Posets of finite GK-dimensional graded pre-Nichols algebras of diagonal type

Abstract

We classify graded pre-Nichols algebras of diagonal type with finite Gelfand–Kirillov dimension over an algebraically closed field of characteristic zero. The characterization is made through an isomorphism of posets with a family of appropriate subsets of the set of positive roots of a semisimple Lie algebra attached to the Nichols algebra. The relation between this Lie algebra and the Nichols algebra is that the algebra of functions of the corresponding unipotent group appears in a central extension of the Nichols algebra, generalizing the corresponding extensions for small quantum groups in de Concini–Kac–Procesi forms of quantum groups. On the way to achieving this result, we also classify graded quotients of algebras of functions of unipotent algebraic groups attached to semisimple Lie algebras.