Degree
Doctor of Philosophy (PhD)
Department
Mathematics
Document Type
Dissertation
Abstract
Discriminant ideals are defined for an algebra R with central subalgebra C and trace tr : R → C. They are indexed by positive integers and more general than discriminants. Usually R is required to be a finite module over C. Unlike the abundace of work on discriminants, there is hardly any literature on discriminant ideals. The levels of discriminant ideals relate to the sums of squares of dimensions of irreducible modules over maximal ideals of C containing these discriminant ideals. We study the lowest level when R is a Cayley-Hamilton Hopf algebra, i.e. C is also a Hopf subalgebra, and all irreducible representations over the kernel of the counit of C are one-dimensional. The level is determined by considering the actions of these one-dimensional modules through tensor product and then we look into the orbits of elements in the zero set of the lowest discriminant ideal under winding automorphisms, which fix C. We then apply the results to the examples of the group algebra of certain central extensions of finitely generated Abelian groups by finite Abelian groups and the quantum Borel subalgebra at roots of unity.
Date
4-18-2024
Recommended Citation
Mi, Zhongkai, "The Lowest Discriminant Ideal of Cayley-Hamilton Hopf Algebras" (2024). LSU Doctoral Dissertations. 6416.
https://repository.lsu.edu/gradschool_dissertations/6416
Committee Chair
Achar, Pramod