Shifted Poisson and Batalin-Vilkovisky structures on the derived variety of complexes

TL;DR

This paper explores shifted Poisson and Batalin-Vilkovisky structures on derived varieties of complexes, extending classical structures and constructing new shifted Poisson structures in derived algebraic geometry.

Contribution

It introduces new shifted Poisson structures on derived varieties of complexes and demonstrates their relation to Batalin-Vilkovisky algebra structures.

Findings

Constructed 1-shifted Poisson structures on derived quotients of complexes.

Extended Kirillov-Kostant Poisson structure to derived varieties.

Showed the 1-shifted structure is homotopy equivalent to a BV algebra.

Abstract

We study the shifted Poisson structure on the cochain complex C*(g) of a graded Lie algebra arising from shifted Lie bialgebra structure on g. We apply this to construct a 1-shifted Poisson structures on an infinitesimal quotient of the derived variety of complexes RCom(V) by a subgroup of the automorphisms of V, and a non-shifted Poisson structure on an appropriately defined derived variety of 1-periodic complexes, extending the standard Kirillov-Kostant Poisson structure on gl*_n. We also show that in the case of RCom(V) the 1-shifted structure is up to homotopy a Batalin-Vilkovisky algebra structure.