Skip Navigation

International Mathematics Research Notices (2007) Vol. 2007 : article ID rnm001, 18 pages, doi:10.1093/imrn/rnm001 published on May 24, 2007
This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Right arrow How to cite this article
Google Scholar
Right arrow Articles by Jeong, K.
Right arrow Articles by Shin, D.-U.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

Copyright © The Author 2007. Published by Oxford University Press.

Abstract Crystals for Quantum Generalized Kac-Moody Algebras

Kyeonghoon Jeong1, Seok-Jin Kang1, Masaki Kashiwara2 and Dong-Uy Shin3,

1 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, San 56-1 Shinrim-Dong, Kwanak-Ku, Seoul, 151–747, Korea
2 Research Institute for Mathematical Sciences, Kyoto University, Kitashirakawa, Sakyo-Ku, Kyoto 606–8502, Japan
3 Department of Mathematics, Chonnam National University, Yongbong-dong, Buk-gu, Kwangju, 500-757, Korea

Correspondence: Correspondence to be sent to: Dong-Uy Shin, Department of Mathematics, Chonnam National University, Yongbong-dong, Buk-gu, Kwangju, 500-757, Korea. e-mail: dushin{at}chonnam.ac.kr

In this article, we introduce the notion of abstract crystals for quantum generalized Kac-Moody algebras and study their fundamental properties. We then prove the crystal embedding theorem and give a characterization of the crystals B({infty}) and B({lambda}).

Communicated by Michio Jimbo



References

  1. Kashiwara M. On crystal bases of the q-analogue of universal enveloping algebras. Duke Mathematical Journal (1991) 63:465–516.[CrossRef][Web of Science]
  2. Lusztig G. Canonical bases arising from quantized enveloping algebras. Journal of the American Mathematical Society (1990) 3:447–498.[CrossRef]
  3. Borcherds R. E. Generalized Kac-Moody algebras. Journal of Algebra (1988) 115:501–512.[CrossRef][Web of Science]
  4. Borcherds R. E. Monstrous moonshine and monstrous Lie superalgebras. Inventiones Mathematicae (1992) 109:405–444.[CrossRef][Web of Science]
  5. Kang S.-J. Quantum deformations of generalized Kac-Moody algebras and their modules. Journal of Algebra (1995) 175:1041–1066.[CrossRef][Web of Science]
  6. Jeong K., Kang S.-J., Kashiwara M. Crystal Bases for quantum generalized Kac-Moody Algebras. Proceedings of the London Mathematical Society (2005) 90(3):395–438.[Abstract/Free Full Text]
  7. Kashiwara M., Schapira P. Categories and Sheaves (2006) Vol. 332. Berlin, Heidelberg: Springer-Verlag. Grundlehren der Mathematischen Wissenschaften.
  8. Shin D.-U. Polyhedral Realization of Crystal Bases for Generalized Kac-Moody Algebras. (2006) Preprint Math. QA/0610711.
  9. Shin D.-U. Polyhedral Realization of the Highest Weight Crystals for Generalized Kac-Moody Algebras. Transactions of the American Mathematical Society (forthcoming).

Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?



This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Right arrow How to cite this article
Google Scholar
Right arrow Articles by Jeong, K.
Right arrow Articles by Shin, D.-U.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?