Graded Lie algebras: Generalization of Hermitian representations
Graded Lie algebras: Generalization of Hermitian representations
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分级李代数：埃尔米特表示的推广
DOI：
10.1063/1.523148
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发表时间：
1977
影响因子：
1.3
通讯作者：
V. Rittenberg
中科院分区：
文献类型：
--
作者：
M. Scheunert;W. Nahm;V. Rittenberg
埃尔米特表示在单李代数表示的研究中起着基础性的作用。我们展示了这个概念如何推广到经典的单分次李代数。星星和等级星星表示通过伴随和等级伴随操作来定义。每个代数允许最多两个伴随和两个等级伴随操作（我们列出了所有经典单分次李代数的各种可能性）。每一个伴随（等级伴随）运算对应一类星星（等级星星）表示。属于同一类的两个星星表示的张量积可完全约化为属于同一类的不可约表示。这个性质非常有用，因为一般来说，经典单分次李代数的有限维表示是不完全可约的。
Hermitian representations play a fundamental role in the study of the representations of simple Lie algebras. We show how this concept generalizes for classical simple graded Lie algebras. Star and grade star representations are defined through adjoint and grade adjoint operations. Each algebra admits at most two adjoint and two grade adjoint operations (we list the various possibilities for all classical simple graded Lie algebras). To each adjoint (grade adjoint) operation corresponds a class of star (grade star) representations. The tensor product of two star representations belonging to one class is completely reducible into irreducible representations belonging to the same class. This property is very useful since in general the finite‐dimensional representations of classical simple graded Lie algebras are not completely reducible.