Book Review
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MathSciNet review:
3855021
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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 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, DOI 10.1090/ulect/021
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 10.1090/jams/842
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 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 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 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 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 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 10.1016/0022-4049(94)90097-3
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 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 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 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 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 10.1142/S0217979292000876
Hans Wenzl, $C^*$ tensor categories from quantum groups, J. Amer. Math. Soc. 11 (1998), no. 2, 261–282. MR 1470857, DOI 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
References
- Proceedings of the International Congress of Mathematicians. Vol. I, II: Edited by I. Satake, Mathematical Society of Japan, Tokyo; Springer-Verlag, Tokyo, 1991. Held in Kyoto, August 21–29, 1990. 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 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 10.1090/jams/842
- 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 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 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 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 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 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
- 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 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 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. ${\mathcal {C}}$-categories, ${\mathcal {C}}$-functors and ${\mathcal {C}}$-morphisms, Publ. Math. Debrecen 24 (1977), no. 3-4, 351–361. MR 0498792
- 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 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 10.1007/s002080050120
- Vladimir G. Turaev, Modular categories and $3$-manifold invariants: Topological and quantum group methods in field theory and condensed matter physics, Internat. J. Modern Phys. B 6 (1992), no. 11-12, 1807–1824. MR 1186845, DOI 10.1142/S0217979292000876
- Hans Wenzl, $C^*$ tensor categories from quantum groups, J. Amer. Math. Soc. 11 (1998), no. 2, 261–282. MR 1470857, DOI 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
