Journal of Symplectic Geometry

Volume 14 (2016)

Number 2

Categorification of Clifford algebras and $\mathrm{U}_q (\mathfrak{sl}(1 \vert 1))$

Pages: 541 – 585



Yin Tian (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)


We construct families of differential graded algebras $R_n$ and $R_n \boxtimes R_n$ for $n \gt 0$, and differential graded categories $DGP(R_n)$ generated by some distinguished projective $R_n$-modules. The category $DGP(R_n)$ gives an algebraic formulation of the contact category of a disk. The $0$-th homology category $H^0 (DGP(R_n))$ of $DGP(R_n)$ is a triangulated category and its Grothendieck group $K_0(R_n)$ is isomorphic to a Clifford algebra. We then categorify the multiplication on $K_0(R_n)$ to a functor $DGP(R_n \boxtimes R_n) \to DGP(R_n)$. We also construct a subcategory of $H^0 (DGP(R_n))$ which categorifies an integral version of $\mathrm{U}_q (\mathfrak{sl}(1 \vert 1))$ as an algebra.

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