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Seminario del 2019
2019
20 giugno
Cyclic Gerstenhaber-Schack cohomology"
"In this talk, we answer a long-standin question by explaining how the diagonal complex computing
the Gerstenhaber-Schack cohomology of a bialgebra (that is, the cohomology theory governing bialgebra deformations)
can be given the structure of an operad with multiplication if the bialgebra is a (not
necessarily finite dimensional) Hopf algebra with invertible antipode; if the antipode is
involutive, the operad is even cyclic. Therefore, the Gerstenhaber-Schack cohomology of
any such Hopf algebra carries a Gerstenhaber resp. Batalin-Vilkovisky algebra structure;
in particular, one obtains a cup product and a cyclic boundary B that generate the Gerstenhaber
bracket, and that allows to define cyclic Gerstenhaber-Schack cohomology. In case
the Hopf algebra in question is finite dimensional, the Gerstenhaber bracket turns out to
be zero in cohomology and hence the interesting structure is not given by this e2-algebra
structure but rather by the resulting e3-algebra structure, which is expressed in terms of
the cup product and B."