Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 060, 58 pages      arXiv:2011.01012

Linear ${\mathbb Z}_2^n$-Manifolds and Linear Actions

Andrew James Bruce, Eduardo Ibarguëngoytia and Norbert Poncin
Department of Mathematics, University of Luxembourg, Maison du Nombre, 6, avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg

Received November 05, 2020, in final form May 30, 2021; Published online June 16, 2021

We establish the representability of the general linear ${\mathbb Z}_2^n$-group and use the restricted functor of points – whose test category is the category of ${\mathbb Z}_2^n$-manifolds over a single topological point – to define its smooth linear actions on ${\mathbb Z}_2^n$-graded vector spaces and linear ${\mathbb Z}_2^n$-manifolds. Throughout the paper, particular emphasis is placed on the full faithfulness and target category of the restricted functor of points of a number of categories that we are using.

Key words: supergeometry; ringed spaces; functors of points; linear group actions.

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