Browsing by Subject "Hopf"
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Item On simple modules for certain pointed Hopf algebras(Texas A&M University, 2007-04-25) Pereira Lopez, MarianaIn 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.Item Two theorems related to group schemes(2010-12) Jones, James Hunter, 1982-; Voloch, José Felipe; Helm, DavidAfter presenting some preliminary information, this paper presents two proofs regarding group schemes. The first relates the category of affine group schemes to the category of commutative Hopf algebras. The second shows that a commutative group scheme of finite order is in fact killed by its order.