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2 results

  • Coideal Subalgebras and Quantum Symmetric Pairs

  • Edited by Susan Montgomery, University of Southern California, Hans-Jurgen Schneider, Universität Munchen
  • Mathematical Sciences Research Institute
  • Book:
    New Directions in Hopf Algebras
    Published online:
    25 June 2025
    Print publication:
    06 May 2002, pp 117-166

  • Summary

    Coideal subalgebras of the quantized enveloping algebra are surveyed, with selected proofs included. The first half of the paper studies generators, Harish-Chandra modules, and associated quantum homogeneous spaces. The second half discusses various well known quantum coideal subalgebras and the implications of the abstract theory on these examples. The focus is on the locally finite part of the quantized enveloping algebra, analogs of enveloping algebras of nilpotent Lie subalgebras, and coideals used to form quantum symmetric pairs. The last family of examples is explored in detail. Connections are made to the construction of quantum symmetric spaces.

    The introduction of quantum groups in the early 1980's has had a tremendous influence on the theory of Hopf algebras. Indeed, quantum groups provide a source of new and interesting examples. We shall discuss the reverse impact: the theory of quantum groups uses the Hopf structure extensively. This special structure is often hidden in the classical setting, while it is prominent and fundamental for quantum analogs.

    Let g be a semisimple Lie algebra and write G for the corresponding connected, simply connected algebraic group. There are two standard types of quantum groups associated to g and G. The first is the quantized enveloping algebra which is a quantum analog of the enveloping algebra of g. The second is the quantized function algebra which is a quantum analog of the algebra of regular functions on G. We will be focusing on a particular aspect of the Hopf theory of both types of quantum groups: the study of (one-sided) coideal subalgebras.

  • Subalgebras Which Appear in Quantum Iwasawa Decompositions

  • Gail Letzter
  • Journal:
    Canadian Journal of Mathematics / Volume 49 / Issue 6 / 01 December 1997
    Published online by Cambridge University Press:
    20 November 2018, pp. 1206-1223
    Print publication:
    01 December 1997
    • Article
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  • Let g be a semisimple Lie algebra. Quantum analogs of the enveloping algebra of the fixed Lie subalgebra are introduced for involutions corresponding to the negative of a diagram automorphism. These subalgebras of the quantized enveloping algebra specialize to their classical counterparts. They are used to form an Iwasawa type decompostition and begin a study of quantum Harish-Chandra modules.