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Products and vector bundles within the category of $$G$$-supermanifolds. (English. Russian original) Zbl 0843.58003
Sib. Math. J. 34, No. 1, 1-9 (1993); translation from Sib. Mat. Zh. 34, No. 1, 5-15 (1993).
Let $$(M, A)$$ be a supermanifold over the basic algebra $$B$$ (graded commutative Banach algebra), where $$M$$ is an underlying topological manifold and $$A$$ is a sheaf of supercommutative $$B$$-algebras. It is known that $$A$$ under some reasonable conditions is not a sheaf of functions. Therefore, many of the usual geometric constructions such as the products and vector bundles are not so immediate as in the usual differential geometry. The authors develop the concepts of a product of $$G$$-supermanifolds and (super)vector bundles in the spirit of Berezin-Leites-Konstant.