Symplectic $  {\mathbb Z}_2^n $-manifolds

Bruce Andrew James; Grabowski Janusz

doi:10.3934/jgm.2021020

Article Contents

2021, Volume 13, Issue 3: 285-311. Doi: 10.3934/jgm.2021020

This issue Previous Article Book review: General theory of Lie groupoids and Lie algebroids, by Kirill C. H. Mackenzie Next Article On twistor almost complex structures

Symplectic $ {\mathbb Z}_2^n $-manifolds

Bruce Andrew James^1, , and
Grabowski Janusz^2,

1.
Department of Mathematics, University of Luxembourg, Esch-sur-Alzette, Luxembourg
2.
Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

^* Corresponding author: Andrew James Bruce
^* Corresponding author: Andrew James Bruce

Received: February 28, 2021

Revised: June 30, 2021

Early access: August 2021

Published: September 2021

The Second author is supported by the Polish National Science Centre grant HARMONIA under the contract number 2016/22/M/ST1/00542

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Abstract

Roughly speaking, $ {\mathbb Z}_2^n $-manifolds are 'manifolds' equipped with $ {\mathbb Z}_2^n $-graded commutative coordinates with the sign rule being determined by the scalar product of their $ {\mathbb Z}_2^n $-degrees. We examine the notion of a symplectic $ {\mathbb Z}_2^n $-manifold, i.e., a $ {\mathbb Z}_2^n $-manifold equipped with a symplectic two-form that may carry non-zero $ {\mathbb Z}_2^n $-degree. We show that the basic notions and results of symplectic geometry generalise to the 'higher graded' setting, including a generalisation of Darboux's theorem.

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Mathematics Subject Classification: Primary: 53D05, 53D15, 58A50; Secondary: 14A22, 17B63, 17B75.

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