BerezinKostantLeites

Berezin-Kostant-Leites (BKL) refers to a standard mathematical formalism for supermanifolds named after Boris Berezin, Bertram Kostant, and Dmitry Leites. It presents supergeometry as a theory of locally ringed spaces with a Z2-graded structure sheaf, providing a rigorous framework for incorporating anticommuting coordinates into differential geometry. The BKL approach is one of the foundational formalisms alongside functor-of-points and DeWitt’s ringed-space perspectives.

In the Berezin-Kostant-Leites construction, a supermanifold of dimension m|n is a pair (|M|, O_M) where |M| is an m-dimensional smooth manifold (the underlying body) and O_M is a sheaf of Z2-graded, commutative C^∞(|M|)-algebras. Locally, around every point p in |M|, there is an isomorphism O_M|_U ≅ C^∞(U) ⊗ Λ^• R^n, where U is an open subset of |M| and Λ^• R^n is the Grassmann algebra on n generators. The even part corresponds to ordinary smooth functions, while the odd part encodes the anticommuting coordinates.

Morphisms in this category consist of a continuous map between underlying spaces together with a compatible morphism of sheaves that preserves the Z2-grading. This formalism emphasizes the ringed-space perspective, providing tools such as the Berezin integral and the Berezinian (superdeterminant) for analysis on supermanifolds.

Typical examples include Euclidean superspace R^{m|n}, whose structure sheaf is generated by n odd coordinates over the base manifold R^m. The BKL framework remains a central reference point in differential geometry and mathematical supersymmetry.