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Crystal Bases for Quantum Generalized Kac–Moody Algebras
Author(s) -
Jeong Kyeonghoon,
Kang SeokJin,
Kashiwara Masaki
Publication year - 2005
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611504015023
Subject(s) - subalgebra , tensor product , mathematics , pure mathematics , crystal (programming language) , basis (linear algebra) , tensor (intrinsic definition) , quantum , algebra over a field , product (mathematics) , subject (documents) , construct (python library) , mathematics subject classification , quantum mechanics , geometry , computer science , physics , library science , programming language
In this paper, we develop the crystal basis theory for quantum generalized Kac–Moody algebras. For a quantum generalized Kac–Moody algebra U q (g), we first introduce the category O int of U q (g)‐modules and prove its semisimplicity. Next, we define the notion of crystal bases for U q (g)‐modules in the category O int and for the subalgebra U q − ( g ) . We then prove the tensor product rule and the existence theorem for crystal bases. Finally, we construct the global bases for U q (g)‐modules in the category O int and for the subalgebra U q − ( g ) . 2000 Mathematics Subject Classification 17B37, 17B67.