Abstract
This paper aims at setting out the basics of mathbb {Z}-graded manifolds theory. We introduce mathbb {Z}-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric algebra to define functions is made clear. Moreover, we define vector fields and exhibit their graded local basis. The paper also reviews some correspondences between differential mathbb {Z}-graded manifolds and algebraic structures.
Highlights
Introduction to graded geometryMaxime Fairon1 AbstractThis paper aims at setting out the basics of Z-graded manifolds theory
In the Berezin–Leites [2] and Kostant [11] approach, they are defined as Z2-graded locally ringed spaces. This means that a supermanifold is a pair (|N |, ON ) such that |N | is a topological space and ON is a sheaf of Z2-graded algebras which satisfies, for all sufficiently small open subsets U, ON (U ) C∞ Rn (U ) ⊗ Rm, B Maxime Fairon mmmfai@leeds.ac.uk 1 School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom where Rm is endowed with its canonical Z2-grading
From the late 90s, the introduction of an integer grading was necessary in some topics related to Poisson geometry, Lie algebroids and Courant algebroids
Summary
In the 70s, supermanifolds were introduced and studied to provide a geometric background to the developing theory of supersymmetry. Its structure sheaf ON is a sheaf of Z-graded algebras which is given locally by ON (U ) C∞ Rn (U ) ⊗ SW, where W = i Wi is a real Z-graded vector space whose component of degree zero satisfies W0 = {0}. The existence of non-nilpotent elements of degree 0, which are products of such generators with non-zero even degree, requires the introduction of formal power series to obtain the locality of every stalk of the sheaf. This condition, which has not always been accurately considered in past works, occurs with Zn2-grading for n 2, as discussed in [6]. Notice that some authors use the expression graded manifolds to talk about supermanifolds with an additional Z-grading (see [22]), but we only use this expression in the present paper to refer to Z-graded manifolds
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