Paolo Saracco

• PhD
• PostDoc Position at Université Libre de Bruxelles

The proof of Theorem 3.14 contains an unsubstantiated claim. To overcome this problem, we add a hypothesis to the statement of 3.14 and we provide a new valid proof. We adjust Theorem 3.15, Corollary 3.16, Proposition 4.23, Theorem 4.26, Corollary 4.29, and Corollary 4.32 accordingly.

We extend a theorem, originally formulated by Blattner–Cohen–Montgomery for crossed products arising from Hopf algebras weakly acting on noncommutative algebras, to the realm of left Hopf algebroids. Our main motivation is an application to universal enveloping algebras of projective Lie–Rinehart algebras: for any given curved (resp. flat) connecti...

A notion of heaps of modules as an affine version of modules over a ring or, more generally, over a truss, is introduced and studied. Basic properties of heaps of modules are derived. Examples arising from geometry (connections, affine spaces) and algebraic topology (chain contractions) are presented. Relationships between heaps of modules, modules...

Let A be a k-algebra over a commutative ring k. By the renowned Tannaka-Krein reconstruction, liftings of the monoidal structure from k-modules to A-modules correspond to bialgebra structures on A and liftings of the closed monoidal structure correspond to Hopf algebra structures on A. In this paper, we determine conditions on A that correspond to...

We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective for Hopf algebroids. As an application, we investigate normal Hopf ideals in commutative Hopf algebroids (affine groupoid s...

We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective for Hopf algebroids. As an application, we investigate normal Hopf ideals in commutative Hopf algebroids (affine groupoid s...

We extend a theorem, originally formulated by Blattner-Cohen-Montgomery for crossed products arising from Hopf algebras weakly acting on noncommutative algebras, to the realm of left Hopf algebroids. Our main motivation is an application to universal enveloping algebras of projective Lie-Rinehart algebras: for any given curved (resp. flat) connecti...

Given a subgroup H of a finite group G, we begin a systematic study of the partial representations of G that restrict to global representations of H. After adapting several results from Dokuchaev et al. (J. Algebra 226(1), 505–532, 2000) (which correspond to the case H = {1G}), we develop further an effective theory that allows explicit computation...

We prove how the universal enveloping algebra constructions for Lie–Rinehart algebras and anchored Lie algebras are naturally left adjoint functors. This provides a conceptual motivation for the universal properties these constructions satisfy. As a supplement, the categorical approach offers new insights into the definitions of Lie–Rinehart algebr...

We study the globalization of partial actions on sets and topological spaces and of partial coactions on algebras by applying the general theory of globalization for geometric partial comodules, as previously developed by the authors. We show that this approach does not only allow to recover all known results in these settings, but it allows to tre...

A notion of heaps of modules as an affine version of modules over a ring or, more generally, over a truss, is introduced and studied. Basic properties of heaps of modules are derived. Examples arising from geometry (connections, affine spaces) and algebraic topology (chain contractions) are presented. Relationships between heaps of modules and modu...

Categorical aspects of the theory of modules over trusses are studied. Tensor product of modules over trusses is defined and its existence established. In particular, it is shown that bimodules over trusses form a monoidal category. Truss versions of the Eilenberg-Watts theorem and Morita equivalence are formulated. Projective and small-projective...

We discuss globalization for geometric partial comodules in a monoidal category with pushouts and we provide a concrete procedure to construct it, whenever it exists. The mild assumptions required by our approach make it possible to apply it in a number of contexts of interests, recovering and extending numerous ad hoc globalization constructions f...

The aim of this paper is to prove the statement in the title. As a by-product, we obtain new globalization results in cases never considered before, such as partial corepresentations of Hopf algebras. Moreover, we show that for partial representations of groups and Hopf algebras, our globalization coincides with those described earlier in literatur...

We study the globalization of partial actions on sets and topological spaces and of partial coactions on algebras by applying the general theory of globalization for geometric partial comodules, as previously developed by the authors. We show that this approach does not only allow to recover all known results in these settings, but it allows to tre...

A differentially recursive sequence over a differential field is a sequence of elements satisfying a homogeneous differential equation with non-constant coefficients (namely, Taylor expansions of elements of the field) in the differential algebra of Hurwitz series. The main aim of this paper is to explore the space of all differentially recursive s...

We prove how the universal enveloping algebra constructions for Lie-Rinehart algebras and anchored Lie algebras are naturally left adjoint functors. This provides a conceptual motivation for the universal properties these constructions satisfy. As a supplement, the categorical approach offers new insights into the definitions of Lie-Rinehart algebr...

