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The Koszul complex, as introduced in 1950, was a differential graded algebra which modelled a principal fibre bundle. Since then it has been an effective tool, both in algebra and in topology, for the calculation of homological and homotopical invariants. After a partial summary of these results we recall more recent generalizations of this complex, and some applications.
The notions of left-right noncommutative Poisson algebra (NPlr-algebra) and left-right algebra with bracket AWBlr are introduced. These algebras are special cases of NLP-algebras and algebras with bracket AWB, respectively, studied earlier. An NPlr-algebra is a noncommutative analogue of the classical Poisson algebra. Properties of these new algebras are studied. In the categories AWBlr and NPlr-algebras the notions of actions, representations, centers, actors and crossed modules are described as...
For any Lie-Rinehart algebra , B(atalin)-V(ilkovisky) algebra structures on the exterior -algebra correspond bijectively to right -module structures on ; likewise, generators for the Gerstenhaber algebra correspond bijectively to right -connections on . When is projective as an -module, given a B-V algebra structure on , the homology of the B-V algebra coincides with the homology of with coefficients in with reference to the right -module structure determined by . When...
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