[Paper Review] Yetter-Drinfeld modules over bosonizations of dually paired Hopf algebras

I. Heckenberger, Henriette Schneider|arXiv (Cornell University)|Nov 20, 2011

Algebraic structures and combinatorial models3 references3 citations

TL;DR

This paper establishes a braided monoidal isomorphism between rational Yetter-Drinfeld modules over the bosonizations of dually paired Hopf algebras in the Yetter-Drinfeld category over a Hopf algebra H with bijective antipode. The isomorphism provides a natural categorical framework that yields a conceptual proof of the existence of reflections in Nichols algebras of semisimple Yetter-Drinfeld modules, and shows that the reflection operator arises from the bosonization of the dual pair via the isomorphism, leading to a new derivation of the Weyl groupoid structure.

ABSTRACT

Let $(R^{\vee},R)$ be a dual pair of Hopf algebras in the category of Yetter-Drinfeld modules over a Hopf algebra $H$ with bijective antipode. We show that there is a braided monoidal isomorphism between rational left Yetter-Drinfeld modules over the bosonizations of $R$ and of $R^{\vee}$, respectively. As an application of this very general category isomorphism we obtain a natural proof of the existence of reflections of Nichols algebras of semisimple Yetter-Drinfeld modules over $H$. Key words: Hopf algebras, quantum groups, Weyl groupoid

Motivation & Objective

• To establish a braided monoidal isomorphism between rational left Yetter-Drinfeld modules over the bosonizations of dually paired Hopf algebras in the Yetter-Drinfeld category over H.
• To provide a conceptual, natural proof of the existence of reflection operators on Nichols algebras of semisimple Yetter-Drinfeld modules.
• To show that the reflection operation corresponds to the application of a braided monoidal functor arising from the duality of the underlying Hopf algebras.
• To demonstrate that the bosonization of the dual pair induces an isomorphism between Nichols algebras of the original and reflected modules.
• To re-derive the Weyl groupoid structure of a Nichols algebra using the categorical isomorphism, offering a new perspective on its construction.

Proposed method

• Construct a monoidal isomorphism between right and left relative Yetter-Drinfeld modules using the duality of the Hopf algebras R and R∨ in the Yetter-Drinfeld category over H.
• Prove that this isomorphism induces a braided monoidal equivalence between the Drinfeld centers of the categories of relative Yetter-Drinfeld modules.
• Show that the isomorphism preserves the subcategories of rational Yetter-Drinfeld modules over the bosonizations R#H and R∨#H.
• Define the functor (Ω, ω) as a braided monoidal isomorphism between the categories of rational left Yetter-Drinfeld modules over R#H and R∨#H.
• Use the isomorphism to relate the coinvariant algebra K_i^M of the Nichols algebra B(M) to the Nichols algebra of the reflected module via bosonization.
• Apply the isomorphism to show that Ω(K_i^M) # B(M_i*) ≅ B(R_i(M)) as braided Hopf algebras in the Yetter-Drinfeld category over H.

Experimental results

Research questions

• RQ1 Does a braided monoidal isomorphism exist between the categories of rational Yetter-Drinfeld modules over the bosonizations of dually paired Hopf algebras in the Yetter-Drinfeld category over H?
• RQ2 Can this isomorphism provide a natural explanation for the existence of reflection operators on Nichols algebras of semisimple Yetter-Drinfeld modules?
• RQ3 How is the reflection of a semisimple Yetter-Drinfeld module related to the duality of the underlying Hopf algebras in the bosonization construction?
• RQ4 Does the isomorphism (Ω, ω) preserve the Hopf algebra structure when applied to coinvariant algebras and their bosonizations?
• RQ5 Can the Weyl groupoid of a Nichols algebra be reconstructed using the categorical isomorphism between the bosonized modules?

Key findings

• There exists a braided monoidal isomorphism (Ω, ω) between the categories of rational left Yetter-Drinfeld modules over the bosonizations R#H and R∨#H.
• The isomorphism (Ω, ω) maps Hopf algebras to Hopf algebras and satisfies Ω(X) = X as H-modules for any X in the category.
• The coinvariant algebra K_i^M of the Nichols algebra B(M) with respect to M_i is a Hopf algebra in the category of rational Yetter-Drinfeld modules over R#H.
• The isomorphism satisfies Ω(K_i^M) # B(M_i*) ≅ B(R_i(M)) as braided Hopf algebras in the Yetter-Drinfeld category over H.
• The reflection operator R_i(M) arises naturally from the isomorphism (Ω, ω) applied to the coinvariant algebra and its dual, providing a new derivation of the Weyl groupoid structure.
• The construction confirms that R_i^2(M) ≅ M and that the generalized Cartan matrix is preserved under reflection, as shown in Corollary 8.11.