[Paper Review] Cyclic homology of Hopf Galois extensions and Hopf algebras

Pascual Jara, Dragoş Ştefan|ArXiv.org|Jul 8, 2003

Algebraic structures and combinatorial models19 references19 citations

TL;DR

This paper develops a cyclic homology theory for Hopf Galois extensions and Hopf algebras using a new category of modular crossed modules over a Hopf algebra H. It constructs a cyclic object Z*(H,M) for each such module M, generalizing standard cyclic homology of H and relative cyclic homology of Galois extensions. The key result computes the cyclic homology of induced modules Ind_K^H M when K is cocommutative and M decomposes into central one-dimensional subcomodules, enabling explicit calculations for group algebras and quantum tori.

ABSTRACT

Let H be a Hopf algebra. By definition a modular crossed H-module is a vector space M on which H acts and coacts in a compatible way. To every modular crossed H-module M we associate a cyclic object Z(H,M). The cyclic homology of Z(H,M) extends the usual cyclic homology of the algebra structure of H, and the relative cyclic homology of an H-Galois extension. For a Hopf subalgebra K we compute, under some assumptions, the cyclic homology of an induced modular crossed module. As a direct application of this computation, we describe the relative cyclic homology of strongly graded algebras. In particular, we calculate the (usual) cyclic homology of group algebras and quantum tori. Finally, when H is the enveloping algebra of a Lie algebra, we construct a spectral sequence that converges to the cyclic homology of H with coefficients in an arbitrary modular crossed module. We also show that the cyclic homology of almost symmetric algebras is isomorphic to the cyclic homology of H with coefficients in a certain modular crossed-module.

Motivation & Objective

• To unify and generalize cyclic homology computations for Hopf algebras, group algebras, and quantum tori by introducing a new framework based on modular crossed modules.
• To extend the cyclic homology theory to relative settings via Hopf Galois extensions, particularly focusing on the quotient A/[A,B] in such extensions.
• To establish a spectral sequence computing the cyclic homology of U(g)-modules, linking it to Lie algebra homology and extending known results on almost symmetric algebras.
• To provide a systematic method for computing the cyclic homology of induced modules Ind_K^H M when K is cocommutative and M is a sum of one-dimensional central subcomodules.
• To show that almost symmetric algebras and U_f(g) are U(g)-Galois extensions, enabling the application of the general theory to compute their cyclic homology.

Proposed method

• Introduces the category CM_m(H) of modular crossed modules over a Hopf algebra H, defined by two compatibility conditions between left H-module and right H-comodule structures.
• Defines a cyclic object Z*(H,M) for each M in CM_m(H), generalizing the standard cyclic homology of H (when M = ad H) and relative cyclic homology of H-Galois extensions (when M = A_B).
• Constructs the induction functor Ind_K^H M = H ⊗_K M for Hopf subalgebras K ⊂ H, equipping it with a structure in CM_m(H) to allow homological computations.
• Computes HC*(Ind_K^H M) when K is cocommutative and M decomposes into one-dimensional subcomodules with central group-like elements, using a decomposition into tensor products.
• Applies the result to strongly graded algebras, showing that their relative cyclic homology is isomorphic to HC*(H, Ind_K^H M) under suitable conditions.
• Constructs a spectral sequence E^2_{p,q} = H_{p+q-2i}(g, F_pM/F_{p-1}M) ⇒ HC_*(M) for U(g)-modules M, linking Lie algebra homology to cyclic homology.

Experimental results

Research questions

• RQ1 How can cyclic homology be generalized to Hopf Galois extensions using a category of modular crossed modules?
• RQ2 What is the cyclic homology of induced modules Ind_K^H M when K is cocommutative and M is a sum of one-dimensional central subcomodules?
• RQ3 Can the cyclic homology of group algebras and quantum tori be computed via this new framework?
• RQ4 How does the cyclic homology of U(g)-modules relate to Lie algebra homology via a spectral sequence?
• RQ5 To what extent do almost symmetric algebras and U_f(g) arise as U(g)-Galois extensions, and how does this allow for cyclic homology computation?

Key findings

• The cyclic homology of Ind_K^H M is computed explicitly when K is cocommutative and M decomposes into one-dimensional subcomodules with central group-like elements, yielding HC*(Ind_K^H M) ≅ ⊕_i HC*(K, M_i) ⊗ HC*(K) for each component.
• The relative cyclic homology of strongly graded algebras is isomorphic to the cyclic homology of the corresponding induced module, enabling direct computation via the main result.
• The cyclic homology of group algebras over a field of characteristic zero is recovered as a special case, consistent with Burghelea’s result.
• The cyclic homology of quantum tori is computed via the theory, showing it is isomorphic to the cyclic homology of the corresponding group algebra.
• A spectral sequence is constructed that converges to HC_*(M) for any U(g)-module M, with E^2_{p,q} = ⊕_i H_{p+q-2i}(g, F_pM/F_{p-1}M), generalizing Kassel’s results.
• Almost symmetric algebras are shown to be U(g)-Galois extensions of k, and their cyclic homology is isomorphic to HC_*(U(g), U_f(g)), extending known results to a broader class of algebras.