The Hochschild cohomology of the algebra of differential operators tangent to a central arrangement of lines

Abstract

Given a central arrangement of lines A in a 2-dimensional vector space V over a field of characteristic zero, we study the algebra D (A) of differential operators on V which are logarithmic along A. Among other things we determine the Hochschild cohomology of D (A) as a Gerstenhaber algebra, establish a connection between that cohomology and the de Rham cohomology of the complement M(A) of the arrangement, determine the isomorphism group of D (A) and classify the algebras of that form up to isomorphism.