An $L_{\infty}$-module structure on annular Khovanov homology

Let L be a link in a thickened annulus. In Grigsby et al. (2018), Grigsby–Licata–Wehrli showed that the annular Khovanov homology of L is equipped with an action of \mathfrak{sl}_{2}(\wedge) , the exterior current algebra of the Lie algebra \mathfrak{sl}_{2} . In this paper, we upgrade this result to the setting of L_{\infty} -algebras and modules. That is, we show that \mathfrak{sl}_{2}(\wedge) is an L_{\infty} -algebra and that the annular Khovanov homology of L is an L_{\infty} -module over \mathfrak{sl}_{2}(\wedge) . Up to L_{\infty} -quasi-isomorphism, this structure is invariant under Reidemeister moves. Finally, we include explicit formulas to compute the higher L_{\infty} -operations.