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Elliptic Lie algebras and tubular algebras

Authors
  • Lin, Yanan
  • Peng, Liangang
Type
Published Article
Journal
Advances in Mathematics
Publication Date
Jan 01, 2004
Accepted Date
Sep 09, 2004
Volume
196
Issue
2
Pages
487–530
Identifiers
DOI: 10.1016/j.aim.2004.09.006
Source
Elsevier
Keywords
License
Unknown

Abstract

This article is to study relations 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, we prove that the elliptic Lie algebra of type D 4 ( 1 , 1 ) , E 6 ( 1 , 1 ) , E 7 ( 1 , 1 ) or E 8 ( 1 , 1 ) is isomorphic to the Ringel–Hall Lie algebra of the root category of the tubular algebra with the same type. As a by-product of our proof, we obtain a Chevalley basis of the elliptic Lie algebra following indecomposable objects of the root category of the corresponding tubular algebra. This can be viewed as an analogue of the Frenkel–Malkin–Vybornov theorem in which they described a Chevalley basis for each untwisted affine Kac–Moody Lie algebra by using indecomposable representations of the corresponding affine quiver.

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