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ISSN 1088-9485(e) ISSN 0273-0979(p)

Book Review

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Book Information

Author(s): Bangming Deng, Jie Du, Brian Parshall and Jianpan Wang
Title: Finite dimensional algebras and quantum groups
Additional book information: Mathematical Surveys and Monographs, 150, American Mathematical Society, Providence, RI, 2008, xxvi+759 pp., US $119 hardcover, ISBN 978-0-8218-4186-0

References:

1.: A. Beilinson, G. Lusztig and R. MacPherson, A geometric setting for the quantum deformation of ${\rm GL}_n$ , Duke Math. J. 61 (1990), 655-677. MR 1074310 (91m:17012)
2.: A. Beilinson and J. Bernstein, Localisation de $\mathfrak{g}$ -modules, C. R. Acad. Sci. Paris Ser. I Math. 292 (1981), 15-18. MR 610137 (82k:14015)
3.: J.-L. Brylinksi and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), 387-410. MR 632980 (83e:22020)
4.: P. Gabriel, Unzerlegbare Darstellungen I, Manuscr. Math. 6 (1972), 71-103. MR 0332887 (48:11212)
5.: N. Iwahori, On the structure of a Hecke ring of a Chevalley group over a finite field, J. Fac. Sci. Univ. Tokyo 10 (1964), 215-236. MR 0165016 (29:2307)
6.: M. Kashiwara, On crystal bases of the -analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465-516. MR 1115118 (93b:17045)
7.: D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 560412 (81j:20066)
8.: D. Kazhdan and G. Lusztig, Schubert varieties and Poincaré duality, Proc. Symp. Pure Math. 36 (1980), 185-203. MR 573434 (84g:14054)
9.: M. Khovanov and A. Lauda, A diagrammatic approach to categorification of quantum groups. I., Represent. Theory 13 (2009), 309-347. MR 2525917
10.: G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447-498. MR 1035415 (90m:17023)
11.: G. Lusztig, Introduction to Quantum Groups, Progress in Mathematics, 110. Birkhäuser, Boston, 1993. MR 1227098 (94m:17016)
12.: G. Lusztig, Hecke Algebras with Unequal Parameters, CRM Monograph Series, 18. American Mathematical Society, Providence, RI, 2003. MR 1974442 (2004k:20011)
13.: C. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), 583-591. MR 1062796 (91i:16024)
14.: R. Rouquier, -Kac-Moody algebras; arXiv:0812.5023.
15.: M. Varagnolo and E. Vasserot, Canonical bases and Khovanov-Lauda algebras, arXiv: 0901.3992.

Additional Information:

Reviewer(s):
Jonathan Brundan
Affiliation: University of Oregon
Email: brundan@uoregon.edu

Review Information:
Journal: Bull. Amer. Math. Soc. 48 (2011), 107-114.

MSC (2010): Primary 17B37, 81R50; Secondary 05E10
DOI: 10.1090/S0273-0979-10-01293-0
PII: S 0273-0979(10)01293-0
Posted: February 25, 2010
Additional notes: The reviewer was supported in part by NSF Grant DMS-0635607.
Copyright of article: Copyright 2010, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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