Abstract
Our purpose in this Chapter is to study the main features of the theory of vector bundles in the category of G-supermanifolds.
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We recall that an exact sequence (say, of modules) is split if N ≃ M ⊕ Q. A splitting of the exact sequence is a morphism i: Q → N such that poi = Id; the existence of at least one of such a morphism is apparently equivalent to the splitness of the sequence. Let us also notice that in ordinary differential geometry all exact sequences of smooth vector bundles do split, due to the existence of smooth partitions of unity [Hus]. 2 In view of Proposition I.2.2, the sequence (1.3) can also be written and is therefore obtained from (1.2) by tensoring with.
Juxtaposition here denotes matrix multiplication.
As a matter of fact, PGL[r|s] is a Lie supergroup, cf. Chapter VII.
We recall that a principal ideal domain is a commutative ring K with no zero divisors such that every ideal is of the form bK for some b ∈ K.
A more general definition of adjoint representation, for a generic Lie supergroup, will be given in the next Chapter.
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© 1991 Springer Science+Business Media Dordrecht
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Bartocci, C., Bruzzo, U., Hernández-Ruipérez, D. (1991). Geometry of supervector bundles. In: The Geometry of Supermanifolds. Mathematics and Its Applications, vol 71. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3504-7_6
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DOI: https://doi.org/10.1007/978-94-011-3504-7_6
Publisher Name: Springer, Dordrecht
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