English

(Co)Homology of Partial Smash Products

Rings and Algebras 2023-11-29 v1 Representation Theory

Abstract

Given a cocommutative Hopf algebra H\mathcal{H} over a commutative ring KK and a symmetric partial action of H\mathcal{H} on a KK-algebra A,A, we obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the smash product A#H,A \# \mathcal{H}, involving the Hochschild homology of AA and the partial homology of H.\mathcal{H}. An analogous third quadrant cohomological spectral sequence is also obtained. The definition of the partial (co)homology of H\mathcal{H} under consideration is based on the category of the partial representations of H.\mathcal{H}. A specific partial representation of H\mathcal{H} on a subalgebra B\mathcal{B} of the partial ``Hopf" algebra Hpar\mathcal{H}_{par} is involved in the definition and we construct a projective resolution of B.\mathcal{B}.

Keywords

Cite

@article{arxiv.2311.16990,
  title  = {(Co)Homology of Partial Smash Products},
  author = {Mikhailo Dokuchaev and Emmanuel Jerez},
  journal= {arXiv preprint arXiv:2311.16990},
  year   = {2023}
}