A∞ deformations of extended Khovanov arc algebras and Stroppel's conjecture

Abstract

Extended Khovanov arc algebras Kmn are graded associative algebras which naturally appear in a variety of contexts, from knot and link homology, low-dimensional topology and topological quantum field theory to representation theory and symplectic geometry. C. Stroppel conjectured in her ICM 2010 address that the bigraded Hochschild cohomology groups of Kmn vanish in a certain range, implying that the algebras Kmn admit no nontrivial A∞ deformations, in particular that the algebras are intrinsically formal. Whereas Stroppel's conjecture is known to hold for the algebras Km1 and K1n by work of Seidel and Thomas, we show that Kmn does in fact admit nontrivial A∞ deformations with nonvanishing higher products for all m,n≥2. We describe both the extended Khovanov arc algebras Kmn and their Koszul duals concretely as path algebras of quivers with relations. This allows us to give an explicit algebraic construction of A∞ deformations of Kmn by using the correspondence between A∞ deformations of a Koszul algebra and filtered associative deformations of its Koszul dual.