Yetter-Drinfeld modules over Nichols systems: reflections, induced objects and maximal subobject. (English) Zbl 07808423
In this paper the author uses the notion of Nichols system as studied in his previous work [Commun. Algebra 51, No. 2, 822–840 (2023; Zbl 1521.16019)] in order to study the category of Yetter-Drinfeld modules over these kind of braided Hopf algebras and generalize results about Verma modules in the representation theory of Lie algebras to Nichols algebras of arbitrary group type. The results rely on properties of a special morphism, called Shapovalov morphism, which generalizes a polynomial called Shapovalov determinant. Using the roots of a Nichols system, he obtains some properties about the geometry of the support of Nichols systems and their Yetter-Drinfeld modules, by looking at iterated reflections. A construction reminiscent of Verma modules in the representation theory of Lie algebras is performed by inducing comodules of the Nichols system to Yetter-Drinfeld modules over the Nichols system. The behavior of reflections on these induced objects is discussed, entailing a characterization of irreducibility via reflectiveness, if the Nichols system is finite dimensional. A description of the maximal subobject and irreducibility of Yetter-Drinfeld modules over Nichols systems is given.
Reviewer: Sonia Natale (Córdoba)
MSC:
| 16T05 | Hopf algebras and their applications |
| 18M15 | Braided monoidal categories and ribbon categories |
Keywords:
braided monoidal category; Hopf algebra; Nichols algebra; Shapovalov determinant; Yetter-Drinfeld moduleCitations:
Zbl 1521.16019References:
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