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Grothendieck groups of the Khovanov–Kuperberg algebras

Louis-Hadrien Robert

Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany

E-mail Address: louis-hadrien.robert@uni-hamburg.de

https://doi.org/10.1142/S0218216515500704Cited by:0 (Source: Crossref)

Abstract

The aim of this paper is to take benefit of the foam nature of the Khovanov–Kuperberg algebras to compute the Grothendieck groups of their categories of finite-dimensional projective modules. The computation relies on the Hattori–Stallings trace and some geometrical properties of foams in the solid torus.

Keywords:

AMSC: 19A49, 16W50, 57R56, 57M25

References

1. H. Bass, Euler characteristics and characters of discrete groups, Invent. Math. 35 (1976) 155–196. Crossref, Web of Science, Google Scholar
2. C. Blanchet, N. Habegger, G. Masbaum and P. Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology 34(4) (1995) 883–927. Crossref, Web of Science, Google Scholar
3. B. Gornik, Note on Khovanov link cohomology, preprint (2004), arXiv: math/0402266. Google Scholar
4. F. Jaeger, A new invariant of plane bipartite cubic graphs, Discrete Math. 101(1–3) (1992) 149–164. Special volume to mark the centennial of Julius Petersen’s “Die Theorie der regulären Graphs”, Part II. Google Scholar
5. M. Khovanov, A functor-valued invariant of tangles, Algebr. Geom. Topol. 2 (2002) 665–741 (electronic). Crossref, Google Scholar
6. M. Khovanov, sl(3) link homology, Algebr. Geom. Topol. 4 (2004) 1045–1081. Crossref, Google Scholar
7. M. Khovanov and G. Kuperberg, Correction to: “Web bases for sl(3) are not dual canonical”, Pacific J. Math. 188(2) (1999) 399. Google Scholar
8. J. Kock, Frobenius Algebras and 2D Topological Quantum Field Theories, London Mathematical Society Student Texts, Vol. 59 (Cambridge University Press, Cambridge, 2004). Google Scholar
9. G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180(1) (1996) 109–151. Crossref, Web of Science, Google Scholar
10. L. Lewark, Les homologies de Khovanov–Rozansky, toiles nouées pondérées et le genre lisse, Ph.D. thesis, Univeristé Paris 7 — Denis Diderot (2013). Google Scholar
11. M. Lorenz, Representations of finite-dimensional Hopf algebras, J. Algebra 188(2) (1997) 476–505. Crossref, Web of Science, Google Scholar
12. M. Mackaay, W. Pan and D. Tubbenhauer, The ${𝔰 𝔩}_{3}$ -web algebra, Math. Z. 277(1–2) (2014) 401–479. Crossref, Web of Science, Google Scholar
13. M. Mackaay and P. Vaz, The universal ${sl}_{3}$ -link homology, Algebr. Geom. Topol. 7 (2007) 1135–1169. Crossref, Web of Science, Google Scholar
14. S. Morrison and A. Nieh, On Khovanov’s cobordism theory for ${𝔰 𝔲}_{3}$ knot homology, J. Knot Theory Ramifications 17(9) (2008) 1121–1173. Link, Web of Science, Google Scholar
15. H. Murakami, T. Ohtsuki and S. Yamada, Homfly polynomial via an invariant of colored plane graphs, Enseign. Math. (2) 44 (3– 4) (1998) 325–360. Google Scholar
16. C. Năstăsescu and F. Van Oystaeyen, Methods of Graded Rings, Lecture Notes in Mathematics, Vol. 1836 (Springer-Verlag, Berlin, 2004). Crossref, Google Scholar
17. L.-H. Robert, A characterisation of indecomposable web-modules over Khovanov–Kuperberg algebras, preprint (2013), arXiv:1309.2793. Google Scholar
18. L.-H. Robert, A large family of indecomposable projective modules for the Khovanov–Kuperberg algebra of $s l_{3}$ -webs, J. Knot Theory Ramifications 22(11) (2013), Article ID:1350062, 25pp. Link, Web of Science, Google Scholar
19. L.-H. Robert, Sur l’homologie $s l_{3}$ des enchevêtrements; algèbre de Khovanov–Kuperberg, Ph.D. thesis, Université Paris 7 — Denis Diderot (2013). Google Scholar
20. G. Sjödin, On filtered modules and their associated graded modules, Math. Scand. 33 (1973) 229–249. Crossref, Google Scholar
21. N. Yu, Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127(1) (1990) 1–26. Crossref, Web of Science, Google Scholar