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Deformation theory via deviations. (English) Zbl 0823.16027
Bureš, J. (ed.) et al., The proceedings of the Winter school Geometry and topology, Srní, Czechoslovakia, January 1992. Palermo: Circolo Matemático di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 32, 97-124 (1994).
The authors recall the bases of a cohomology theory controlling the deformations of an algebraic structure, especially for associative algebras (Hochschild cohomology) and for bialgebras (studying what they call the hypercomplex with Hochschild and coHochschild differentiation). Then they construct a cohomology theory for the so-called Drinfel’d algebras (quasi-bialgebras in Gerstenhaber and Schack’s terminology) and verify that it controls the deformations of these algebras, in particular that the obstructions to integrability of partial deformations lies in the $$H^ 3$$. The main tools are deviations that are defined at the beginning of the paper and an interesting use of a translation of the equations into deviation diagrams which enable the use of topological arguments.
For the entire collection see [Zbl 0794.00022].

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16S80 Deformations of associative rings 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)