Bialgebra cohomology, deformations, and quantum groups.

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ID: 97350
1990
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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.
Reference Key
gerstenhaber1990bialgebraproceedings Use this key to autocite in the manuscript while using SciMatic Manuscript Manager or Thesis Manager
Authors Gerstenhaber, M;Schack, S D;
Journal Proceedings of the National Academy of Sciences of the United States of America
Year 1990
DOI
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Keywords
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