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Drinfel’d algebra deformations, homotopy comodules and the associahedra. (English) Zbl 0891.16025

By a Drinfel’d algebra the authors mean a quasi-bialgebra, i.e. an (associative unital) algebra \((V,\cdot)\) which is also an almost coassociative counital coalgebra \((V,\delta)\) (such that \((1\otimes\delta)\delta\cdot\Phi=\Phi\cdot(\delta\otimes 1)\delta\)) with both structures compatible in the bialgebra sense. The authors introduce a deformation theory for Drinfel’d algebras that generalizes the Gerstenhaber-Schack deformation theory for bi-algebras. Since the description of deformations of associative algebras resp. bialgebras by cohomology groups intimately connects associativity and the definition of the underlying complex of the cohomology, it is clear, that the condition of “almost coassociativity” extremely complicates the construction of a suitable complex and cohomology. A modified cobar construction, the notion of a homotopy comodule structure, and a differential graded Lie algebra structure on the simplicial chain complex of the associahedra are used to handle the new cohomology and lead to a very technical paper. Although the authors have shifted some of the technical calculations to an appendix they write at the end of the proof that their cohomology controls the deformations of the Drinfel’d algebra: “This straighforward computation (of the obstructions) would stretch the length of the paper beyond any reasonable limit and is therefore omitted”.

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16S80 Deformations of associative rings
17B37 Quantum groups (quantized enveloping algebras) and related deformations
18G60 Other (co)homology theories (MSC2010)
17B56 Cohomology of Lie (super)algebras
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