Book Review
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MathSciNet review:
2731654
Full text of review:
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Book Information:
Authors:
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.,
ISBN 978-0-8218-4186-0,
US $119 hardcover
A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of $\textrm {GL}_n$, Duke Math. J. 61 (1990), no. 2, 655–677. MR 1074310, DOI 10.1215/S0012-7094-90-06124-1
Alexandre Beĭlinson and Joseph Bernstein, Localisation de $g$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15–18 (French, with English summary). MR 610137
J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), no. 3, 387–410. MR 632980, DOI 10.1007/BF01389272
Peter Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71–103; correction, ibid. 6 (1972), 309 (German, with English summary). MR 332887, DOI 10.1007/BF01298413
Nagayoshi Iwahori, On the structure of a Hecke ring of a Chevalley group over a finite field, J. Fac. Sci. Univ. Tokyo Sect. I 10 (1964), 215–236 (1964). MR 165016
M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
David Kazhdan and George Lusztig, Schubert varieties and Poincaré duality, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 185–203. MR 573434
Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309–347. MR 2525917, DOI 10.1090/S1088-4165-09-00346-X
G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series, vol. 18, American Mathematical Society, Providence, RI, 2003. MR 1974442, DOI 10.1090/crmm/018
Claus Michael Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583–591. MR 1062796, DOI 10.1007/BF01231516
R. Rouquier, $2$-Kac-Moody algebras; arXiv:0812.5023.
M. Varagnolo and E. Vasserot, Canonical bases and Khovanov-Lauda algebras, arXiv: 0901.3992.
References
- A. Beilinson, G. Lusztig and R. MacPherson, A geometric setting for the quantum deformation of $\textrm {GL}_n$, Duke Math. J. 61 (1990), 655–677. MR 1074310 (91m:17012)
- 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)
- J.-L. Brylinksi and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), 387–410. MR 632980 (83e:22020)
- P. Gabriel, Unzerlegbare Darstellungen I, Manuscr. Math. 6 (1972), 71–103. MR 0332887 (48:11212)
- 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)
- M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465–516. MR 1115118 (93b:17045)
- D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184. MR 560412 (81j:20066)
- D. Kazhdan and G. Lusztig, Schubert varieties and Poincaré duality, Proc. Symp. Pure Math. 36 (1980), 185–203. MR 573434 (84g:14054)
- M. Khovanov and A. Lauda, A diagrammatic approach to categorification of quantum groups. I., Represent. Theory 13 (2009), 309–347. MR 2525917
- G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447–498. MR 1035415 (90m:17023)
- G. Lusztig, Introduction to Quantum Groups, Progress in Mathematics, 110. Birkhäuser, Boston, 1993. MR 1227098 (94m:17016)
- G. Lusztig, Hecke Algebras with Unequal Parameters, CRM Monograph Series, 18. American Mathematical Society, Providence, RI, 2003. MR 1974442 (2004k:20011)
- C. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), 583–591. MR 1062796 (91i:16024)
- R. Rouquier, $2$-Kac-Moody algebras; arXiv:0812.5023.
- M. Varagnolo and E. Vasserot, Canonical bases and Khovanov-Lauda algebras, arXiv: 0901.3992.
Review Information:
Reviewer:
Jonathan Brundan
Affiliation:
University of Oregon
Email:
brundan@uoregon.edu
Journal:
Bull. Amer. Math. Soc.
48 (2011), 107-114
DOI:
https://doi.org/10.1090/S0273-0979-10-01293-0
Published electronically:
February 25, 2010
Additional Notes:
The reviewer was supported in part by NSF Grant DMS-0635607.
Review copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
