Half-quantum groups at roots of unity, path algebras and representation type

TL;DR

This paper classifies the representation type of finite dimensional half-quantum groups at roots of unity, showing most are of wild type and relating their algebraic structure to path algebras of Cayley graphs.

Contribution

It establishes the wild representation type for most half-quantum groups at roots of unity and links their algebraic structure to path algebras of Cayley graphs, providing explicit isomorphisms.

Findings

Most half-quantum groups at roots of unity are of wild representation type.

The underlying algebra is isomorphic to a quotient of a path algebra of a Cayley graph.

An explicit isomorphism is described using a quantum Fourier transform.

Abstract

We show that finite dimensional half-quantum groups at roots of unity corresponding to simple Lie algebras having symmetric Cartan matrix are of wild representation type, except for sl_2. Moreover, the underlying associative algebra is isomorphic to an admissible quotient of the path algebra of the Cayley graph of an abelian group. A quantum type Fourier transform enables to describe an explicit isomorphism.