Searcharxiv⌕ Search

arXiv subjects

Quinn T. Kolt

Publications and source records attributed to Quinn T. Kolt.

4 recordsLinked to original sources
arXiv2025

On the formal ribbon extension of a quasitriangular Hopf algebra

Any finite-dimensional quasitriangular Hopf algebra $H$ can be formally extended to a ribbon Hopf algebra $\tilde H$ of twice the dimension. We investigate this extension and its representations. We show that every indecomposable $H$-module has precisely two compatible $\tilde H$-actions. We investigate the behavior of simple, projective, and Müger central $\tilde H$-modules in terms of these $\tilde H$-actions. We also observe that, in the semisimple case, this construction agrees with the pivotalization/sphericalization construction introduced by Etingof, Nikshych, and Ostrik (2003). As an example, we investigate the formal ribbon extension of odd-index doubled Nichols Hopf algebras $D\mathcal K_n$.

math.QA↗
arXiv2024

Modular data of non-semisimple modular categories

We investigate non-semisimple modular categories with an eye towards a structure theory, low-rank classification, and applications to low dimensional topology and topological physics. We aim to extend the well-understood theory of semisimple modular categories to the non-semisimple case by using representations of factorizable ribbon Hopf algebras as a case study. We focus on the Cohen-Westreich modular data, which is obtained from the Lyubashenko-Majid modular representation restricted to the Higman ideal of a factorizable ribbon Hopf algebra. The Cohen-Westreich $S$-matrix diagonalizes the mixed fusion rules and reduces to the usual $S$-matrix for semisimple modular categories. The paper includes detailed studies on small quantum groups $U_qsl(2)$ and the Drinfeld doubles of Nichols Hopf algebras, especially the $\mathrm{SL}(2, \mathbb{Z})$-representation on their centers, Cohen-Westreich modular data, and the congruence kernel theorem's validity.

math.QA↗
arXiv2024

Indecomposable Hopf $*$-algebra representations with invariant inner product

We generalize a result of Araki (1985) on indecomposable group representations with invariant (necessarily indefinite) inner product and irreducible subrepresentation to Hopf $*$-algebras. Moreover, we characterize invariant inner products on the projective indecomposable representations of small quantum groups $U_qsl(2)$ at odd roots of unity and on the indecomposable representations of generalized Taft algebras $H_{n,d}(q)$.

math.QA↗