Morita Invariance for Infinitesimal Deformations

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Let A and B be two Morita equivalent finite dimensional associative algebras over a field 𝕜. It is well known that Hochschild cohomology is invariant under Morita equivalence. Since infinitesimal deformations are connected with the second Hochschild cohomology group, we explicitly describe the transfer map connecting HH2(A) with HH2(B). This allows us to transfer Morita equivalence between A and B to that between infinitesimal deformations of them. As an application, when 𝕜 is algebraically closed, we consider the quotient path algebra associated to A and describe the presentation by quiver and relations of the infinitesimal deformations of A.