Source: J. Gen. Lie Theory Appl. Volume 5
(2011), Article ID
G100202, 9 pages.
Foams are surfaces with branch lines at which three sheets merge. They have been
used in the categorification of $\mathrm{sl}(3)$ quantum knot invariants and
also in physics. The $2D$-TQFT of surfaces, on the other hand, is classified by
means of commutative Frobenius algebras, where saddle points correspond to
multiplication and comultiplication. In this paper, we explore algebraic
operations that branch lines derive under TQFT. In particular, we investigate
Lie bracket and bialgebra structures. Relations to the original Frobenius
algebra structures are discussed both algebraically and diagrammatically.
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References
M. Asaeda and C. Frohman, A note on the Bar-Natan skein module, Internat. J. Math., 18 (2007), pp. 1225–1243.
D. Bar-Natan, Khovanov's homology for tangles and cobordisms, Geom. Topol., 9 (2005), pp. 1443–1499.
J. S. Carter, A. S. Crans, M. Elhamdadi, E. Karadayi, and M. Saito, Cohomology of Frobenius algebras and the Yang-Baxter equation, Commun. Contemp. Math., 10 (2008), pp. 791–814.
J. S. Carter, D. E. Flath, and M. Saito, The Classical and Quantum 6$j$-Symbols, vol. 43 of Mathematical Notes, Princeton University Press, Princeton, NJ, 1995.
S. Chung, M. Fukuma, and A. Shapere, Structure of topological lattice field theories in three dimensions, Internat. J. Modern Phys. A, 9 (1994), pp. 1305–1360.
U. Kaiser, Frobenius algebras and skein modules of surfaces in 3-manifolds, in Algebraic Topology–-Old and New, vol. 85 of Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 2009, pp. 59–81.
L. H. Kauffman, M. Saito, and M. C. Sullivan, Quantum invariants of templates, J. Knot Theory Ramifications, 12 (2003), pp. 653–681.
M. Khovanov, $\rsl(3)$ link homology, Algebr. Geom. Topol., 4 (2004), pp. 1045–1081.
–––, Link homology and Frobenius extensions, Fund. Math., 190 (2006), pp. 179–190.
J. Kock, Frobenius algebras and 2D topological quantum field theories, vol. 59 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 2004.
G. Kuperberg, Spiders for rank $2$ Lie algebras, Comm. Math. Phys., 180 (1996), pp. 109–151.
M. Mackaay and P. Vaz, The universal ${\rm sl}\sb 3$-link homology, Algebr. Geom. Topol., 7 (2007), pp. 1135–1169.
V. G. Turaev and O. Y. Viro, State sum invariants of $3$-manifolds and quantum $6j$-symbols, Topology, 31 (1992), pp. 865–902.
