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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
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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.

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.