Pereira Lopez, Mariana (2006-12). On simple modules for certain pointed Hopf algebras. Doctoral Dissertation. Thesis uri icon

abstract

  • In 2003, Radford introduced a new method to construct simple modules for the Drinfel??????d double of a graded Hopf algebra. Until then, simple modules for such algebras were usually constructed by taking quotients of Verma modules by maximal submodules. This new method gives a more explicit construction, in the sense that the simple modules are given as subspaces of the Hopf algebra and one can easily find spanning sets for them. I use this method to study the representations of two types of pointed Hopf algebras: restricted two-parameter quantum groups, and the Drinfel??????d double of rank one pointed Hopf algebras of nilpotent type. The groups of group-like elements of these Hopf algebras are abelian; hence, they fall among those Hopf algebras classified by Andruskiewitsch and Schneider. I study, in particular, under what conditions a simple module can be factored as the tensor product of a one dimensional module with a module that is naturally a module for a special quotient. For restricted two-parameter quantum groups, given ???, a primitive ??????th root of unity, the factorization of simple u???,y,???,z (sln)-modules is possible, if and only if gcd((y ?????? z)n, ??????) = 1. I construct simple modules using the computer algebra system Singular::Plural and present computational results and conjectures about bases and dimensions. For rank one pointed Hopf algebras, given the data D = (G, ????, a), the factorization of simple D(HD)-modules is possible if and only if |????(a)| is odd and |????(a)| = |a| = |????|. Under this condition, the tensor product of two simple D(HD)-modules is completely reducible, if and only if the sum of their dimensions is less or equal than |????(a)| + 1.
  • In 2003, Radford introduced a new method to construct simple modules for
    the Drinfel??????d double of a graded Hopf algebra. Until then, simple modules for such
    algebras were usually constructed by taking quotients of Verma modules by maximal
    submodules. This new method gives a more explicit construction, in the sense that
    the simple modules are given as subspaces of the Hopf algebra and one can easily
    find spanning sets for them. I use this method to study the representations of two
    types of pointed Hopf algebras: restricted two-parameter quantum groups, and the
    Drinfel??????d double of rank one pointed Hopf algebras of nilpotent type. The groups of
    group-like elements of these Hopf algebras are abelian; hence, they fall among those
    Hopf algebras classified by Andruskiewitsch and Schneider. I study, in particular,
    under what conditions a simple module can be factored as the tensor product of
    a one dimensional module with a module that is naturally a module for a special
    quotient. For restricted two-parameter quantum groups, given ???, a primitive ??????th root
    of unity, the factorization of simple u???,y,???,z (sln)-modules is possible, if and only if
    gcd((y ?????? z)n, ??????) = 1. I construct simple modules using the computer algebra system
    Singular::Plural and present computational results and conjectures about bases
    and dimensions. For rank one pointed Hopf algebras, given the data D = (G, ????, a),
    the factorization of simple D(HD)-modules is possible if and only if |????(a)| is odd and
    |????(a)| = |a| = |????|. Under this condition, the tensor product of two simple D(HD)-modules is completely reducible, if and only if the sum of their dimensions is less or
    equal than |????(a)| + 1.

publication date

  • December 2006