Quantum groups of finite representation type

Abstract: Algebras of finite representation type - that is, those who have only finitely many indecomposable finite dimensional representations up to isomorphism - have been of central interest in representation theory. Classically, they appeared from work in modular representations; a (finite) group has finite representation type iff its p-Sylow subgroup is cyclic. On the quantum side, results of Farnsteiner describe the structure of finite group schemes (finite dimensional co-commutative Hopf algebras). Among the first examples of non-commutative and non-cocommutative quantum groups (Hopf algebras) are the Sweedler algebra and Taft algebras. These are pointed (their simple modules are 1-dimensional - they are points), and they are also of finite type.

To study this in the generality of infinite dimensional quantum groups (which includes gl_n, quantum sl, etc.), one defines an algebra to be of finite type if given any dimension vector, there are only finitely many indecomposables of this dimension vector; by the well known Brower-Thrall problems, this is equivalent to the above for finite dimensional algebras. We give an overview of various examples of infinte quantum groups of finite type, and give a complete classification of the pointed quantum groups of finite representation type. We re-obtain results known for the finite dimensional case (including Taft algebras and their generalizations), and show that these include several interesting Hopf algebras, such as those whose categories of comodules form the category of chain complexes or the category of double chain complexes, and in general, the list includes these and certain kind of twists and deformations of theirs.

rings and algebrasrepresentation theory

Audience: researchers in the topic

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