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

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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.