We give a construction of homotopy algebras based on "higher derived brackets". More precisely, the data include a Lie superalgebra with a projector on an Abelian subalgebra satisfying a certain axiom, and an odd element . Given this, we introduce an infinite sequence of higher brackets on the image of the projector, and explicitly calculate their Jacobiators in terms of 2 . This allows to control higher Jacobi identities in terms of the "order" of 2 . Examples include Stasheff's strongly homotopy Lie algebras and variants of homotopy Batalin-Vilkovisky algebras. There is a generalization with replaced by an arbitrary odd derivation. We discuss applications and links with other constructions.

Let ∆ be an arbitrary linear differential operator of the second order acting on functions on a (super)manifold M. In local coordinates ∆ = 1 2 S ab ∂ b ∂ a + T a ∂ a + R. The principal symbol of ∆ is the symmetric tensor field S ab , or the quadratic function S = 1 2 S ab p b p a on T * M. The principal symbol can be understood as a symmetric "bracket" on functions:where ε =∆ is the parity of the operator ∆; in coordinates {f, g} = S ab ∂ b f ∂ a g(−1)ãf . In the following by a bracket in a commutative algebra we mean an arbitrary symmetric bi-derivation. The problem is to describe all operators ∆ with a given S ab , or, which is the same, all operators generating a given bracket {f, g}. Without loss of generality we set R = ∆(1) := 1 in the sequel. Initially we suppose that the operators act on scalar functions; operator pencils acting on densities of arbitrary weights will naturally appear in the course of study. Everything is applicable to supermanifolds as well as to usual manifolds. For odd operators in the super case questions about identities of the Jacobi type arise. The problem is closely related with the geometry of the Batalin-Vilkovisky formalism in quantum field theory (description of the "generating operators" for an odd bracket).The first non-trivial observation is that Hörmander's subprincipal symbol sub ∆ = (∂ b S ba (−1)b (ε+1) −2T a )p a can be interpreted as an "upper connection" in the bundle Vol M. Precisely, γ a = ∂ b S ba (−1)b (ε+1) − 2T a has the transformation law γ a ′ = γ a + S ab ∂ b ln J ∂x a ′ ∂x a , where J = Dx ′ Dx (the Jacobian), and it specifies a "contravariant derivative" ∇ a ρ = (S ab ∂ b + γ a )ρ on volume forms. The coordinate-dependent Hamiltonian γ = sub ∆ = γ a p a plays the role of a local connection form. If the matrix S ab is invertible, then we can lower the index a and get a usual connection. (Let us stress that ∆ acts on functions, and a priori there is no extra structure on our manifold. The bundle Vol M and an upper connection in it arise from the operator itself.) Thus, ∆ is defined by a set of data: a bracket on functions and an associated upper connection in Vol M.Define the algebra of densities V(M) as the algebra of formal linear combinations of densities of arbitrary weights w ∈ R. In V(M) there is a unit 1 and a natural invariant scalar multiplication. The scalar product is given by the formula: ψ, χ = M Res(t −2 ψ(x, t)χ(x, t)) Dx.

Abstract. We define graded manifolds as a version of supermanifolds endowed with an extra Z-grading in the structure sheaf, called weight (not linked with parity). Examples are ordinary supermanifolds, vector bundles, double vector bundles (in particular, iterated constructions like T T M ), etc. I give a construction of doubles for graded QS-and graded QP -manifolds (graded manifolds endowed with a homological vector field and a Schouten/Poisson bracket). Relation is explained with Drinfeld's Lie bialgebras and their doubles. Graded QS-manifolds can be considered, roughly, as "generalized Lie bialgebroids". The double for them is closely related with the analog of Drinfeld's double for Lie bialgebroids recently suggested by Roytenberg. Lie bialgebroids as a generalization of Lie bialgebras, over some base manifold, were defined by Mackenzie and P. Xu. Graded QP -manifolds give an odd version for all this, in particular, they contain "odd analogs" for Lie bialgebras, Manin triples, and Drinfeld's double.

Double Lie algebroids were discovered by Kirill Mackenzie from the study of double Lie groupoids and were defined in terms of rather complicated conditions making use of duality theory for Lie algebroids and double vector bundles. In this paper we establish a simple alternative characterization of double Lie algebroids in a supermanifold language. Namely, we show that a double Lie algebroid in Mackenzie's sense is equivalent to a double vector bundle endowed with a pair of commuting homological vector fields of appropriate weights. Our approach helps to simplify and elucidate Mackenzie's original definition; we show how it fits into a bigger picture of equivalent structures on 'neighbor' double vector bundles. It also opens ways for extending the theory to multiple Lie algebroids, which we introduce here.Date: 3 (16) June 2012.

Abstract. We study power expansions of the characteristic function of a linear operator A in a p|q-dimensional superspace V . We show that traces of exterior powers of A satisfy universal recurrence relations of period q. 'Underlying' recurrence relations hold in the Grothendieck ring of representations of GL(V ). They are expressed by vanishing of certain Hankel determinants of order q + 1 in this ring, which generalizes the vanishing of sufficiently high exterior powers of an ordinary vector space. In particular, this allows to explicitly express the Berezinian of an operator as a rational function of traces. We analyze the Cayley-Hamilton identity in a superspace. Using the geometric meaning of the Berezinian we also give a simple formulation of the analog of Cramer's rule.