Semisimple Graded Lie Algebras
Semisimple Graded Lie Algebras
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半简单分级李代数
DOI：
10.1063/1.522421
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发表时间：
1975
影响因子：
1.3
通讯作者：
V. Rittenberg
中科院分区：
文献类型：
--
作者：
A. Pais;V. Rittenberg
在分级李代数中引入了度量的概念。在非奇异的度量条件下定义了半单阶李代数。证明了对于这类代数，度量张量产生一个二次卡西米尔算子。此外，对于这门课，分级表示是不可约的，其权重与李代数的根有关（“根-权定理”）。解决了求所有半单阶李代数的问题。对于SU(N), N≤2，对于O(N), N≤5，以及所有的异常群都没有。对于所有其他的半简单李代数，有且只有一个。这些是根据Sp（2N）矩阵的方便实现显式构造的。对SU(2)进行了详细的讨论，发现了一个新的群[GSU(2)]，它留下了一个c‐数/q‐数混合二次型不变量。我们也为这个群定义了不可约张量算子。SU(N), N≤2，提供了非半简单分级的例子。
The concept of metric is introduced for graded Lie algebras. Semisimple graded Lie algebras are defined in terms of metric conditions of nonsingularity. It is shown that for this class of algebras the metric tensor generates a quadratic Casimir operator. Also for this class, the grading representation is irreducible and its weights are related to the roots of the Lie algebra (’’root‐weight theorem’’). The problem is solved to find all semisimple graded Lie algebras. For SU(N), N≳2, for O(N), N≳5, and for all exceptional groups there are none. For all other semisimple Lie algebras there is one and only one. These are explicity constructed in terms of a convenient realization of Sp(2N) matrices. SU(2) is discussed in some detail and a new group [GSU(2)] is found which leaves a mixed c‐number/q‐number quadratic form invariant. We also define irreducible tensor operators for this group. SU(N), N≳2, provides examples of nonsemisimple gradings.