Homology and cohomology with coefficients, of an algebra over a quadratic operad. (English) Zbl 0967.18004
The notion of homology for an algebra over an operad was introduced by Ginzburg and Kapranov. Their work was continued by Kimura-Voronov and Markl-Fox for cohomology with coefficients.
The author extends their work, defining homology with coefficients. This extension is not obvious and requires the introduction of suitable coefficients, called corepresentations by the author, which are in general different from the cohomology coefficients.
All constructions, and in particular the differential in homology and cohomology, are presented very explicitly, using formulae in terms of the “comp-operations” of the operad. The case of the dual of the Leibniz algebra is given as a detailed example.
The author extends their work, defining homology with coefficients. This extension is not obvious and requires the introduction of suitable coefficients, called corepresentations by the author, which are in general different from the cohomology coefficients.
All constructions, and in particular the differential in homology and cohomology, are presented very explicitly, using formulae in terms of the “comp-operations” of the operad. The case of the dual of the Leibniz algebra is given as a detailed example.
Reviewer: Nathalie Wahl (Oxford)
MSC:
| 18D50 | Operads (MSC2010) |
| 17A32 | Leibniz algebras |
| 18G60 | Other (co)homology theories (MSC2010) |
Keywords:
homology for algebra over operad; operads; cohomology with coefficients; homology with coefficients; corepresentations; comp-operations; Leibniz algebraReferences:
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