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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
A categorification of $\mathbold{U}_T(\mathfrak{sl}(1|1))$ and its tensor product representations

Yin Tian

Geometry & Topology 18 (2014) 1635–1717
Abstract

We define the Hopf superalgebra UT(sl(1|1)), which is a variant of the quantum supergroup Uq(sl(1|1)), and its representations V 1n for n > 0. We construct families of DG algebras A, B and Rn, and consider the DG categories DGP(A), DGP(B) and DGP(Rn), which are full DG subcategories of the categories of DG A–, B– and Rn–modules generated by certain distinguished projective modules. Their 0 th homology categories HP(A), HP(B) and HP(Rn) 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 UT(sl(1|1)), UT(sl(1|1)) UT(sl(1|1)) and V 1n, respectively. We categorify the multiplication and comultiplication on UT(sl(1|1)) to a bifunctor HP(A) × HP(A) HP(A) and a functor HP(A) HP(B), respectively. The UT(sl(1|1))–action on V 1n is lifted to a bifunctor HP(A) × HP(Rn) HP(Rn).

Keywords
Hopf superalgebra, categorification, tight contact structure, Heegaard Floer homology
Mathematical Subject Classification 2010
Primary: 18D10
Secondary: 16D20, 57M50
References
Publication
Received: 26 January 2013
Revised: 13 January 2014
Accepted: 24 January 2014
Published: 7 July 2014
Proposed: Yasha Eliashberg
Seconded: Peter Ozsvath, Leonid Polterovich
Authors
Yin Tian
Department of Mathematics
University of Southern California
3620 S Vermont Ave.
KAP 104
Los Angeles, CA 90089
USA