Topological invariants from quantum group \(\mathcal {U}_{\xi }\mathfrak {sl}(2|1)\) at roots of unity.(English)Zbl 1398.57022
The paper under review constructs link invariants and \(3\)-manifold invariants using nilpotent irreducible finite dimensional representations of the quantum group \(U_{\xi}\mathfrak{sl}(2|1)\), for \(\xi\) an odd root of unity. The construction is based upon the notions of modified dimension, modified trace and relative \(G\)-modular category.
Reviewer: Volodymyr Mazorchuk (Uppsala)
MSC:
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
Cite
References:
| [1] | Abdesselam, B; Arnaudon, D; Bauer, M, Centre and representations of \(\cal{U}_q\mathfrak{sl}(2|1)\) at roots of unity, J. Phys. A Math. Gen, 30, 867-880, (1997) ·Zbl 0992.17009 ·doi:10.1088/0305-4470/30/3/012 |
| [2] | Costantino, F; Geer, N; Patureau-Mirand, B, Quantum invariants of \(3\)-manifolds via link surgery presentations and non-semi-simple categories, J. Topol., 7, 1005-1053, (2014) ·Zbl 1320.57016 ·doi:10.1112/jtopol/jtu006 |
| [3] | Gainutdinov, A.M., Runkel, I.: Projective objects and the modified trace in factorisable finite tensor categories. arXiv:1703.00150 (2017) ·Zbl 1407.16031 |
| [4] | Geer, N; Kujawa, J; Patureau-Mirand, B, Generalized trace and modified dimension functions on ribbon categories, Selecta Math., 17, 453-504, (2011) ·Zbl 1248.18006 ·doi:10.1007/s00029-010-0046-7 |
| [5] | Geer, N; Kujawa, J; Patureau-Mirand, B, Ambidextrous objects and trace fuctions for nonsemisimple categories, Proc. Am. Math. Soc., 141, 2963-2978, (2013) ·Zbl 1280.18005 ·doi:10.1090/S0002-9939-2013-11563-7 |
| [6] | Geer, N; Patureau-Mirand, B, Multivariable link invariants arising from \(\mathfrak{sl}(2|1)\) and the Alexander polynomial, J. Pure Appl. Algebra, 210, 283-298, (2007) ·Zbl 1121.57005 ·doi:10.1016/j.jpaa.2006.09.015 |
| [7] | Geer, N; Patureau-Mirand, B, An invariant supertrace for the category of representations of Lie superalgebras, Pacific J. Math, 238, 331-348, (2008) ·Zbl 1210.17010 ·doi:10.2140/pjm.2008.238.331 |
| [8] | Geer, N; Patureau-Mirand, B; Turaev, V, Modified quantum dimensions and re-normalized links invariants, Compos. Math., 145, 196-212, (2009) ·Zbl 1160.81022 ·doi:10.1112/S0010437X08003795 |
| [9] | Ha, N.P.: Topological unrolled quantum groups. Work in progress |
| [10] | Heckenberger I.: Nichols algebras of diagonal type and arithmetic root systems, Habilitationsarbeit, Leipzig (2005) ·Zbl 1138.17307 |
| [11] | Kac, VG, Lie superalgebra, Adv. Math., 26, 8-96, (1977) ·Zbl 0366.17012 ·doi:10.1016/0001-8708(77)90017-2 |
| [12] | Khoroshkin, SM; Tolstoy, VN, Universal \(\cal{R}\)-matrix for quantized (super)algebras, Commun. Math, 141, 599-617, (1991) ·Zbl 0744.17015 ·doi:10.1007/BF02102819 |
| [13] | Lentner, S; Nett, D, New \(\cal{R}\)-matrices for small quantum groups, Algebras Represent. Theory, 18, 1649-1673, (2015) ·Zbl 1345.17012 ·doi:10.1007/s10468-015-9555-6 |
| [14] | Patureau-Mirand, B.: Invariants Topologiques Quantiques Non Semi-simples. Universite de Bretagne Sud, Lorient (2012) |
| [15] | Turaev V.G.: Quantum Invariants of Knots and 3-manifolds. Studies in Mathematiques (1994) ·Zbl 0812.57003 |
| [16] | Yamane, H.: Quantized Enveloping Algebras Associatied with Simple Lie Superalgebras and tHeir Universal \(\cal{R}\)-matrices. Publ. RIMS, Kyoto Univ, Kyoto (1994) ·Zbl 0821.17005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
