Topological invariants from quantum group \(\mathcal {U}_{\xi }\mathfrak {sl}(2|1)\) at roots of unity

Article
  • 25 Downloads

Abstract

In this article we construct link invariants and 3-manifold invariants from the quantum group associated with the Lie superalgebra \(\mathfrak {sl}(2|1)\). The construction is based on nilpotent irreducible finite dimensional representations of quantum group \(\mathcal {U}_{\xi }\mathfrak {sl}(2|1)\) where \(\xi \) is a root of unity of odd order. These constructions use the notion of modified trace and relative \( G \)-modular category of previous authors.

Keywords

Lie superalgebra Quantum group Link invariant 3-manifold 

Mathematics Subject Classification

57M27 17B37 

Notes

Acknowledgements

I would like to thank B. Patureau-Mirand, my thesis advisor, who helped me with this work, and who gave me the motivation to study mathematics. I would like to thank the referee for his constructive remarks. I would also like to thank my professors and friends in the laboratory LMBA of the Université de Bretagne Sud.

References

  1. 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)CrossRefMATHGoogle Scholar
  2. 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(4), 1005–1053 (2014)CrossRefMATHGoogle Scholar
  3. 3.
    Gainutdinov, A.M., Runkel, I.: Projective objects and the modified trace in factorisable finite tensor categories. arXiv:1703.00150 (2017)
  4. 4.
    Geer, N., Kujawa, J., Patureau-Mirand, B.: Generalized trace and modified dimension functions on ribbon categories. Selecta Math. 17, 453–504 (2011)CrossRefMATHGoogle Scholar
  5. 5.
    Geer, N., Kujawa, J., Patureau-Mirand, B.: Ambidextrous objects and trace fuctions for nonsemisimple categories. Proc. Am. Math. Soc. 141, 2963–2978 (2013)CrossRefMATHGoogle Scholar
  6. 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)CrossRefMATHGoogle Scholar
  7. 7.
    Geer, N., Patureau-Mirand, B.: An invariant supertrace for the category of representations of lie superalgebras. Pacific J. Math 238, 331–348 (2008)CrossRefMATHGoogle Scholar
  8. 8.
    Geer, N., Patureau-Mirand, B., Turaev, V.: Modified quantum dimensions and re-normalized links invariants. Compos. Math. 145, 196–212 (2009)CrossRefMATHGoogle Scholar
  9. 9.
    Ha, N.P.: Topological unrolled quantum groups. Work in progressGoogle Scholar
  10. 10.
    Heckenberger I.: Nichols algebras of diagonal type and arithmetic root systems, Habilitationsarbeit, Leipzig (2005)Google Scholar
  11. 11.
    Kac, V.G.: Lie superalgebra. Adv. Math. 26, 8–96 (1977)CrossRefMATHGoogle Scholar
  12. 12.
    Khoroshkin, S.M., Tolstoy, V.N.: Universal \(\cal{R}\)-matrix for quantized (super)algebras. Commun. Math 141, 599–617 (1991)CrossRefMATHGoogle Scholar
  13. 13.
    Lentner, S., Nett, D.: New \(\cal{R}\)-matrices for small quantum groups. Algebras Represent. Theory 18(6), 1649–1673 (2015)CrossRefMATHGoogle Scholar
  14. 14.
    Patureau-Mirand, B.: Invariants Topologiques Quantiques Non Semi-simples. Universite de Bretagne Sud, Lorient (2012)Google Scholar
  15. 15.
    Turaev V.G.: Quantum Invariants of Knots and 3-manifolds. Studies in Mathematiques (1994)Google Scholar
  16. 16.
    Yamane, H.: Quantized Enveloping Algebras Associatied with Simple Lie Superalgebras and tHeir Universal \(\cal{R}\)-matrices. Publ. RIMS, Kyoto Univ, Kyoto (1994)MATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques de Bretagne AtlantiqueUniversité de Bretagne Sud, Centre de Recherche, Campus de TohannicVannesFrance

Personalised recommendations