Book Review
The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.
MathSciNet review: 3855021
Full text of review: PDF This review is available free of charge.
Book Information:
Authors: Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik
Title: Tensor categories
Additional book information: Mathematical Surveys and Monographs, Vol. 205, American Mathematical Society, Providence, RI, 2015, xvi+343 pp., ISBN 978-1-4704-2024-6, US$65.00
- Proceedings of the International Congress of Mathematicians. Vol. I, II, Mathematical Society of Japan, Tokyo; Springer-Verlag, Tokyo, 1991. Held in Kyoto, August 21–29, 1990; Edited by Ichirô Satake. MR 1159197
- John C. Baez and James Dolan, Categorification, Higher category theory (Evanston, IL, 1997) Contemp. Math., vol. 230, Amer. Math. Soc., Providence, RI, 1998, pp. 1–36. MR 1664990, DOI https://doi.org/10.1090/conm/230/03336
- Bojko Bakalov and Alexander Kirillov Jr., Lectures on tensor categories and modular functors, University Lecture Series, vol. 21, American Mathematical Society, Providence, RI, 2001. MR 1797619
- B. Bartlett, C. L. Douglas, C. J. Schommer-Pries, and J. Vicary, Modular categories as representations of the $3$-dimensional bordism $2$-category, ArXiv e-prints (2015).
- Paul Bruillard, Siu-Hung Ng, Eric C. Rowell, and Zhenghan Wang, Rank-finiteness for modular categories, J. Amer. Math. Soc. 29 (2016), no. 3, 857–881. MR 3486174, DOI https://doi.org/10.1090/S0894-0347-2015-00842-6
- Pavel Etingof and Shlomo Gelaki, Some properties of finite-dimensional semisimple Hopf algebras, Math. Res. Lett. 5 (1998), no. 1-2, 191–197. MR 1617921, DOI https://doi.org/10.4310/MRL.1998.v5.n2.a5
- Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik, On fusion categories, Ann. of Math. (2) 162 (2005), no. 2, 581–642. MR 2183279, DOI https://doi.org/10.4007/annals.2005.162.581
- V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. MR 908150, DOI https://doi.org/10.2307/1971403
- Vaughan F. R. Jones, Scott Morrison, and Noah Snyder, The classification of subfactors of index at most 5, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 2, 277–327. MR 3166042, DOI https://doi.org/10.1090/S0273-0979-2013-01442-3
- André Joyal and Ross Street, Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20–78. MR 1250465, DOI https://doi.org/10.1006/aima.1993.1055
- M. M. Kapranov and V. A. Voevodsky, $2$-categories and Zamolodchikov tetrahedra equations, Algebraic groups and their generalizations: quantum and infinite-dimensional methods (University Park, PA, 1991) Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 177–259. MR 1278735, DOI https://doi.org/10.1016/0022-4049%2894%2990097-3
- Alexander Kirillov Jr. and Viktor Ostrik, On a $q$-analogue of the McKay correspondence and the ADE classification of $\mathfrak {sl}_2$ conformal field theories, Adv. Math. 171 (2002), no. 2, 183–227. MR 1936496, DOI https://doi.org/10.1006/aima.2002.2072
- S. Mac Lane, The PNAS way back then, Proceedings of the National Academy of Sciences 94 (1997), no. 12, 5983–5985.
- Gregory Moore and Nathan Seiberg, Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989), no. 2, 177–254. MR 1002038
- Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, Non-abelian anyons and topological quantum computation, Rev. Modern Phys. 80 (2008), no. 3, 1083–1159. MR 2443722, DOI https://doi.org/10.1103/RevModPhys.80.1083
- Adrian Ocneanu, Quantized groups, string algebras and Galois theory for algebras, Operator algebras and applications, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 136, Cambridge Univ. Press, Cambridge, 1988, pp. 119–172. MR 996454
- B. Pareigis, Non-additive ring and module theory. II. ${\cal C}$-categories, ${\cal C}$-functors and ${\cal C}$-morphisms, Publ. Math. Debrecen 24 (1977), no. 3-4, 351–361. MR 498792
- N. Reshetikhin and V. G. Turaev, Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547–597. MR 1091619, DOI https://doi.org/10.1007/BF01239527
- Vladimir Turaev and Hans Wenzl, Semisimple and modular categories from link invariants, Math. Ann. 309 (1997), no. 3, 411–461. MR 1474200, DOI https://doi.org/10.1007/s002080050120
- Vladimir G. Turaev, Modular categories and $3$-manifold invariants, Internat. J. Modern Phys. B 6 (1992), no. 11-12, 1807–1824. Topological and quantum group methods in field theory and condensed matter physics. MR 1186845, DOI https://doi.org/10.1142/S0217979292000876
- Hans Wenzl, $C^*$ tensor categories from quantum groups, J. Amer. Math. Soc. 11 (1998), no. 2, 261–282. MR 1470857, DOI https://doi.org/10.1090/S0894-0347-98-00253-7
- Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772
Review Information:
Reviewer: Eric C. Rowell
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: rowell@math.tamu.edu
Journal: Bull. Amer. Math. Soc. 55 (2018), 545-551
DOI: https://doi.org/10.1090/bull/1632
Published electronically: May 23, 2018
Review copyright: © Copyright 2018 American Mathematical Society