A$_\infty$ deformations of extended Khovanov arc algebras and Stroppel's conjecture

Severin Barmeier; Zhengfang Wang

Paper

A$_\infty$ deformations of extended Khovanov arc algebras and Stroppel's conjecture

Abstract

Extended Khovanov arc algebras

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

vanish in a certain range, implying that the algebras

admit no nontrivial A

deformations, in particular that the algebras are intrinsically formal. Whereas Stroppel's Conjecture is known to hold for the algebras

and

by work of Seidel and Thomas, we show that

does in fact admit nontrivial A

deformations with nonvanishing higher products for all

. We describe both

and its Koszul dual concretely as path algebras of quivers with relations and give an explicit algebraic construction of A

deformations of

by using the correspondence between A

deformations of a Koszul algebra and filtered associative deformations of its Koszul dual. These deformations can also be viewed as A

deformations of Fukaya--Seidel categories associated to Hilbert schemes of surfaces based on recent work of Mak and Smith.