Research ArticleNo Access

Relationship of the Hennings and Witten–Reshetikhan–Turaev invariants for higher rank quantum groups

Winston Cheong

Math Department, Kansas State University, 138 Cardwell Hall, 1228 N M.L.K. Jr. Dr Manhattan, KS 66506-2602, USA

E-mail Address: winstonc@ksu.edu

Search for more papers by this author

Alexander Doser

Math Department, Northwestern University, Lunt Hall, 2033 Sheridan Rd, Evanston, IL 60208, USA

E-mail Address: alexanderdoser2023@u.northwestern.edu

Search for more papers by this author

McKinley Gray

Math Department, University of Illinois Chicago, Earhart Hall, 851 S Morgan St, Chicago, IL 60607, USA

E-mail Address: mgray6@uic.edu

Search for more papers by this author

, and

Stephen F. Sawin

https://orcid.org/0000-0001-9218-4780

Math Department, Fairfield University, 1073 N Benson Rd, Fairfield, CT 06824, USA

E-mail Address: ssawin@fairfield.edu

Corresponding author.

Publisher’s Note: The revised date has been corrected due to production error in the earlier published version.

Search for more papers by this author

https://doi.org/10.1142/S0218216523500487Cited by:0 (Source: Crossref)

Abstract

The Hennings invariant for the small quantum group associated to an arbitrary simple Lie algebra at a root of unity is shown to agree with the Witten–Reshetikhin–Turaev (WRT) three-manifold invariant arising from Chern–Simons field theory for the same Lie algebra and the same root of unity on all integer homology three-spheres, at roots of unity where both are defined. This partially generalizes the work of Chen et al. [On the relation between the WRT invariant and the Hennings invariant, Math. Proc. Cambridge Philos. Soc. 146(1) (2009) 151–163; Three-manifold invariants associated with restricted quantum groups, Math. Z. 272(3–4) (2012) 987–999] which relates the Hennings and WRT invariants for $SL (2)$ and $SO (3)$ for arbitrary rational homology three-spheres.

Keywords:

AMSC: 17B37, 57M27

References

1. The Stack Project Authors, https://stacks.math.columbia.edu/tag/0AMQ. Google Scholar
2. Q. Chen, S. Kuppum and P. Srinivasan, On the relation between the WRT invariant and the Hennings invariant, Math. Proc. Cambridge Philos. Soc. 146(1) (2009) 151–163. Crossref, Web of Science, Google Scholar
3. Q. Chen, C.-C. Yu and Y. Zhang, Three-manifold invariants associated with restricted quantum groups, Math. Z. 272(3–4) (2012) 987–999. Crossref, Web of Science, Google Scholar
4. R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Commun. Math. Phys. 129 (1990) 393–429. Crossref, Web of Science, Google Scholar
5. K. Habiro, Cyclotomic completions of polynomial rings, Publ. Res. Inst. Math. Sci. 40(4) (2004) 1127–1146. Crossref, Web of Science, Google Scholar
6. K. Habiro, Bottom tangles and universal invariants, Algebra Geom. Topol. 6 (2006) 1113–1214. Crossref, Web of Science, Google Scholar
7. K. Habiro and T. T. Q. Le, Unified quantum invariants for integral homology spheres associated with simple Lie algebras, Geom. Topol. 20 (2016) 2687–2835. https://doi.org/10.2140/gt.2016.20.2687, arXiv:1503:03549v1. Crossref, Web of Science, Google Scholar
8. M. A. Hennings, Invariants of links and 3-manifolds obtained from Hopf algebras, J. London Math. Soc. 2(54) (1996) 594–624. Crossref, Web of Science, Google Scholar
9. L. H. Kauffman, Gauss codes, quantum groups and ribbon Hopf algebras, Rev. Math. Phys. 5(4) (1993) 735–773. Link, Web of Science, Google Scholar
10. R. J. Lawrence, A universal link invariant using quantum groups, in Differential Geometric Methods in Theoretical Physics (Chester, 1988) (World Scientific Publication Teaneck, New Jersey, 1989), pp. 55–63. Google Scholar
11. G. Lusztig, Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra, J. Amer. Math. Soc. 3(1) (1990) 257–296. Google Scholar
12. G. Lusztig, Introduction to Quantum Groups, Progress in Mathematics, Vol. 110 (Birkhäuser, Boston Inc., Boston, MA, 1993). Google Scholar
13. T. Ohtsuki, Invariants of $3$ -manifolds derived from universal invariants of framed links, Math. Proc. Cambridge Philos. Soc. 117(2) (1995) 259–273. Crossref, Web of Science, Google Scholar
14. N. Reshetikhin and V. G. Turaev, Invariants of $3$ -manifolds via link polynomials and quantum groups, Invent. Math. 103(3) (1991) 547–597. Crossref, Web of Science, Google Scholar
15. N. Yu. Reshetikhin, Quasitriangular Hopf algebras and invariants of tangles, Leningrad. Math. J. 1 (1990) 491–513. Google Scholar
16. S. Sawin, Jones–Witten invariants for nonsimply connected Lie groups and the geometry of the Weyl alcove, Adv. Math. 165(1) (2002) 1–34, arXiv:math.QA/9905010. Crossref, Web of Science, Google Scholar
17. S. Sawin, Quantum groups at roots of unity and modularity, J. Knot Theory Ramifications 15(10) (2006) 1245–1277. Link, Web of Science, Google Scholar
18. M. Sweedler, Hopf Algebras (W. A. Benjamin, Inc., New York, 1969). Google Scholar
19. V. G. Turaev, Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics, Vol. 18 (Walter de Gruyter & Co., Berlin, 1994). Crossref, Google Scholar
20. E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121(3) (1989) 351–399. Crossref, Web of Science, Google Scholar