The Berezinian in some monoidal categories
type: Item
instance of: scholarly article
title: The Berezinian in some monoidal categories
zbMATH Open document ID: 0616.18006
DOI: 10.1007/BF01060941
published in: Ukrainian Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: The aim of the article is to generalize the notions of supertrace and of Berezinian (determinant of Berezin) from the category of \({\mathbb{Z}}/2\)- graded finite-dimensional vector spaces to certain monoidal categories \({\mathcal Q}\)- categories of comodules over the Hopf algebras H with symmetry. The author establishes the connection between the supertrace and Berezinian, defines the notion of supermanifold as the ringed space, locally isomorphic to the space (U,\({\mathcal O}_ U)\), where \(U\subset {\mathbb{R}}^ m\) and \({\mathcal O}_ U=S(V)\otimes C_ U^{\infty}\), S(V) being the symmetric algebra of some odd object of the category \({\mathcal Q}\). Then the densities and the integral of densities on the supermanifold (X,\({\mathcal O}_ X)\) are defined.
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 58A50
Mathematics Subject Classification ID: 58C35
Mathematics Subject Classification ID: 58C50
zbMATH DE Number: 3998062
zbMATH Keywords: graded manifold
zbMATH Keywords: supertrace
zbMATH Keywords: Berezinian
zbMATH Keywords: monoidal categories
zbMATH Keywords: Hopf algebras
zbMATH Keywords: ringed space
zbMATH Keywords: densities
zbMATH Keywords: integral of densities
zbMATH Keywords: supermanifold
author: Volodymyr Lyubashenko
MaRDI profile type: Publication
cites work: Q3043319
cites work: Q4693151
cites work: Functions of noncommuting operators that generate a graded Lie algebra
cites work: Q4130629
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description: scientific article; zbMATH DE number 3998062