Construct explicit formal deformations of A_N from HH^2(A_N)

Construct explicit formal deformations of the algebra A_N = ⟨x,y⟩/(yx − xy − x^N) arising from elements of the second Hochschild cohomology HH^2(A_N), leveraging the fact that HH^3(A_N) = 0 makes integration unobstructed. Develop explicit formulas analogous to the derivation representatives constructed for HH^1(A_N).

Background

The authors compute HH^2(A_N) and note, by classical deformation theory, that its elements integrate to formal deformations of A_N, with no obstruction because HH^3(A_N)=0.

However, unlike their explicit representatives for HH^1(A_N), they do not provide explicit constructions for the corresponding deformations of A_N and leave this task open.