Tubular algebras, Derived Categories and Elliptic Lie Algebras
Speaker:Prof. Yanan Lin
School of Mathematical Sciences
Xiamen University
Date & Time:15 Nov 2012 (Thursday) 11:00
Venue:JM15
Organized by:Department of Mathematics

Abstract

This talk is to show relationship between tubular algebras of Ringel and elliptic Lie algebras in the sense of Saito–Yoshii. Using the explicit structure of the derived categories of tubular algebras given by Happel–Ringel, L. Peng and reporter proved that the elliptic Lie algebra of type D_4^(1,1), E_q^(1,1), q=6,7,8, is isomorphic to the Ringel–Hall Lie algebra of the root category of the tubular algebra with the same type.

In order to describe singularities on surfaces and their deformations, K. Saito in introduced one kind of extended affine root systems. In particular, he intensively studied 2-extended affine root systems using Dynkin diagrams with markings (also called elliptic Dynkin diagrams). Various attempts have been made to construct Lie algebras whose nonisotropic roots form elliptic root systems. In 2000, K. Saito and D. Yoshii constructed certain Lie algebras by using the Borcherds lattice vertex, called them simply-laced elliptic Lie algebras and showed that they are isomorphic to the corresponding 2-toroidal algebras.

Given a field k and a quiver, one can define the path algebra over k which is an associative algebra such that all paths in the quiver form a k-basis and the multiplication is defined by composition of paths in a natural way. A relation over k associated to the quiver is a k-linear combination of some paths with length > 1 and with same starting vertex and same ending vertex. An (associative) algebra A is determined by a quiver with relations if it is the quotient algebra of the path algebra of this quiver modulo by the ideal generated by these relations. In this case, the quiver is called the ordinary quiver of this algebra. A representation of a quiver with relations is a collection of k-vector spaces attached to the vertices and linear transformations attached to the arrows such that the k-linear combination of the composition of the transformations corresponding to each relations is zero. Then each A-module naturally corresponds to a representation. And in this way, the A-module category is equivalent to the representation category of the quiver with relations. Tubular algebras are certain special class of associative algebras of global dimension 2 which can be determined by quivers with relations. Their module categories and derived categories are of wide interest, for example, Geigle and Lenzing found that the derived category of a tubular algebra is isomorphic to the derived category of coherent sheaves on a weighted projective line.

Inspired by the results Frobenius morphisms on algebras and derived categories in the sense of Deng and Du, Z. Chen and reporter realized some non-simply-laced elliptic Lie algebras.

Biography

Prof. Yanan Lin is currently the Dean and Professor of School of Mathematical Sciences of Xiamen University. Prof. Lin obtained his Ph.D. in Bielefeld University. He has received various awards, such as, the First Prize in Natural Sciences by the Ministry of Education (教育部高等學校自然科學獎一等獎), Distinguished Teacher in Fuijian Province (福建省高校教學名師), the Fourth National Distinguished Teacher Award (全國第四屆教學名師獎), and the First prize of Teaching Achievement Award in Fuijian and Xiamen Province (廈門大學教學成果一等獎/福建省教學成果一等獎). Prof. Lin’s publications include, Recollements of extension algebras, Tilting modulaes over path algebras of Dynkin type and comlete slice modules, Elliptic Lie algebras and tubular algebras etc.