[Paper Review] Deformation bicomplex of module-algebras

Donald Yau|ArXiv.org|Jul 24, 2007

Advanced Topics in Algebra13 references3 citations

TL;DR

This paper constructs a deformation bicomplex for module-algebras over a bialgebra H, generalizing Hochschild cohomology to simultaneously deform both the algebra and H-module structures. It establishes that the cohomology of the total complex controls formal deformations, with vanishing of H² implying rigidity. The framework is extended to module-coalgebras, comodule-(co)algebras, and (co)module-bialgebras, each with analogous deformation complexes and cohomological control of deformations.

ABSTRACT

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.

Motivation & Objective

• To develop a deformation theory for H-module-algebras that simultaneously deforms both the algebra and H-module structures.
• To generalize classical deformation theory of associative algebras and bialgebras to include module and comodule structures.
• To extend the deformation complex to module-coalgebras, comodule-(co)algebras, and (co)module-bialgebras.
• To identify infinitesimal deformations with 2-cocycles in the deformation bicomplex and relate cohomology to rigidity.
• To construct cup-products on the deformation complexes, endowing each row and column with differential graded algebra structures.

Proposed method

• Define the deformation bicomplex $ C_{\mathrm{MA}}^{**}(A) $ as $ \mathrm{Hoch}^{*}(H, \operatorname{Hom}(A^{\otimes*}, A)) $, with entries $ \operatorname{Hom}(H^{\otimes q}, \operatorname{Hom}(A^{\otimes p}, A)) $.
• Construct the total complex $ C_{\mathrm{MA}}^{*}(A) $ by shifting degree: $ C_{\mathrm{MA}}^{n}(A) = \bigoplus_{p+q=n+1} C_{\mathrm{MA}}^{p,q}(A) $.
• Equip each row $ C_{\mathrm{MA}}^{*,q}(A) $ with a Hochschild $ \cup $-product via the associative algebra structure on $ \operatorname{Hom}(H^{\otimes q}, A) $.
• Equip each column $ C_{\mathrm{MA}}^{p,*}(A) $ with a $ \cup $-product via the algebra structure on $ \operatorname{Hom}(A^{\otimes p}, A) $, particularly non-trivial for $ p=1 $.
• Generalize the construction to module-coalgebras, comodule-(co)algebras, and (co)module-bialgebras, yielding bicomplexes and tricomplexes.
• Define deformations as power series $ \Theta = \sum \theta_n t^n $ with coefficients in the deformation complex, satisfying compatibility conditions for module and algebra structures.

Experimental results

Research questions

• RQ1 How can one simultaneously deform both the algebra and module structures on an H-module-algebra?
• RQ2 What is the appropriate cohomological framework to classify such simultaneous deformations?
• RQ3 How do the deformation bicomplexes for module-coalgebras, comodule-algebras, and (co)module-bialgebras generalize classical deformation theories?
• RQ4 What is the role of the $ \cup $-product in the deformation complex, and does it endow the complex with a differential graded algebra structure?
• RQ5 Under what cohomological conditions is a module-bialgebra rigid, i.e., all deformations equivalent to the trivial one?

Key findings

• The deformation bicomplex $ C_{\mathrm{MA}}^{**}(A) $ for an H-module-algebra A is isomorphic to $ \mathrm{Hoch}^{*}(H, \operatorname{Hom}(A^{\otimes*}, A)) $, with the 0th row being the Hochschild complex of A.
• Each row $ C_{\mathrm{MA}}^{*,q}(A) $ is isomorphic to a Hochschild cochain complex with coefficients in $ \operatorname{Hom}(H^{\otimes q}, A) $, and thus inherits a differential graded associative algebra structure via the $ \cup $-product.
• The infinitesimal of any deformation of an H-module-algebra is a 2-cocycle in the total complex $ C_{\mathrm{MA}}^{*}(A) $, and its cohomology class determines the deformation's equivalence class.
• If $ H^{2}(C_{\mathrm{MA}}^{*}(A)) = 0 $, then every deformation of the H-module-algebra is equivalent to the trivial deformation, implying rigidity.
• The deformation tricomplex $ C_{\mathrm{CB}}^{***}(A) $ for an H-comodule-bialgebra A has three differentials and admits $ \cup $-products in all three directions, each making the respective cochain complex into a DGA.
• The total complex $ C_{\mathrm{CB}}^{*}(A) $ of the comodule-bialgebra deformation tricomplex satisfies $ C_{\mathrm{CB}}^{1}(A) = \operatorname{Bider}(A) $ and $ C_{\mathrm{CB}}^{2}(A) = \operatorname{Hom}(A, H \otimes A) \oplus \operatorname{Hom}(A^{\otimes 2}, A) \oplus \operatorname{Hom}(A, A^{\otimes 2}) $, and vanishing of $ H^{2}_{\mathrm{CB}}(A) $ implies rigidity of A as a comodule-bialgebra.