Characters of typical irreducible finite-dimensional q(n) -modules
Characters of typical irreducible finite-dimensional q(n) -modules
复制标题
典型不可约有限维 q(n) 模的特征
DOI：
10.1007/bf01077312
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发表时间：
1986
影响因子：
0.4
通讯作者：
I. Penkov
中科院分区：
文献类型：
--
作者：
I. Penkov
级数q的经典李超代数（它们是Gell-Mann—micheli - radicati[10]的简单(f, d)-代数的自然扩展）通常被称为“奇异”。原因之一是，与基本经典级数$1 (n I m),~[(nlm),=~（n [2m）相比，李超代数q (n)在[6]意义上不矛盾。由于证明了即使是基本经典级数的表示理论也远非平凡，所以在研究经典李超代数的有限维不可约表示的特征时，VG Kac既没有考虑级数q，也没有考虑相应的简单级数~ q (~ q (n)[3]是[10]中tho （f, d)-代数的另一种符号或[6]中q (n I)）。关于q (m)的基本结果是由A， N. Sergeev发现的。特别是，他明确地描述了通用包络代数U (q (m))[11]的中心，并找到了标准表示[4]的张量代数的不可约子模的特征公式。在本文中，我们证明了回溯到Bernshtein- Gel'fand-Gel'fand[2]的有限维不可约表示的特征的代数方法（并由Kac应用于基本经典级数[7]），也导致了级数在一定的自然变化后的目标~。在这里（就像Kac所做的那样），我们得到了在最高权重集合中某个“处处密集”（更准确地说是Zariski开）集合的决定性答案。根据Kac[7]，我们称这些权重为典型最高权重。一个自然的问题。调查剩余（非典型）表示的方法目前仍然开放。
The classical Lie superalgebras of the series q (which are natural extensions of the simple (f, d)-algebras of Gell-Mann--Michel-Radicati [10]) are usually called" strange." One of the reasons is that in contrast with the basic classical series $1 (n I m),~[(nlm),=~(n [2m), the Lie superalgebras q (n) are not contragradient in the sense of [6]. And since it turned out that even the theory of representations of the basic classical series is far from banal, in investigating the characters of finite-dimensional irreducible representations of the classical Lie superalgebras, VG Kac did not consider either the series q or the corresponding simple series~ q (~ q (n)[3] is another notation for tho (f, d)-algebras in [10] or Q (n I) in [6]). Essential results on q (m) were found by A, N. Sergeev. In particular, he explicitly described the center of the universal enveloping algebra U (q (m))[11] and found a formula for the characters for irreducible submodules of the tensor algebra of the standard representation [4].In the present paper we show that the algebraic approach to the characters of the irreducible finite-dimensional representations going back to Bernshtein--Gel'fand-Gel'fand [2](and applied by Kac to the basic classical series [7]), leads to the goal after certain natural changes for the series~ also. Here (just as Kac did) we get a decisive answer for a certain" everywhere dense"(more precisely Zariski open) set in the set of highest weights. Following Kac [7], we call these weights typical highest weights. The problem of a natural. method of investigating the remaining (nontypical) representations remains open for now.