Manchester Geometry Seminar 2010/2011

I shall speak about graded manifolds and will show how they lead to new interesting examples of geometric and algebraic structures.

"Non-linear" analogs of Lie algebroids and Lie algebras arise from their descriptions in terms of supermanifolds. It is known that a Lie algebra structure on a vector space V (or a Lie algebroid structure on a vector bundle) is equivalent to a homological vector field on the opposite space Π V of weight +1 with respect to the grading defined by the linear structure. If we now replace vector spaces or vector bundles by manifolds with a non-negative Z-grading in the structure sheaf, we arrive at "non-linear (or higher) Lie algebroids" and "non-linear Lie algebras". A remarkable feature of this non-linear graded geometry is the existence of a certain faithful linear model. For example, a non-linear Lie algebroid gives rise to a genuine vector bundle whose space of sections is a cochain complex with two bracket operations of opposite parity. I will give a construction of this bundle and describe the algebraic identities satisfied by the differential and the brackets. They define a new type of algebras. (They are related with L-infinity algebras, but are different from them.)