A parameterization of the canonical bases of affine modified quantized enveloping algebras

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• 作者：Jie Xiao ; Minghui Zhao
• 关键词：Ringel ; Hall algebras ; Root categories ; Modified quantized enveloping algebras ; Canonical bases ; 16G20 ; 17B37
• 刊名：Chinese Annals of Mathematics - Series B
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：37
• 期：2
• 页码：235-258
• 全文大小：299 KB
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• 作者单位：Jie Xiao (1)
  Minghui Zhao (2)
  
  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China
  2. College of Science, Beijing Forestry University, Beijing, 100083, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Applications of Mathematics
  Chinese Library of Science
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1860-6261

文摘

For a symmetrizable Kac-Moody Lie algebra g, Lusztig introduced the corresponding modified quantized enveloping algebra \(\dot U\) and its canonical basis \(\dot B\) given by Lusztig in 1992. In this paper, in the case that g is a symmetric Kac-Moody Lie algebra of finite or affine type, the authors define a set \(\tilde M\) which depends only on the root category R and prove that there is a bijection between \(\tilde M\) and \(\dot B\), where R is the T 2-orbit category of the bounded derived category of the corresponding Dynkin or tame quiver. The method in this paper is based on a result of Lin, Xiao and Zhang in 2011, which gives a PBW-type basis of U+. Keywords Ringel-Hall algebras Root categories Modified quantized enveloping algebras Canonical bases