Representations of Quantum Algebras and Combinatorics of Young Tableaux
The main part of the book is devoted to a study of representations of quantum algebras. The author uses the quantum deformation of the Kac-Moody Lie algebra Ak(1) as the main working example. The first three chapters contain a brief review of concepts needed for a description of Ak(1) by generators and relations. The next three chapters contain a discussion of crystal bases for integrable representations of Ak(1) and a description of behavior of crystal bases under tensor products. The existence of crystal bases is proved in chapters 7, 8 and 9. The proof is based on relations between crystal bases and the Lusztig canonical bases. The next two chapters contain a discussion of the third possible approach to crystal bases, which uses a combinatorial construction based on (multi)-Young diagrams, due to Misra and Miwa. The final part of the book contains the proof of a conjecture (due to Leclerc, Lascoux and Thibon) concerning the representation theory of cyclotomic Hecke algebras of classical type. The book offers a nice introduction to combinatorics related representation theories and their applications.