Generalised bialgebras and entwined monads and comonads

Abstract

Communicated by C. Kassel MSC: 18C20; 18D50; 16T10; 16T15J.-L. Loday has defined generalised bialgebras and proved structure theorems in this setting which can be seen as general forms of the Poincaré-Birkhoff-Witt and the Cartier-Milnor-Moore theorems. It was observed by the present authors that parts of the theory of generalised bialgebras are special cases of results on entwined monads and comonads and the corresponding mixed bimodules. In this article the Rigidity Theorem of J.-L. Loday is extended to this more general categorical framework.

Confluence laws and Hopf-Borel type theorem for operads

“…Remark 1.2.3. Confluence laws are equivalent to the data of a S-module morphism α : A → C * A as presented in [LMW15]. The equivalence comes from the equality:…”

“…The general framework for this theorem was introduced by Loday in [Lod08]. Rigidity theorems were then further studied, for instance in [LMW15] and applications can be found for example in [BCR15] to compute explicit bases of algebras. While studying the general framework and its rewriting for particular symmetric operads, it became clear to the authors that the three hypotheses of this theorem had to be clarified and some further clarifications were needed in the proof.…”

Representations of Crossed Modules and Other Generalized Yetter-Drinfel’d Modules

“…The definition of a crossed module of shelves/racks is redundant: it suffices to have a generalized augmented shelf/rack, that is, a shelf/rack S, an S-set or S-rack-set R, and an S-equivariant map π : R → S (in the sense of (22)). For this data, relation (21) can be taken as the definition of a shelf/rack operation on R, called the induced operation; with this choice, π becomes a shelf morphism, and S acts on R by shelf (auto)morphisms.…”

“…Using the S-equivariance relation (22) for π, one readily checks condition (15) with α 2 = γ 1 = γ 2 = 1 and α 1 = 0 (observe that σ A,A is in general not idempotent in this setting, and the choice α 1 = 1 from the previous examples would not work; cf. Remark 2.11).…”

“…The idea of mixing compatible acting and coacting structures can be found in the literature under various guises: J. Beck's mixed distributive laws [3], the AC-bialgebras of T.F. Fox and M. Markl [12], the algebracoalgebra entwining structures of T. Brzeziński and S. Majid [4], J.-L. Loday's generalized bialgebras [25] interpreted in terms of bimodules over a bimonad by M. Livernet, B. Mesablishvili, and R. Wisbauer [29,22], to cite just a few. The framework chosen in each case depends on the classical constructions and results one wants to extend to a generalized setting: the triplecotriple philosophy of [12] is well adapted for (co)homology constructions, the category of vector spaces is sufficient for developing quantum principal bundle theory and generalized gauge theory in [4], while operads provide a convenient setting for generalizing Poincaré-Birkhoff-Witt, Cartier-Milnor-Moore, and the Rigidity theorems in [25].…”