A categorification of TEXTBACKSLASHmathboldTEXTBACKSLASHUT(𝔰𝔩(1|1)) and its tensor product representations

Tian, Yin¹

¹ Department of Mathematics, University of Southern California, 3620 S Vermont Ave., KAP 104, Los Angeles, CA 90089, USA

Geometry & topology, Tome 18 (2014) no. 3, pp. 1635-1717.

Résumé

We define the Hopf superalgebra $U_T\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right)$, which is a variant of the quantum supergroup $U_q\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right)$, and its representations $V_1^{\otimes n}$ for $n>0$. We construct families of DG algebras $A$, $B$ and $R_n$, and consider the DG categories $DGP\left(A\right)$, $DGP\left(B\right)$ and $DGP\left(R_n\right)$, which are full DG subcategories of the categories of DG $A$–, $B$– and $R_n$–modules generated by certain distinguished projective modules. Their $0^{\text{ th}}$ homology categories $HP\left(A\right)$, $HP\left(B\right)$ and $HP\left(R_n\right)$ are triangulated and give algebraic formulations of the contact categories of an annulus, a twice punctured disk and an $n$ times punctured disk. Their Grothendieck groups are isomorphic to $U_T\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right)$, $U_T\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right){\otimes }_{\mathbb{Z}}{U}_T\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right)$ and $V_1^{\otimes n}$, respectively. We categorify the multiplication and comultiplication on $U_T\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right)$ to a bifunctor $HP\left(A\right)\times HP\left(A\right)\to HP\left(A\right)$ and a functor $HP\left(A\right)\to HP\left(B\right)$, respectively. The $U_T\left(\mathfrak{s}\mathfrak{l}\left(1|1\right)\right)$–action on $V_1^{\otimes n}$ is lifted to a bifunctor $HP\left(A\right)\times HP\left(R_n\right)\to HP\left(R_n\right)$.

Détail

DOI : 10.2140/gt.2014.18.1635

Classification : 18D10, 16D20, 57M50
Keywords: Hopf superalgebra, categorification, tight contact structure, Heegaard Floer homology

Bibliographie

[1] G Benkart, D Moon, Planar rook algebras and tensor representations of gl(1|1),

[2] J Bernstein, V Lunts, Equivariant sheaves and functors, 1578, Springer, Berlin (1994)

[3] S Bigelow, E Ramos, R Yi, The Alexander and Jones polynomials through representations of rook algebras, J. Knot Theory Ramifications 21 (2012) 1250114, 18 | DOI

[4] J Chuang, R Rouquier, Derived equivalences for symmetric groups and sl2–categorification, Ann. of Math. 167 (2008) 245 | DOI

[5] L Crane, I B Frenkel, Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases, J. Math. Phys. 35 (1994) 5136 | DOI

[6] D Flath, T Halverson, K Herbig, The planar rook algebra and Pascal’s triangle, Enseign. Math. (2) 55 (2009) 77

[7] E Giroux, Structures de contact sur les variétés fibrées en cercles audessus d’une surface, Comment. Math. Helv. 76 (2001) 218 | DOI

[8] K Honda, Contact structures, Heegaard–Floer homology and triangulated categories, in preparation

[9] K Honda, On the classification of tight contact structures, I, Geom. Topol. 4 (2000) 309 | DOI

[10] K Honda, Gluing tight contact structures, Duke Math. J. 115 (2002) 435 | DOI

[11] K Honda, W H Kazez, G Matić, Pinwheels and bypasses, Algebr. Geom. Topol. 5 (2005) 769 | DOI

[12] K Honda, W H Kazez, G Matić, Right-veering diffeomorphisms of compact surfaces with boundary, II, Geom. Topol. 12 (2008) 2057 | DOI

[13] K Honda, W H Kazez, G Matić, The contact invariant in sutured Floer homology, Invent. Math. 176 (2009) 637 | DOI

[14] Y Huang, Bypass attachments and homotopy classes of 2–plane fields in contact topology,

[15] L H Kauffman, H Saleur, Free fermions and the Alexander–Conway polynomial, Comm. Math. Phys. 141 (1991) 293

[16] B Keller, On differential graded categories, from: "International Congress of Mathematicians, Vol. II" (editors M Sanz-Solé, J Soria, J L Varona, J Verdera), Eur. Math. Soc., Zürich (2006) 151

[17] M Khovanov, How to categorify one-half of quantum gl(1|2),

[18] M Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359 | DOI

[19] M Khovanov, A D Lauda, A categorification of quantum sl(n),

[20] M Khovanov, A D Lauda, A diagrammatic approach to categorification of quantum groups, I, Represent. Theory 13 (2009) 309 | DOI

[21] M Khovanov, A D Lauda, A diagrammatic approach to categorification of quantum groups, II, Trans. Amer. Math. Soc. 363 (2011) 2685 | DOI

[22] A D Lauda, A categorification of quantum sl(2), Adv. Math. 225 (2010) 3327 | DOI

[23] R Lipshitz, P Ozsváth, D Thurston, Bordered Heegaard–Floer homology : Invariance and pairing,

[24] S Makar-Limanov, Morse surgeries of index 0 on tight manifolds, Preprint (1997)

[25] C Manolescu, P Ozsváth, S Sarkar, A combinatorial description of knot Floer homology, Ann. of Math. 169 (2009) 633 | DOI

[26] D Mathews, Chord diagrams, contact-topological quantum field theory and contact categories, Algebr. Geom. Topol. 10 (2010) 2091 | DOI

[27] D V Mathews, Sutured TQFT, torsion and tori, Internat. J. Math. 24 (2013) 1350039, 35 | DOI

[28] P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58 | DOI

[29] P Ozsváth, Z Szabó, Heegaard–Floer homology and contact structures, Duke Math. J. 129 (2005) 39 | DOI

[30] J A Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University (2003)

[31] N Y Reshetikhin, V G Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990) 1

[32] R Rouquier, 2–Kac–Moody algebras,

[33] L Rozansky, H Saleur, Quantum field theory for the multivariable Alexander–Conway polynomial, Nuclear Phys. B 376 (1992) 461 | DOI

[34] A Sartori, Categorification of tensor powers of the vector representation of Uq(gl(1|1)),

[35] L Solomon, Representations of the rook monoid, J. Algebra 256 (2002) 309 | DOI

[36] Y Tian, Categorical braid group actions on representations of TEXTBACKSLASHmathboldTEXTBACKSLASHUT(sl(1|1)), in preparation

[37] Y Tian, A categorification of TEXTBACKSLASHmathboldTEXTBACKSLASHUq(sl(1|1)) as an algebra

[38] B Webster, Knot invariants and higher representation theory I : Diagrammatic and geometric categorification of tensor products,

[39] B Webster, Knot invariants and higher representation theory II : The categorification of quantum knot invariants,

[40] C A Weibel, An introduction to homological algebra, 38, Cambridge Univ. Press (1994) | DOI

[41] E Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351

[42] R Zarev, Bordered Floer homology for sutured manifolds,

[43] R Zarev, Equivalence of gluing maps for SFH, in preparation