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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: It is well known that diagram lemmas for abelian groups (and more generally in abelian categories) used in algebraic topology, can be suitably extended to “non-abelian” structures such as groups, rings, loops, etc. Moreover, they are equivalent to properties which arise in the axiomatic study of these structures. For the five lemma this is well known, and in the present paper we establish this for the snake lemma and the lemma, which, when suitably formulated, turn out to be equivalent to each other for all (pointed) algebraic structures, and also in general categories of a special type. In particular, we show that among varieties of universal algebras, they are satisfied precisely in the so-called (pointed) ideal determined varieties. [Copyright &y& Elsevier]

Abstract: In this paper, we determine the lower central and derived series for the braid groups of the projective plane. We are motivated in part by the study of Fadell–Neuwirth short exact sequences, but the problem is interesting in its own right. The n-string braid groups of the projective plane were originally studied by Van Buskirk during the 1960s, and are of particular interest due to the fact that they have torsion. The group (resp. ) is isomorphic to the cyclic group of order 2 (resp. the generalised quaternion group of order 16) and hence their lower central and derived series are known. If , we first prove that the lower central series of is constant from the commutator subgroup onwards. We observe that is isomorphic to , where denotes the free group of rank k, and denotes the quaternion group of order 8, and that is an extension of an index 2 subgroup K of by . As for the derived series of , we show that for all , it is constant from the derived subgroup onwards. The group being finite and soluble for , the critical cases are . We are able to determine completely the derived series of . The subgroups , and are isomorphic respectively to , and , and we compute the derived series quotients of these groups. From onwards, the derived series of , as well as its successive derived series quotients, coincide with those of . We analyse the derived series of and its quotients up to , and we show that is a semi-direct product of by . Finally, we give a presentation of . [Copyright &y& Elsevier]

Abstract: A duality of two categories is introduced. It generalizes the Malcev description of torsion free finite rank abelian groups. An equivalence of two categories is also introduced. It generalizes the Kurosh description of p-primitive groups. The composition of the duality and the equivalence is the duality earlier introduced by W. Wickless and the author. It is shown that the last duality preserves exactness for short exact sequences of homomorphisms. [Copyright &y& Elsevier]