[Paper Review] Weak Crossed Biproducts and Weak Projections

J. M. Fernández Vilaboa, R. González Rodrı́guez+1 more|ArXiv.org|Jun 9, 2009

Algebraic structures and combinatorial models1 references4 citations

TL;DR

This paper introduces weak crossed biproducts as a general framework for constructing weak bialgebras with weak projections, extending classical crossed product structures to the weak Hopf algebra setting. It proves that every weak projection in a weak bialgebra induces a weak crossed biproduct structure, and explicitly computes this structure for groupoids with exact factorizations.

ABSTRACT

We present the universal theory of weak crossed biproducts, and we prove that every weak projection of weak bialgebras induces an example of this crossed structure. As an example, we give the construction of a weak projection of a weak bialgebra associated to a groupoid that admits an exact factorization.

Motivation & Objective

• To develop a general theory of weak crossed biproducts that extends classical crossed product constructions to the setting of weak bialgebras and weak Hopf algebras.
• To address the limitations of existing crossed product frameworks in the weak setting, where standard unit and counit properties fail.
• To characterize weak crossed biproducts via universal properties and identify conditions under which they arise from weak projections.
• To provide explicit constructions of weak crossed biproducts for groupoids admitting exact factorizations.
• To unify various known weak bialgebra constructions—such as bicrossproducts and quantum groupoid structures—under a single categorical framework.

Proposed method

• Define weak crossed biproducts in a braided monoidal category using a preunit ν and precounit υ, with associated idempotent ∇C⊗B on C⊗B.
• Construct the algebra and coalgebra structures on C⊗B using twisted product and coproduct maps satisfying cocycle-like and twisted compatibility conditions.
• Utilize the categorical framework of Bespalov and Drabant to generalize cross product and coproduct structures without explicit cocycles.
• Introduce morphisms f:B→D and g:D→B such that g∘f=idB, with f an algebra and coalgebra morphism and g a coalgebra morphism and B-module morphism.
• Establish the universal property of the weak crossed biproduct via isomorphism α:C×B→D, where C×B is the image of ∇C⊗B.
• Verify that the induced structure satisfies the weak bialgebra axioms and that the morphism g is a weak projection via verification of ∇C⊗B properties and counit/multiplication compatibility.

Experimental results

Research questions

• RQ1 How can classical crossed product and coproduct structures be generalized to the weak Hopf algebra setting where units and counits are not multiplicative or comultiplicative?
• RQ2 What universal properties characterize weak crossed biproducts in the context of weak bialgebras?
• RQ3 Under what conditions does a weak projection in a weak bialgebra give rise to a weak crossed biproduct structure?
• RQ4 How can the weak crossed biproduct be explicitly computed for groupoids that admit an exact factorization?
• RQ5 What role do idempotent morphisms and preunits/precounits play in structuring weak crossed biproducts in braided monoidal categories?

Key findings

• Every weak projection (f,g) of a weak bialgebra D onto a weak Hopf algebra B induces a weak crossed biproduct structure on D.
• The weak crossed biproduct is constructed via a preunit ν and precounit υ, with associated idempotent ∇C⊗B on C⊗B, where C×B is the image of ∇C⊗B.
• The isomorphism α:C×B→D realizes D as a weak crossed biproduct with algebra and coalgebra structures given by the smash product and cosmash coproduct, respectively.
• The morphism f:B→D is an algebra and coalgebra morphism, and g:D→B is a coalgebra morphism and a right B-module morphism, satisfying g∘f=idB.
• The conditions δD∘f=(f⊗f)∘δB and τB^C=(εC⊗δB)∘∇C⊗B ensure that f is comultiplicative and g is compatible with the coalgebra structure.
• For a groupoid with an exact factorization, the weak crossed biproduct structure is explicitly computed using the induced preunit and precounit from the factorization.