Braided cohomology of quasi-triangular bialgebras and braided Morita invariance

Technical Summary: Braided Cohomology of Quasi-Triangular Bialgebras and Braided Morita Invariance

Problem Statement
The paper addresses the generalization of symmetric cohomology, originally defined for groups by Staic and extended to cocommutative Hopf algebras by Shiba, Sanada, and Itaba, to the setting of quasi-triangular bialgebras. In the classical and cocommutative settings, the cochain complex involves degreewise actions of symmetric groups derived from the flip morphisms in the symmetric monoidal category of modules. However, for quasi-triangular bialgebras, the relevant structure is a braiding (an $R$-matrix) rather than a symmetry, and the underlying category is a braided monoidal category rather than a symmetric one. The authors aim to define a "braided cohomology" for braided coalgebras in arbitrary linear monoidal categories and investigate whether this cohomology is invariant under braided Morita equivalence (i.e., linear braided monoidal equivalence of module categories).

Methodology
The authors employ a categorical framework centered on the augmented braided category $(\Delta B)_+$, a crossed simplicial group introduced by Krasauskas and Fiedorowicz–Loday. The methodology proceeds through the following steps:

1. Categorical Construction: They define a braided coalgebra $(C, c)$ in a linear monoidal category $\mathcal{C}$, where $C$ is an object and $c$ is a braiding isomorphism satisfying specific compatibility conditions with the coalgebra structure.
2. Functorial Association: To every braided coalgebra, they associate a strict monoidal functor $\tilde{T}_\bullet(C, c): (\Delta B)_+^{op} \to [\mathcal{C}, \mathcal{C}]$. This functor maps the augmented braided category to the category of endofunctors of $\mathcal{C}$.
3. Complex Definition: Using this functor, they construct the braided cochain complex $KB^\bullet(C, c, M)$ with coefficients in an object $M$. This is achieved by:
   • Restricting the functor to the subcategory $(\Delta_{inj}B)_+$ (braided injections).
   • Evaluating at the unit object and applying the Hom-functor $\text{Hom}_\mathcal{C}(-, M)$ to land in the category of $k$-modules ($\mathcal{V}$).
   • Defining the differential as the alternating sum of face maps.
   • Taking the subcomplex of elements invariant under a specific action of the braid group $B_{n+1}$ (specifically, elements $\phi$ such that $K(\tau_i^n)(\phi) = -\phi$), which generalizes the symmetric invariants used in classical symmetric cohomology.
4. Relative Morphisms: To study invariance, the authors introduce the notion of relative morphisms between strict monoidal functors. They establish a bijective correspondence between relative morphisms from $\tilde{T}_\bullet(C, c)$ to $\tilde{T}_\bullet(D, c')$ and specific morphisms $\alpha: D \to F(C)$ in the target category that satisfy conditions analogous to coalgebra morphisms but adapted to the monoidal functor $F$.

Key Contributions and Results

• Definition of Braided Cohomology: The paper defines the braided cochain complex and braided cohomology for braided coalgebras in any linear monoidal category. This framework encompasses quasi-triangular bialgebras (viewed as braided coalgebras in their module categories), differential graded settings, and Yetter-Drinfel'd modules.
• Theorem A (Characterization of Relative Morphisms): The authors prove that relative morphisms between the functors associated with braided coalgebras are in bijective correspondence with morphisms $\alpha$ satisfying specific structural equations involving the monoidal structure of the functor and the braiding. This reduces the study of natural transformations between complex functors to the study of single morphisms in the underlying category.
• Theorem B (Braided Morita Invariance): For quasi-triangular $k$-bialgebras $(A, R)$ and $(B, R')$, if there exists a $k$-linear braided monoidal equivalence $F: (A\text{-Mod}, c_R) \to (B\text{-Mod}, c_{R'})$ that commutes with the forgetful functors (up to natural isomorphism), then there is an isomorphism of cochain complexes $KB^\bullet(A, R, M) \cong KB^\bullet(B, R', F(M))$. Consequently, the braided cohomology is invariant under such equivalences.
• Corollary C (Finite-Dimensional Case): The condition regarding the commutation with forgetful functors is automatically satisfied for finite-dimensional quasi-triangular bialgebras. Thus, the braided cohomology of finite-dimensional quasi-triangular bialgebras is a braided Morita invariant.
• Explicit Computation: The authors provide an explicit calculation for the Sweedler Hopf algebra, a non-cocommutative example. They demonstrate that for different $R$-matrices parameterized by $\lambda \in k$, the first braided cohomology groups $KB^1(A, R_\lambda, M)$ are isomorphic as $k$-vector spaces, explicitly constructing the isomorphism via the braided Morita equivalence induced by a bialgebra twist.

Significance and Claims
The paper claims to provide a unified framework that generalizes symmetric cohomology from cocommutative Hopf algebras to the broader class of quasi-triangular bialgebras by replacing symmetric groups with braid groups and flip morphisms with arbitrary braidings.

• Generalization: It extends the scope of symmetric cohomology to non-cocommutative settings, including examples like the Sweedler Hopf algebra, Kac-Paljutkin Hopf algebra, and Drinfel'd doubles.
• Invariance: It establishes that braided cohomology is a robust invariant under braided Morita equivalence, a concept crucial for classifying quasi-triangular structures up to monoidal equivalence of their module categories.
• Categorical Connection: The work reveals a direct connection between the augmented braided category and the cohomology of braided coalgebras, addressing a problem posed by Singh in the context of braid group representations.
• Future Directions: The authors note that their constructions are applicable in non-additive situations (e.g., simplicial sets, spectra) and suggest that the braided cohomology admits an $\infty$-categorical generalization, though this is not developed in the current work.

The paper does not claim to solve the classification of all quasi-triangular bialgebras but rather provides a cohomological tool that remains stable under the natural equivalences (braided Morita equivalence) used to classify these structures. The results are presented as theoretical generalizations of existing constructions in group and Hopf algebra cohomology.