A Serre-type Theorem for the Elliptic Lie Algebras with Rank $≥ 2$

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