Yetter-Drinfeld modules over Nichols systems and their reflections

TL;DR

This paper develops reflection functors for Yetter-Drinfeld modules over Nichols systems, explores their geometric and algebraic properties, and connects the reflection theory to classical structures like Dynkin diagrams and Shapovalov determinants.

Contribution

It introduces reflection functors for Yetter-Drinfeld modules over Nichols systems and relates their properties to the geometry of Nichols systems and classical Lie algebra theory.

Findings

Explicit formula for the Shapovalov morphism in group type Nichols systems

Characterization of irreducibility via Shapovalov determinants

Connection between reflection theory and Dynkin diagrams

Abstract

We construct reflection functors for Yetter-Drinfeld modules over Nichols systems and discuss their fundamental properties. We will obtain properties about the geometry of the support of Nichols systems and their Yetter-Drinfeld modules, by looking at iterated reflections. We will also study the maximal subobject of Yetter-Drinfeld modules over Nichols systems and find a special morphism, that we name Shapovalov morphism, whose kernel coincides with this maximal subobject. Moreover, we will use this morphism to characterize properties about the reflections of the Yetter-Drinfeld modules. We calculate an explicit formula of the Shapovalov morphism in the case where the Nichols system is of group type. We will use the formula to calculate its kernel in the components of degree $2$ and to ascribe the theory of reflections of Yetter-Drinfeld modules over Nichols systems of diagonal type with the reflection theory of Dynkin diagrams. We will also apply and specify the theory to the Yetter-Drinfeld modules over Nichols systems that are obtained by inducing comodules of the Nichols systems, a construction that is reminiscent of Verma modules in the representation theory of Lie algebras. For Nichols systems of diagonal type we will obtain the result, that such an induced objects irreducibility can be characterized by a polynomial that is given by the positive roots of its Nichols system. This polynomial appeared in previous works and is known as Shapovalov determinant. Finally, we will apply the theory to some specific examples of Nichols algebras of nonabelian group type and to some examples of Nichols algebras of diagonal type.