Dissertation
Equivariant Morita equivalence of Loewy-graded comodule algebras
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2026
DOI: 10.25820/etd.008446
Abstract
We study the representation theory of tensor categories associated to non-semisimple Hopf algebras through the exact module categories for which ${}_H\M$ acts on. Every such category is necessarily equivalent to the finite-dimensional representations of a finite-dimensional AM-exact $H$-comodule algebra $A$. When $H$ has the dual Chevalley property, a method inspired by the lifting method for Hopf algebras was developed to classify the isomorphism classes of AM-exact $H$-comodule algebras. We conjecture that this method works up to $H_0$-equivariant Morita equivalence of the largest semisimple subalgebra in $A$. In particular, we prove that when $H$ is a coradically graded Hopf algebra, every Loewy-graded $H$-comodule algebra $A$ and $H(0)$-equivariant Morita equivalence in degree zero is realized by the restriction of an $H$-equivariant Morita equivalence of Loewy-graded $H$-comodule algebras. We also show that when $A(0)$ is $H(0)$-equivariant Morita equivalent to a coideal subalgebra of $H(0)$, then $A$ is $H$-equivariant Morita equivalent to a coideal subalgebra.
As an application, we discuss the exact module categories over the Kac-Paljutkin Hopf algebra, $\hkp$, through explicit computation. We provide a new classification of indecomposable exact ${}_{\hkp}\M$-module categories and show that for every finite-dimensional Nichols algebra $\B(V)\in \yd{\hkp}$, every Loewy-graded AM-exact $\B(V)\# \hkp$-comodule algebras is equivariant Morita equivalent to a coideal subalgebra of $\B(V)\#\hkp$.
Details
- Title: Subtitle
- Equivariant Morita equivalence of Loewy-graded comodule algebras
- Creators
- Jacob Van Grinsven
- Contributors
- Ryan Kinser (Advisor)Frauke Bleher (Committee Member)Benjamin Cooper (Committee Member)Mohammad Farajzadeh Tehrani (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2026
- DOI
- 10.25820/etd.008446
- Publisher
- University of Iowa
- Number of pages
- vi, 66 pages
- Copyright
- Copyright 2026 Jacob Van Grinsven
- Language
- English
- Date submitted
- 04/20/2026
- Description illustrations
- Illustrations
- Description bibliographic
- Includes bibliographical references (pages 65-66).
- Public Abstract (ETD)
Representation theory allows us to study mathematical objects by asking questions about symmetry. In particular, the symmetries of any object are organized in the abstract mathematical structure of a group. Groups appear not only in mathematics but also in physics where (for example) groups can be used to describe the structure of elementary particles and conservation laws. Although a group encodes these symmetries, we find that the notion of a group is somewhat rigid, only describing classical symmetries and failing to see “quantum symmetries”.
A finite-dimensional Hopf algebra provides a linear-algebraic generalization of a finite group that has enough structure to describe these generalized symmetries. The Hopf algebra tracks both how these symmetries compose as well as how they decompose. With this perspective, the symmetries organized by Hopf algebras can detect the simultaneous symmetries of pairs of interacting objects via the tensor product. For groups, the tensor product detects simultaneous symmetries on pairs of objects but for quantum groups (noncommutative, non-cocommutative Hopf algebras), we find delicate structures consisting of interdependent systems of quantum symmetries.
The work in this paper studies these generalized symmetries through the study of comodule algebras, which are algebraic structures accompanied with some compatible quantum symmetry. Mombelli provided (in [17]) a general method for describing the structure of these comodule algebras for a large class of Hopf algebras. We note that many different comodule algebras may describe the same generalized symmetry. Thus this method, when performed naively, could result in significant redundant work. With this in mind, the main result of this paper suggests that if we are only concerned with the generalized symmetries induced by comodule algebras, we can work “up to equivalence” along the first step of Mombelli’s lifting procedure. In particular we show that for graded comodule algebras, the largest semisimple subalgebra can be chosen freely (with respect to the coradical of H). In light of this result, we conjecture that the remaining step of the comodule lifting procedure works “up to equivalence” as well.
- Academic Unit
- Mathematics
- Record Identifier
- 9985176975602771
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