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In this paper we find a generalization of the snake lemma and Schanuel's lemma in Hv-modules. We define the isomorph sequences and determine the conditions to split the exact sequences in Hv-modules. Some interesting results on these concepts are given. [ABSTRACT FROM AUTHOR]

The largest class of multivalued systems satisfying the module-like axioms are the Hv -modules. The main tools concerning the class of Hv -modules with the ordinary modules are the fundamental relations. Based on the relation ε* , exact sequences in Hv -modules are defined. In this paper, we introduce the Hv -module M[A] and determine its heart and the connection between equivalence relations ε* M[A] and ε* A . Moreover, we define the M[-] and -[M] functors and investigate the exactness and some concepts related to them. Finally, we prove the five short lemma in Hv -modules. [ABSTRACT FROM AUTHOR]

Let V be a vertex operator algebra. The fusion products in this paper are defined by logarithmic intertwining operators. Under this setting, we prove the pure exactness of any extension of the V -module V . Namely, if 0 → Q → ϵ P → ρ V → 0 is a short exact sequence of V -modules and W is a V -module, then 0 → Q ⊠ W → ϵ ⊠ 1 W P ⊠ W → ρ ⊠ 1 W V ⊠ W → 0 is also exact, where we view it as a sequence of g ( V ) -modules. [ABSTRACT FROM AUTHOR]

Abstract: By means of a mixture of linear and nonlinear techniques, the paper treats quasi-Banach function spaces as -modules and studies the basic algebraical constructions and their interactions, placing the emphasis on tensor products. Sample result: if is an exact sequence of -modules and homomorphisms, where Y and Z are function spaces, and V is another function space, then the tensorized sequence is exact too. [Copyright &y& Elsevier]