On the defining relations of the simply-laced elliptic Lie algebras. (English) Zbl 1013.17019

Takebayashi, Tadayoshi

Proc. Japan Acad., Ser. A 77, No. 7, 119-121 (2001).

From the introduction: K. Saito and D. Yoshii [Pub. Res. Inst. Math. Sci. 36, 385-421 (2000; Zbl 0987.17012)] introduced the simply-laced elliptic Lie algebra \(\widehat{\mathfrak g}(R)\) for the simply-laced elliptic root system \(R\), whose derived algebra \({\mathfrak g}(R) := [\widetilde{\mathfrak g}(R),\widetilde{\mathfrak g}(R)]\) is isomorphic to 2-toroidal Lie algebra which is the universal central extension of the tensor of a Lie algebra with the Laurent series of two variables. According to the work of Borcherds, they consider a Lie algebra \(V_Q/DV_Q\) as a quotient of the vertex algebra \(V_Q\) attached to an even lattice \(Q\), and constructed the elliptic Lie algebra \(\widetilde{\mathfrak g}(R)\) as a subalgebra of \(V_Q/DV_Q\). If \(R\) is a simply-laced finite or affine root system, then \({\mathfrak g}(R)\) is isomorphic to a finite or affine Kac-Moody algebra, respectively. The defining relations of the generators of \(\widetilde{\mathfrak g}(R)\) in terms of the elliptic diagram have been described in Saito-Yoshii (loc. cit.). In this article, we rewrite the defining relations more simply by considering the extended elliptic diagram consisting of all pairs of \(\alpha_i\), \(\alpha^*_i\) \((0\leq i\leq 1)\) for the sake of explicitness, although the results are already intrinsically in Saito-Yoshii (loc. cit.).

MSC:

17B65	Infinite-dimensional Lie (super)algebras
17B67	Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras

Keywords:

elliptic Cartan matrix; simply-laced elliptic Lie algebra; elliptic root system; generators; defining relations

Citations:

Zbl 0987.17012

References:

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[5]	Saito, K., and Yoshii, D.: Extended affine root systems IV (Simply-laced elliptic Lie algebras). Publ. RIMS, Kyoto Univ., 36 , 385-421 (2000). · Zbl 0987.17012