A Serre-type Theorem for the Elliptic Lie Algebras with Rank ≥ 2
A Serre-type Theorem for the Elliptic Lie Algebras with Rank ≥ 2
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DOI：
10.2977/prims/1145475810
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发表时间：
2004-06
影响因子：
1.2
通讯作者：
H. Yamane
中科院分区：
文献类型：
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作者：
H. Yamane
2000年，K. Saito和D. Yoshii给出了简列椭圆李代数的Serre-type定理。我们将该定理推广到秩≥2的（约简标记的）椭圆根所关联的椭圆李代数。80年代初，K. Saito [S]提出了广义根系，特别是椭圆型根系的概念。从那时起，已经进行了几次尝试，以构造具有其“实根”形成这些根系统的性质的李代数（参见[SY，引言]）。在上个世纪的最后一年，K. Saito和D. Yoshii [SY]引入了简系椭圆李代数g(R)的三种“普遍”表示，即与简系椭圆根R相关联的椭圆李代数。我们可以说g(R)在具有上述性质的李代数中是极大的（另见第二段）。让我们来解释一下演示。第一个使用Borcherds晶格顶点代数。这可以说是最漂亮和有用的，因为它不依赖于标记G (R)，并且明确地给出了G (R)的基及其结构常数。第二个使用（仿射型）海森堡代数。这也是有用的，特别是研究g(R)的表示理论，因为它给出了g(R)的三角分解。第三种是Serre-type定理，也就是齐藤（K. Saito）所提出的communication定理。2002年5月9日收。2000数学学科分类：小学17B65；大阪大学信息科学与技术研究生院纯数学与应用数学教研室，日本丰中560-0043；电子邮件:yamane@ist.osaka-u.ac.jp
In 2000, K. Saito and D. Yoshii gave a Serre-type theorem for the simply-laced elliptic Lie algebras. We extend the theorem to that for the elliptic Lie algebras associated with the (reduced marked) elliptic root systems with rank ≥ 2. Introduction In the early eighties, K. Saito [S] introduced the concept of the generalized root systems and, in particular, the elliptic root systems. Since then, several attempts have been done to construct Lie algebras having the property that their “real roots” form those root systems (see [SY, Introduction]). In the final year of the last century, K. Saito and D. Yoshii [SY] introduced three kinds of “universal” presentations of the simply-laced elliptic Lie algebras g(R), that is, the elliptic Lie algebras associated with the simply-laced elliptic root systems R. We can say that the g(R) is maximal among the Lie algebras having the above property (see also the second paragraph). Let us explain the presentations. The first one uses the Borcherds lattice vertex algebras. This can be said to be most beautiful and useful, because it does not depend on a marking G of R and gives a basis of g(R) and its structure constants explicitly. The second one uses (affine-type) Heisenberg algebras. This is also useful, especially to study the representation theory of g(R) since it gives a triangular decomposition of g(R). The third one is a Serre-type theorem, that is to say a presentation of Communicated by K. Saito. Received May 9, 2002. 2000 Mathematics Subject Classification(s): Primary 17B65; Secondary 22E65 ∗Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka 560-0043, Japan. e-mail: yamane@ist.osaka-u.ac.jp