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Communications in Mathematical Physics

, Volume 167, Issue 3, pp 635–670 | Cite as

Classification of bicovariant differential calculi on quantum groups of type A, B, C and D

  • Konrad Schmüdgen
  • Axel Schüler
Article

Abstract

Under the assumptions thatq is not a root of unity and that the differentialsdu j i of the matrix entries span the left module of first order forms, we classify bicovariant differential calculi on quantum groupsA n−1 ,B n ,C n andD n . We prove that apart one dimensional differential calculi and from finitely many values ofq, there are precisely2n such calculi on the quantum groupA n−1 =SL q (n) forn≧3. All these calculi have the dimensionn2. For the quantum groupsB n ,C n andD n we show that except for finitely manyq there exist precisely twoN2-dimensional bicovariant calculi forN≧3, whereN=2n+1 forB n andN=2n forC n ,D n . The structure of these calculi is explicitly described and the corresponding ad-invariant right ideals of ker ε are determined. In the limitq→1 two of the 2n calculi forA n−1 and one of the two calculi forB n ,C n andD n contain the ordinary classical differential calculus on the corresponding Lie group as a quotient.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Konrad Schmüdgen
    • 1
  • Axel Schüler
    • 1
  1. 1.Fachbereich Mathematik/InformatikUniversität LeipzigLeipzigGermany

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