Elliptic Lie algebras and tubular algebras

Thomas Brüstle Fakultät für Mathematik,Universität Bielefeld

       e-mail: bruestle@mathematik.uni-bielefeld.de

It is well-known that the positive part 𝔤^{+} of a Kac-Moody algebra 𝔤 is isomorphic to Ringel’s composition algebra 𝔠(Q), defined by a quiver Q with the same root system as 𝔤. Using vertex operator algebras, one can construct a Lie algebra 𝔤(L)for every generalized root lattice L. If L is elliptic and simply laced, Saito and Yoshii [2] gave a description of the algebras 𝔤(L) by generators and relations. Our aim is to compare in some cases the positive part 𝔤^{+}(L) with the composition algebra 𝔠(A) of some tubular algebra A = kQ/I. Here the quiver Q encodes the Serre relations of 𝔤(L) and the ideal I of the path algebra kQ encodes the additional relations that occur in [2]. These additional relations (which Saito and Yoshii consider to be strange, since they involve more than two simples) are very natural from the point of view of finite-dimensional algebras.

To explain our results in more detail, consider a tubular algebra A = kQ/I. The Euler form q_{A} of A is non-negative of Dynkin type 𝔻_{4}, 𝔼_{6}, 𝔼_{7} or 𝔼_{8} and has a two-dimensional radical. Thus the root lattice L of the quadratic form q_{A} is elliptic and simply laced. Instead of Ringel’s original approach by finite fields, we consider Lusztig’s version of the Hall algebra H(A) using constructible maps with convolution product. The composition algebra 𝔠(A) is the Lie subalgebra of H(A) generated by the characteristic maps of the simple A−modules. Collecting constructible maps on modules with the same dimension vector, we obtain an ℕ^{Q_{0}}−grading on 𝔠(A). The lattice L induces an ℤ^{Q_{0}}−grading on 𝔤(L), and we denote by 𝔤^{+}(L) the Lie subalgebra formed by the elements with degree in ℕ^{Q_{0}}. Then our main result is the following:

Theorem. Let A be a tubular algebra with root lattice L as above. Then there is an epimorphism 𝔤^{+}(L) → 𝔠(A) of ℕ^{Q_{0}}−graded Lie algebras.

We believe that the epimorphism in the theorem is in fact an isomorphism, and we suggest to apply the methods introduced by Frenkel, Malkin and Vybornov in [1]. Let us point out that Y. Lin and L. Peng announced the corresponding result for the whole Lie algebra g(L) and the composition algebra defined over D^{b}(A).

[1] Frenkel, Malkin, Vybornov, Tame quivers, Selecta Math. 7 (2001), no. 1, 1–56.

[2] K. Saito, D. Yoshii, Extended affine root system. IV. Simply-laced elliptic Lie algebras, Publ. Res. Inst. Math. Sci. 36 (2000), no. 3, 385–421.