[Paper Review] Hopf Algebra Equivariant Cyclic Homology and Cyclic Homology of Crossed Product Algebras

R. Akbarpour, Masoud Khalkhali|ArXiv.org|Nov 29, 2000

Algebraic structures and combinatorial models4 references3 citations

TL;DR

This paper generalizes Getzler and Jones' work on cyclic homology of crossed product algebras to Hopf algebra actions, introducing a cylindrical module $A\natural\mathcal{H}$ and proving isomorphisms between cyclic modules of crossed product algebras and diagonals of these cylindrical modules. When the antipode $S$ of the Hopf algebra $\mathcal{H}$ is invertible, it establishes a spectral sequence approximating the cyclic homology of $A\rtimes\mathcal{H}$, with explicit computations of the $\mathsf{E}^0$, $\mathsf{E}^1$, and $\mathsf{E}^2$ terms.

ABSTRACT

We introduce the cylindrical module $A atural \mathcal{H}$, where $\mathcal{H}$ is a Hopf algebra and $A$ is a Hopf module algebra over $\mathcal{H}$. We show that there exists an isomorphism between $\mathsf{C}_{\bullet}(A^{op} times \mathcal{H}^{cop})$ the cyclic module of the crossed product algebra $A^{op} times \mathcal{H}^{cop} $, and $Δ(A atural \mathcal{H}) $, the cyclic module related to the diagonal of $A atural \mathcal{H}$. If $S$, the antipode of $\mathcal{H}$, is invertible it follows that $\mathsf{C}_{\bullet}(A times \mathcal{H}) \simeq Δ(A^{op} atural \mathcal{H}^{cop})$. When $S$ is invertible, we approximate $HC_{\bullet}(A times \mathcal{H})$ by a spectral sequence and give an interpretation of $ \mathsf{E}^0, \mathsf{E}^1$ and $\mathsf{E}^2 $ terms of this spectral sequence.

Motivation & Objective

• To extend the theory of cyclic homology for crossed product algebras from group actions to Hopf algebra actions.
• To define a cylindrical module $A\natural\mathcal{H}$ for a Hopf module algebra $A$ over a Hopf algebra $\mathcal{H}$, and show its diagonal is a cyclic module.
• To establish isomorphisms between cyclic modules of crossed product algebras and diagonals of cylindrical modules under invertible antipode.
• To construct a spectral sequence approximating $HC_\bullet(A\rtimes\mathcal{H})$ when the antipode $S$ is invertible, and compute its initial terms.
• To apply the results to semisimple Hopf algebras and algebras with locally nilpotent derivations, recovering known isomorphisms in periodic cyclic cohomology.

Proposed method

• Introduce the cylindrical module $A\natural\mathcal{H}$ for a Hopf module algebra $A$ over a Hopf algebra $\mathcal{H}$, using the action of $\mathcal{H}$ on $A$.
• Prove that the diagonal $\Delta(A\natural\mathcal{H})$ forms a cyclic module by verifying the required face, degeneracy, and cyclic operators satisfy the paracyclic identities.
• Establish an isomorphism between $\mathsf{C}_\bullet(A^{op}\rtimes\mathcal{H}^{cop})$ and $\Delta(A\natural\mathcal{H})$ using the Eilenberg-Zilber theorem for cylindrical modules.
• When the antipode $S$ is invertible, derive an isomorphism $\mathsf{C}_\bullet(A\rtimes\mathcal{H}) \simeq \Delta(A^{op}\natural\mathcal{H}^{cop})$ via duality and module structure.
• Construct the equivariant cyclic module $C_\bullet^{\mathcal{H}^{cop}}(A^{op})$ and use it to build a spectral sequence approximating $HC_\bullet(A^{op}\rtimes\mathcal{H}^{cop})$.
• Compute the $\mathsf{E}^0$, $\mathsf{E}^1$, and $\mathsf{E}^2$ terms of the spectral sequence using homological algebra and the action of $\mathcal{H}$ on $A$.

Experimental results

Research questions

• RQ1 How can the cyclic homology of a crossed product algebra $A\rtimes\mathcal{H}$ be computed when $\mathcal{H}$ is a Hopf algebra acting on $A$?
• RQ2 What is the role of the antipode $S$ in determining isomorphisms between cyclic modules of crossed product algebras and diagonals of cylindrical modules?
• RQ3 Can a spectral sequence be constructed to approximate $HC_\bullet(A\rtimes\mathcal{H})$ when $S$ is invertible, and what are the initial terms of this sequence?
• RQ4 How do the $\mathsf{E}^0$, $\mathsf{E}^1$, and $\mathsf{E}^2$ terms of the spectral sequence relate to the homology of $\mathcal{H}$-invariant and coinvariant subcomplexes?
• RQ5 What are the implications of the spectral sequence for algebras with locally nilpotent derivations or semisimple Hopf algebras?

Key findings

• The diagonal $\Delta(A\natural\mathcal{H})$ of the cylindrical module $A\natural\mathcal{H}$ is isomorphic to the cyclic module $\mathsf{C}_\bullet(A^{op}\rtimes\mathcal{H}^{cop})$.
• When the antipode $S$ is invertible, there is an isomorphism $\mathsf{C}_\bullet(A\rtimes\mathcal{H}) \simeq \Delta(A^{op}\natural\mathcal{H}^{cop})$, extending previous results to general Hopf algebras.
• A spectral sequence approximating $HC_\bullet(A^{op}\rtimes\mathcal{H}^{cop})$ is constructed, with $\mathsf{E}^0$-term given by $\mathsf{E}^0_{pq} = C_p^{\mathcal{H}^{cop}}(A^{op}) \otimes \Omega_q(\mathcal{H}^{cop})$.
• The $\mathsf{E}^1$-term is isomorphic to the homology of the complex $C_\bullet^{\mathcal{H}^{cop}}(A^{op})$ with coefficients in the standard complex of $\mathcal{H}^{cop}$, and the $\mathsf{E}^2$-term is the homology of the $\mathcal{H}^{cop}$-coinvariant subcomplex.
• For semisimple Hopf algebras, the spectral sequence collapses at $\mathsf{E}^1$, and $HC_\bullet(A\rtimes\mathcal{H};\mathsf{W}) \cong HC_\bullet(C_\bullet^{\mathcal{H}}(A);\mathsf{W})$, where $C_\bullet^{\mathcal{H}}(A)$ is the equivariant chain complex.
• In the case of a locally nilpotent derivation $\delta$ on $A$, with $\mathcal{H} = k[x]$, the spectral sequence yields a 6-term exact sequence in periodic cyclic cohomology, and the result $HP^\ast(A\rtimes\mathcal{H}) \simeq HP^\ast(A)$ follows from vanishing of higher terms.