Ashis Mandal

We consider associative algebras equipped with derivations. A pair consisting of an associative algebra and a distinguished derivation is called an AssDer pair. We study central extensions and formal one-parameter deformations of AssDer pairs in terms of cohomology. Finally, we define 2-derivations on associative 2-algebras and show that the category of associative 2-algebras with 2-derivations is equivalent to the category of 2-term associative infinity algebras with homotopy derivations.

We explicitly give all the equivalent classes of deformations of the 5-dimensional Heisenberg Lie algebra over the field of complex numbers or the real numbers. We show that there are altogether 20 infinitesimal deformations (families), 18 of them being extendable to real deformations and 2 of them are only infinitesimal.

Lie algebroids over algebraic spaces is a generalisation of the notion of Lie algebroids over a manifold (real or complex), and can also be treated on a very large class of examples including spaces not necessarily smooth manifolds or also contain singularities as well. Our work on the structure universal enveloping algebroids motivates to find other important homological results can be deduced by the use of Lie-Rinehart algebras and its application. On the other hand we are interested in finding results in Poison geometry for a class of spaces not necessarily smooth manifolds and also contain singularities.