On Triangulated Categories and Enveloping Algebras

By using the approach to Hall algebras arising in homologically finite triangulated categories in [9], we find an “almost” associative multiplication structure for indecomposable objects in a 2-period triangulated category. As an application, we give a new proof of the theorem of Peng and Xiao in [6] which provides a way of constructing all symmetrizable Kac-Moody Lie algebras from two periodic triangulated categories. Introduction Let U be the universal enveloping algebra of a simple Lie algebra of type A,D or E over Q. It is one of the main problems studied in the last twenty years to find interpretations for the algebras U in terms of representations of oriented Coxeter graphs [4]. Such interpretations embody the principle of categorification. The work of Gabriel [1] strongly suggested the possibility of such an interpretation. He showed there exist a bijection between the isomorphism classes of all indecomposable modules over a hereditary algebra of Dynkin type and the positive roots of the corresponding semisimple Lie algebra. It has been explicitly obtained by Ringel [7] in the case of the positive part of U through the Hall algebra approach. Parallely, Lusztig has shown that the negative part of U can be geometrically realized using constructible functions on the space of representations of a preprojective algebra [4]. One may naturally consider to recover the whole Lie algebra and the whole (quantized) enveloping algebra [7]. Lusztig showed that an arbitrarily large finite-dimensional quotient of U can be realized in terms of constructible functions on the triple variety [4]. A different construction was given by Nakajima [5]. On the other hand, Peng and Xiao define a Lie multiplication between isomorphism classes of indecomposable objects in a 2-period triangulated category (i.e. the translation T satisfies T 2 = 1) over a finite field k with the cardinality q [6]. It induces a Lie algebra over Z/(q− 1). However, It is unknown which associative multiplication induces the Lie multiplication over Z/(q − 1). Recently, Toën gave a multiplication formula which defines an associative algebra (called the derived Hall algebra) corresponding to a dg category [8]. In [9], we extended to prove that Toën’s formula can be applied to define an associative algebra for any triangulated category with some homological finiteness conditions. Date: October 28, 2007. 2000 Mathematics Subject Classification. Primary 16G20, 17B67; Secondary 17B35, 18E30.