Maximal orders are noncommutative analogs of integrally closed rings. They have nice structures and their representation theory is more or less understood. There has been a great interest in proving that quantum algebras at roots of unity that appear in Lie theory and topology are maximal orders with the aim of classifying their irreducible representations. By work of De Concini, Kac, Lyubashenko, and Procesi it is known that quantum groups and quantum algebras of functions on complex simple Lie groups at roots of unity are maximal orders. The skein algebra of a surface is a generalization of the quantum algebras of functions on SL_2.
We will show that the skein algebra of a connected surface with non-empty boundary is a maximal order. Partly based on joint work with S. Huang, J. Paprocki, M. Yakimov, and T. Yu.
