The Berezinian in some monoidal categories (Q1821191)

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.