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Bicovariant quantum algebras and quantum Lie algebras

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Abstract

A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun\((\mathfrak{G}_q )\) toU q g, given by elements of the pure braid group. These operators—the “reflection matrix”Y≡L + SL − being a special case—generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation forY inSO q (N).

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Authors and Affiliations

  1. Department of Physics, University of California, 1 Cyclotron Road, 94720, Berkeley, California, USA

    Peter Schupp, Paul Watts & Bruno Zumino

  2. Theoretical Physics Group, Physics Division, Lawrence Berkeley Laboratory, 1 Cyclotron Road, 94720, Berkeley, California, USA

    Peter Schupp, Paul Watts & Bruno Zumino

Authors
  1. Peter Schupp
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  2. Paul Watts
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  3. Bruno Zumino
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Communicated by A. Jaffe

This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY90-21139

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Schupp, P., Watts, P. & Zumino, B. Bicovariant quantum algebras and quantum Lie algebras. Commun.Math. Phys. 157, 305–329 (1993). https://doi.org/10.1007/BF02099762

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