Theodore Voronov

Higher derived brackets and homotopy algebras

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 Δ. Given this, we introduce an infinite sequence of higher brackets on the image of the projector, and explicitly calculate their Jacobiators in terms of Δ2. This allows to control higher Jacobi identities in terms of the “order” of Δ2. Examples include Stasheffs strongly homotopy Lie algebras and variants of homotopy Batalin–Vilkovisky algebras. There is a generalization with Δ replaced by an arbitrary odd derivation. We discuss applications and links with other constructions.

Q-manifolds and higher analogs of Lie algebroids

We show how the relation between Q‐manifolds and Lie algebroids extends to “higher” or “non‐linear” analogs of Lie algebroids. We study the identities satisfied by a new algebraic structure that arises as a replacement of operations on sections of a Lie algebroid. When the base is a point, we obtain a generalization of Lie superalgebras.

Berezinians, exterior powers and recurrent sequences

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 express the Berezinian of an operator as a ratio of two polynomial invariants. 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

Q-Manifolds and Mackenzie Theory

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.

Darboux transformations for differential operators on the superline

We give a full description of Darboux transformations of any order for arbitrary (nondegenerate) differential operators on the superline. We show that every Darboux transformation of such operators factorizes into elementary Darboux transformations of order one. Similar statement holds for operators on the ordinary line.

Differential operators on the superline, Berezinians, and Darboux transformations

We consider differential operators on a supermanifold of dimension 1|1. We define non-degenerate operators as those with an invertible top coefficient in the expansion in the ‘superderivative’ D (which is the square root of the shift generator, the partial derivative in an even variable, with the help of an odd indeterminate). They are remarkably similar to ordinary differential operators. We show that every non-degenerate operator can be written in terms of ‘super Wronskians’ (which are certain Berezinians). We apply this to Darboux transformations (DTs), proving that every DT of an arbitrary non-degenerate operator is the composition of elementary first-order transformations. Hence every DT corresponds to an invariant subspace of the source operator and, upon a choice of basis in this subspace, is expressed by a super Wronskian formula. We consider also dressing transformations, i.e., the effect of a DT on the coefficients of the non-degenerate operator. We calculate these transformations in examples and make some general statements.

Felix Alexandrovich Berezin and His Work

This is a survey of Berezin’s work focused on three topics: representation theory, general concept of quantization, and supermathematics.

Geometry of differential operators, odd Laplacians, and homotopy algebras

Abstract We give a complete description of differential operators generating a given bracket. In particular we consider the case of Jacobi-type identities for odd operators and brackets. This is related with homotopy algebras using the derived bracket construction.

Dual Forms on Supermanifolds and Cartan Calculus

Abstract: We introduce and study the complex of “stable forms” on supermanifolds. Stable forms on a supermanifold M are represented by Lagrangians of “copaths” (formal systems of equations, which may or may not specify actual surfaces) on M×ℝD. Changes of D give rise to stability isomorphisms. The resulting (direct limit) {Cartan-de Rham} complex made of stable forms extends both in positive and negative degree. Its positive half is isomorphic to the complex of forms defined as Lagrangians of paths, studied earlier. Including the negative half is crucial, in particular, for homotopy invariance. For stable forms we introduce (non-obvious) analogs of exterior multiplication by covectors and contraction with vectors and find the anticommutation relations that they obey. Remarkably, the version of the Clifford algebra so obtained is based on the super anticommutators rather than the commutators and (before stabilization) it includes some central element σ. An analog of Cartans homotopy identity is proved, which also contains this “stability operator”σ.

On a non-Abelian Poincaré lemma

We show that a well-known result on solutions of the Maurer--Cartan equation extends to arbitrary (inhomogeneous) odd forms: any such form with values in a Lie superalgebra satisfying