Generalized Kac-Moody algebras
Generalized Kac-Moody algebras
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DOI：
10.1016/0021-8693(88)90275-x
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发表时间：
1988-06
影响因子：
0.9
通讯作者：
R. Borcherds
中科院分区：
文献类型：
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作者：
R. Borcherds
我们研究了一类具有几乎正定的逆变双线性形式的李代数。这些代数推广了Kac-Moody代数，可以认为是具有虚单根的Kac-Moody代数。关于Kac-Moody代数的大多数事实推广到这些新代数；例如，我们证明了一个版本的Kac-Weyl字符公式，它与通常的公式相似，只是它对虚单根有一个额外的校正项。这些新代数有几种出现的方式。图自同构下任意Kac-Moody代数的不动点代数通常不是Kac-Moody代数，而是这些更一般的代数之一。还有一种广义的Kac-Moody代数与任何不超过26维的偶数洛伦兹晶格或不超过10维的洛伦兹晶格相关，我们给出了这些代数根的多重性的简单公式（但不幸的是我不知道卡坦矩阵是什么）！数字26和10来自“无鬼”定理。
We study a class of Lie algebras which have a contravariant bilinear form which is almost positive definite. These algebras generalize Kac-Moody algebras and can be thought of as Kac-Moody algebras with imaginary simple roots. Most facts about Kac-Moody algebras generalize to these new algebras; for example, we prove a version of the Kac-Weyl character formula, which is like the usual one except that it has an extra correction term for the imaginary simple roots. There are several ways in which these new algebras turn up. The fixed point algebra of any Kac-Moody algebra under a diagram automorphism is not usually a Kac-Moody algebra, but is one of these more general algebras. There is also a generalized Kac-Moody algebra associated to any even Lorentzian lattice of dimension at most 26 or any Lorentzian lattice of dimension at most 10, and we give a simple formula for the multiplicities of the roots of these algebras (but unfortunately I do not know what the Cartan matrices are)! The numbers 26 and 10 come from the “no ghost” theorem.