Versal deformations of Leibniz algebras

In this work we consider deformations of Leibniz algebras over a field of characteristic zero. The main problem in deformation theory is to describe all non-equivalent deformations of a given object. We give a method to solve this problem completely, namely work out a construction of a versal deformation for a given Leibniz algebra, which induces all non-equivalent deformations and is unique on the infinitesimal level.Comment: 29 page

Higher categorified algebras versus bounded homotopy algebras

We define Lie 3-algebras and prove that these are in 1-to-1 correspondence with the 3-term Lie infinity algebras whose bilinear and trilinear maps vanish in degree (1,1) and in total degree 1, respectively. Further, we give an answer to a question of [Roy07] pertaining to the use of the nerve and normalization functors in the study of the relationship between categorified algebras and truncated sh algebras.Comment: 21 pages, 1 figur

On Lie algebroid over algebraic spaces

We consider Lie algebroids over algebraic spaces (in short we call it as $a$-spaces) by considering the sheaf of Lie-Rinehart algebras. We discuss about properties of universal enveloping algebroid $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$ of a Lie algebroid $\mathcal{L}$ over an $a$-space $(X, \mathcal{O}_X)$. This is done by sheafification of the presheaf of universal enveloping algebras of Lie-Rinehart algebras. We review the extent to which the structure of the universal enveloping algebroid of Lie algebroids (over special $a$-spaces) resembles a bialgebroid structure, and present a version of Poincare-Birkhoff-Witt theorem and Cartier-Milnor-Moore theorem for this type of structure.Comment: Some minor changes done. Mainly, we shorten it and kept main idea