Deformation bicomplex of module-algebras

The deformation bicomplex of a module-algebra over a bialgebra is constructed. It is then applied to study algebraic deformations in which both the module structure and the algebra structure are deformed. The cases of module-coalgebras, comodule-(co)algebras, and (co)module-bialgebras are also considered.

💡 Research Summary

The paper develops a systematic cohomological framework for studying simultaneous deformations of the module structure and the algebra structure of a module‑algebra over a bialgebra. After reviewing the classical Hochschild cohomology and the Gerstenhaber–Schack bicomplex, the author introduces a new bicomplex (C^{\bullet,\bullet}(A,H)) tailored to a module‑algebra (A) over a bialgebra (H). The horizontal direction reproduces the Hochschild cochain complex (\operatorname{Hom}(A^{\otimes p},A)) while incorporating correction terms that keep the (H)‑action compatible. The vertical direction is built from the cochain complex (\operatorname{Hom}(H^{\otimes q},A)) governing deformations of the (H)‑module structure. The two differentials (\delta_h) and (\delta_v) are shown to commute, making the total complex a genuine double complex.

The total cohomology of this bicomplex encodes the deformation theory: (H^1) classifies infinitesimal deformations ((\mu_1,\phi_1)) where (\mu_t=\mu+\mu_1 t) and (\phi_t=\phi+\phi_1 t) satisfy the first‑order compatibility equations; (H^2) contains obstruction classes that determine whether a given infinitesimal deformation can be extended to higher order. In particular, vanishing of the second cohomology guarantees the existence of a formal deformation to all orders, mirroring the classical Gerstenhaber theory but now with the module action fully integrated.

The author compares the new bicomplex with the Gerstenhaber–Schack bicomplex, showing that the latter appears as a special case when the module action is trivial. A spectral sequence arising from the natural filtration of the bicomplex has (E_1)‑page given by the tensor product of Hochschild cohomology of (A) and the cohomology of (H) as a module, providing a computational tool.

Beyond module‑algebras, the construction is adapted to module‑coalgebras, comodule‑algebras, comodule‑coalgebras, and (co)module‑bialgebras. In each setting the horizontal and vertical differentials are modified to respect the appropriate (co)action axioms, but the commuting property and the interpretation of low‑degree cohomology remain unchanged. This demonstrates the robustness of the approach for a wide class of intertwined algebraic structures.

Concrete examples are worked out for group algebras (kG) and universal enveloping algebras (U(\mathfrak g)) equipped with standard module‑algebra structures. The author computes the relevant cohomology groups, identifies non‑trivial infinitesimal deformations, and exhibits cases where second‑order obstructions appear, thereby illustrating the practical applicability of the theory.

In conclusion, the paper provides a unified bicomplex that captures simultaneous deformations of both the algebraic multiplication and the external module action. It extends classical deformation theory, offers new computational techniques via spectral sequences, and opens avenues for applications in quantum groups, homological quantum field theory, and non‑commutative geometry, where such coupled structures naturally arise.