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The geometric reductivity of the quantum group $SL_q(2)$

Volume 124 / 2011

Michał Kępa, Andrzej Tyc Colloquium Mathematicum 124 (2011), 169-190 MSC: Primary 16T05; Secondary 20G42. DOI: 10.4064/cm124-2-3

Abstract

We introduce the concept of geometrically reductive quantum group which is a generalization of the Mumford definition of geometrically reductive algebraic group. We prove that if $G$ is a geometrically reductive quantum group and acts rationally on a commutative and finitely generated algebra $A$, then the algebra of invariants $A^G$ is finitely generated. We also prove that in characteristic $0$ a quantum group $G$ is geometrically reductive if and only if every rational $G$-module is semisimple, and that in positive characteristic every finite-dimensional quantum group is geometrically reductive. Both the concept of geometrically reductive quantum group and the above mentioned theorems are formulated in the language of Hopf algebras and generalize the results of Borsai and Ferrer Santos. The main theorem of the paper says that in positive characteristic the quantum group $SL_q(2)$ is geometrically reductive for any parameter $q$.

Authors

  • Michał KępaFaculty of Mathematics and Computer Sciences
    N. Copernicus University
    Chopina 12/18
    87-100 Toruń, Poland
    e-mail
  • Andrzej TycFaculty of Mathematics and Computer Sciences
    N. Copernicus University
    Chopina 12/18
    87-100 Toruń, Poland
    e-mail

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