By a theorem of Majid, every monoidal category with a neutral quasi-monoidal functor to finitely generated and projective \(\Bbbk \)-modules gives rise to a coquasi-bialgebra. We prove that if the category is also rigid, then the associated coquasi-bialgebra admits a preantipode, providing in this way an analogue for coquasi-bialgebras of Ulbrich’s...

We investigate the property of being Frobenius for some functors strictly related with Hopf modules over a bialgebra and how this property reflects on the latter. In particular, we characterize one-sided Hopf algebras with anti-(co)multiplicative one-sided antipode as those for which the free Hopf module functor is Frobenius. As a by-product, this...

We prove that a quasi-bialgebra admits a preantipode if and only if the associated free quasi-Hopf bimodule functor is Frobenius, if and only if the related (opmonoidal) monad is a Hopf monad. The same results hold in particular for a bialgebra, tightening the connection between Hopf and Frobenius properties.

Categorical aspects of the theory of modules over trusses are studied. Tensor product of modules over trusses is defined and its existence established. In particular, it is shown that bimodules over trusses form a monoidal category. Truss versions of the Eilenberg-Watts theorem and Morita equivalence are formulated. Projective and small-projective...

Given a subgroup H of a finite group G, we begin a systematic study of the partial representations of G that restrict to global representations of H. After adapting several results from [DEP00] (which correspond to the case where H is trivial), we develop further an effective theory that allows explicit computations. As a case study, we apply our t...

A differentially recursive sequence over a differential field is a sequence of elements satisfying a homogeneous differential equation with non-constant coefficients (namely, Taylor expansions of elements of the field) in the differential algebra of Hurwitz series. The main aim of this paper is to explore the space of all differentially recursive s...

We propose the notion of globalization for geometric partial comodules in a monoidal category and show its existence in many cases of interest, such as partial actions of monoids on sets, geometric partial coactions in abelian categories and partial comodule algebras. Our results subsume several globalization theorems appearing in literature.

We prove that a quasi-bialgebra admits a preantipode if and only if the associated free quasi-Hopf bimodule functor is Frobenius, if and only if the relative (opmonoidal) monad is a Hopf monad. The same results hold in particular for a bialgebra, tightening the connection between Hopf and Frobenius properties.

In this paper we investigate the formal notions of differentiation and integration in the context of commutative Hopf algebroids and Lie algebroid, or more precisely Lie-Rinehart algebras. Specifically, we construct a contravariant functor from the category of commutative Hopf algebroids with a fixed base algebra to that of Lie-Rinehart algebras ov...

We investigate the property of being Frobenius for some functors strictly related with Hopf modules over a bialgebra and how this property reflects on the latter. In particular, we characterize one-sided Hopf algebras with anti-(co)multiplicative one-sided antipode as those for which the free Hopf module functor is Frobenius. As a by-product, this...

Freely available at: http://hdl.handle.net/2318/1664506. This thesis concerns the study of algebraic objects whose structure resembles more or less closely that of a Hopf algebra and of some crucial aspects of “duality” between them. It contains (part of) the fruits of a three years work with A. Ardizzoni and L. El Kaoutit that already appeared a...

We prove in a very general framework several versions of the classical Poincar\'e-Birkhoff-Witt Theorem, which extend results from [BeGi, BrGa, CS, HvOZ, WW]. Applications and examples are discussed in the last part of the paper.

Given a finitely generated and projective Lie-Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is h...

Given a finitely generated and projective Lie-Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is h...

The space of linearly recursive sequences of complex numbers admits two distinguished topologies. Namely, the adic topology induced by the ideal of those sequences whose first term is $0$ and the topology induced from the Krull topology on the space of complex power series via a suitable embedding. We show that these topologies are not equivalent.

The aim of this paper is to establish a contravariant adjunction between the category of quasi-bialgebras and a suitable full subcategory of dual quasi-bialgebras, adapting the notion of finite dual to this framework. Various functorial constructions involving non-associative algebras and non-coassociative coalgebras are then carried out. Several e...

The Structure Theorem for Hopf modules states that if a bialgebra A is a Hopf algebra (i.e. it is endowed with a so-called antipode) then every Hopf module M is of the form M coA ⊗A, where M coA denotes the space of coinvariant elements in M. Actually, it has been shown that this result characterizes Hopf algebras: A is a Hopf algebra if and only i...