A Hall-Fusion Bialgebra

We describe what might be called the “Hall-fusion” bialgebra constructed from a promonoidal double, and mention the corresponding face version for probicategories.

💡 Research Summary

The paper introduces a new algebraic construction called the “Hall‑fusion” bialgebra, obtained from a promonoidal double and, in a second version, from a pair of finite probicategories. The author begins with a finite skeletal Vectₖ‑category 𝔄 whose hom‑spaces are zero unless the source and target coincide. Two promonoidal structures are placed on 𝔄 and its opposite 𝔄ᵒᵖ: a functor
p : 𝔄ᵒᵖ ⊗ 𝔄ᵒᵖ ⊗ 𝔄 → Vect_fd
and a functor
q : 𝔄 ⊗ 𝔄 ⊗ 𝔄ᵒᵖ → Vect_fd.
These are the categorical analogues of a “multiplication” that takes three objects and returns a finite‑dimensional vector space. Their tensor product yields a promonoidal structure H on the product category 𝔄 ⊗ 𝔄ᵒᵖ.

From H the author defines a k‑linear algebra B whose basis elements are symbols e_J^{a,b} indexed by pairs (a,b) with a∈𝔄ᵒᵖ and b∈𝔄. The multiplication is given by
e_J^{a,c}·e_K^{b,d}=∑_{u,v} dim p(a,b,u)·dim q(c,d,v)·e_J^{u,v},
where the sums run over objects u∈𝔄ᵒᵖ and v∈𝔄, and dim p(a,b,u) denotes the dimension of the finite‑dimensional vector space p(a,b,u). The unit is the distinguished element e_K^{I,J}. The coproduct is defined by the familiar Hall‑type formula
Δ(e_J^{a,b})=∑u e_J^{a,u}⊗e_J^{u,b},
and the counit ε(e_J^{a,b}) is the Kronecker delta δ{a,b}.

For these operations to satisfy the bialgebra axioms, a compatibility condition between p and q is required: for every triple (a,b,u) the tensor product p(a,b,u)⊗q(a,b,v) must be naturally isomorphic to the hom‑space 𝔄ᵒᵖ(u,v). In other words, p and q must be dual to each other in a precise categorical sense. When this holds, the multiplication and comultiplication are compatible, and B becomes a bialgebra.

The paper then discusses the existence of an antipode. Suppose there is a functor S : 𝔄ᵒᵖ → 𝔄 with S²=Id_𝔄 (an involutive “antipode” on the underlying category). One can define an antipode on B by
S(e_J^{a,b}) = e_K^{S(b),S(a)}.
If the promonoidal data are chosen so that p(a,b,u)=q(S(b),S(a),S(u)) and the distinguished object I equals S(J), the von Neumann axiom
m ∘ (1⊗S⊗1) ∘ Δ³ = 1
holds. Consequently B satisfies all the axioms of a Hopf algebra, up to the usual technicalities. The author notes that the most immediate examples arise when 𝔄 is the k‑linearisation of a finite groupoid; in that case the construction reproduces familiar “double” Hopf algebras associated with groupoid algebras.

In Section 2 the author presents a “face” version, which replaces the promonoidal categories by finite probicategories (a bicategorical analogue of promonoidal structures). Two probicategories 𝔅 and 𝔅ᵒᵖ are taken over the same finite set N of 0‑cells. For each triple of 0‑cells (i,j,k) there are hom‑categories 𝔅_{ij}, 𝔅_{jk} and promultiplication functors p_{ijk} and q_{ijk}. Face idempotents e_i and e_j are defined as sums of the identity 1‑cells over the appropriate hom‑categories. The product of two basis elements e_J^{a,c} and e_K^{b,d} (where a,c∈𝔅_{ij} and b,d∈𝔅_{jk}) is again given by a double sum over intermediate objects u,v∈𝔅_{ik}, weighted by the dimensions of p_{ijk}(a,b,u) and q_{ijk}(c,d,v). The coproduct and counit are defined exactly as in the vertex version. Under the same duality condition between p and q, this yields a bialgebra; an antipode can be introduced analogously if an involutive functor S exists on the probicategories.

The author remarks that the finiteness requirement on objects can be relaxed provided the promultiplication functors have finite support, allowing the construction to extend to infinite categories or graphs with locally finite structure.

Overall, the paper provides a categorical framework that unifies Hall algebras, fusion constructions, and double‑type Hopf algebras via promonoidal (or probicategorical) data. It shows how the algebraic operations arise naturally from the combinatorics of the underlying categorical structures, and it outlines conditions under which a full Hopf algebra structure (including antipode) can be obtained. The work suggests further directions, such as applying the construction to more general bicategories, exploring connections with 2‑dimensional topological quantum field theories, and investigating representation‑theoretic aspects of the resulting Hopf algebras.