Bialgebra cohomology, deformations, and quantum groups.

Abstract

We introduce cohomology and deformation theories for a bialgebra A (over a commutative unital ring k) such that the second cohomology group is the space of infinitesimal deformations. Our theory gives a natural identification between the underlying k-modules of the original and the deformed bialgebra. Certain explicit deformation formulas are given for the construction of quantum groups-i.e., Hopf algebras that are neither commutative nor cocommutative (whether or not they arise from quantum Yang-Baxter operators). These formulas yield, in particular, all GLq(n) and SLq(n) as deformations of GL(n) and SL(n). Using a Hodge decomposition of the underlying cochain complex, we compute our cohomology for GL(n). With this, we show that every deformation of GL(n) is equivalent to one in which the comultiplication is unchanged, not merely on elements of degree one but on all elements (settling in the strongest way a decade-old conjecture) and in which the quantum determinant, as an element of the underlying k-module, is identical with the usual one. Section 1. Bialgebra Cohomology and the LaplacianLet A be a k-bialgebra with multiplication ,u and comultiplication A. We write A®" for the q-fold tensor product A k. ..

The Hilton-Heckmann argument for the anti-commutativity of cup products

“…It applies to the cohomology of groups, to the Hochschild cohomology of associative algebras, to the Cartier cohomology of coalgebras, and to other somewhat more exotic theories, such as the cohomology theory introduced by M. Gerstenhaber and S. D. Schack in [5] for Hopf bimodules over a Hopf algebra.…”

Hopf structures on the Hopf quiverQ(⟨g⟩,g)

“…The converse procedure is called deformation. According to Gerstenhaber and Schack [1990], a coalgebra-preserving deformation is called preferred. If we want to classify all the Hopf structures on the whole path coalgebra of a Hopf quiver, or the bialgebras of type one [Nichols 1978], then we only need to carry out preferred deformation procedure.…”

“…Aside from quiver techniques, our arguments also rely on Bergman's diamond lemma [1978], which helps to present the Hopf algebras by generators with relations. This is useful in carrying out the preferred deformation procedure in the sense of Gerstenhaber and Schack [1990].…”

Closedness of Star Products and Cohomologies

“…and here also an appropriate cohomology can be introduced [34][35][36]. In the case of Hopf algebras, the deformed algebras will have the same unit and counit, but in general not the same antipode.…”