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Session S16 - Quantum symmetries

No date set.

Hopf actions of some quantum groups on path algebras

Amrei Oswald

University of Iowa, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

We investigate examples of quantum symmetry by studying Hopf actions of $U_q(\mathfrak{b})$, $U_q(\mathfrak{sl}_2)$, generalized Taft algebras, and the small quantum group on path algebras. We begin by parametrizing these actions using linear algebraic data. Then, we attempt to describe the "building blocks" of these actions by viewing path algebras as tensor algebras in the tensor category $\mathsf{rep}(H)$ for the appropriate quantum group $H$ and attempting to classify the minimal, faithful tensor algebras. We construct an equivalence between categories of bimodules in $\mathsf{rep}(H)$ and a subcategory of certain finite-dimensional representations of associative algebras, explicitly given in terms of quivers with relations. This allows us to determine whether classification of indecomposable bimodules in these categories is feasible based on the representation types of the quivers.

Joint work with Ryan Kinser (University of Iowa).

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