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Bialgebra cohomology, deformations, and quantum groups.

M Gerstenhaber1, S D Schack

  • 1Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA.

Proceedings of the National Academy of Sciences of the United States of America
|January 1, 1990
PubMed
Summary

We developed new theories for bialgebra deformations, identifying the second cohomology group with infinitesimal deformations. This work provides explicit formulas for quantum groups like GLq(n) and SLq(n), confirming a long-standing conjecture about their structure.

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ON CANONICAL CONSTRUCTIONS. I.

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Area of Science:

  • Algebraic Topology
  • Quantum Algebra
  • Mathematical Physics

Background:

  • Bialgebras and Hopf algebras are fundamental algebraic structures.
  • Deformation theory studies how mathematical objects change.
  • Quantum groups are non-commutative and non-cocommutative Hopf algebras with applications in various fields.

Purpose of the Study:

  • To introduce cohomology and deformation theories for bialgebras.
  • To provide explicit deformation formulas for constructing quantum groups.
  • To investigate deformations of general linear (GL) and special linear (SL) groups.

Main Methods:

  • Development of cohomology theory for bialgebras.
  • Application of Hodge decomposition to cochain complexes.
  • Explicit construction of deformation formulas for quantum groups.

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Main Results:

  • The second cohomology group of a bialgebra is identified as the space of infinitesimal deformations.
  • Natural identification between original and deformed bialgebra k-modules.
  • All GLq(n) and SLq(n) are obtained as deformations of GL(n) and SL(n).

Conclusions:

  • Every deformation of GL(n) is equivalent to one with an unchanged comultiplication.
  • The quantum determinant is identical to the usual determinant in the deformed structure.
  • Settles a decade-old conjecture regarding the structure of quantum group deformations.