC=J2, however, is a quadratic form and is an element of the enveloping algebra U of L=su(2). Its eigenvalues, constant within an irreducible representation, are non-linear and of the formj(j+ 1).