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In this paper we study the second Hochschild cohomology group of thepreprojective algebra of type $D_4$ over an algebraically closed field $K$ ofcharacteristic 2. We also calculate the second Hochschild cohomology group of anon-standard algebra which arises as a socle deformation of this preprojectivealgebra and so show that the two algebras are not derived equivalent. Thisanswers a question raised by Holm and Skowro\'nski.

Distinguishing derived equivalence classes using the second Hochschild cohomology group