Restricted Lie Algebras via Monadic Decomposition

Ardizzoni, Alessandro; Goyvaerts, Isar; Menini, Claudia

doi:10.1007/s10468-017-9734-8

Cited by 4 publications

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References 13 publications

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“…Through the same characterization, exceeding the initial expectations, in Theorem 3. 4 we find out that the (co)comparison functor attached to an adjunction whose associated (co)monad is (co)separable is always a coreflection (reflection) up to retracts. This result allows us to obtain in Theorem 3.5 the following semi-analogue of [15,Proposition 3.5] proved by X.-W. Chen: Given a functor G : D → C with a left adjoint F , then G is semiseparable if and only if the associated monad GF is separable and the comparison functor K GF : D → C CF is a bireflection up to retracts.…”

Section: Introductionmentioning

confidence: 88%

“…2) If G is a (co)reflection up to retracts and U is separable, then U is an equivalence up to retracts if and only if U • G is a (co)reflection up to retracts.Proof. Set G ′ := U • G. 1) Since U is conservative, if G ′ is a coreflection, by[4, Corollary 4.9], which is a consequence of [9, Lemma 1.2], we get that U is an equivalence. Conversely, if U is an equivalence then it is in particular a coreflection and hence, by Remark 1.10, G ′ is a coreflection as a composition of coreflections.…”

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Heavily separable functors

Ardizzoni

Menini

2020

Journal of Algebra

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Prompted by an example related to the tensor algebra, we introduce and investigate a stronger version of the notion of separable functor that we call heavily separable. We test this notion on several functors traditionally connected to the study of separability. IntroductionGiven a field k, the functor P : Bialg k → Vec k , assigning to a k-bialgebra B the k-vector space of its primitive elements, admits a left adjoint T, assigning to a vector space V the tensor algebra TV endowed with its canonical bialgebra structure such that the elements in V becomes primitive. By investigating the properties of the adjunction (T, P), together with its unit η and counit ǫ, we discovered that there is a natural retraction γ : TP → Id of η, i.e. γ • η = Id, fulfilling the condition γγ = γ • PǫT. The existence of a natural retraction of the unit of an adjunction is, by Rafael Theorem, equivalent to the fact that the left adjoint is a separable functor. It is then natural to wonder if the above extra condition on the retraction γ corresponds to a stronger notion of separability. In the present paper, we show that an affirmative answer to this question is given by what we call a heavily separable (h-separable for short) functor and we investigate this notion in case of functors usually connected to the study of separability.Explicitly, in Section 1 we introduce the concept of h-separable functor and we recover classical results in the h-separable case such as their behaviour with respect to composition (Lemma 1.4). In Section 2, we obtain a Rafael type Theorem 2.1. As a consequence we characterize the hseparability of a left (respectively right) adjoint functor either with respect to the forgetful functor from the Eilenberg-Moore category of the associated monad (resp. comonad) in Proposition 2.3 or by the existence of an augmentation (resp. grouplike morphism) of the associated monad (resp. comonad) in Corollary 2.7. In Theorem 2.8, we prove that the induced functor attached to an A-coring is h-separable if and only if this coring has an invariant grouplike element.Section 3 is devoted to the investigation of the h-separability of the induction functor ϕ * and of the restriction of scalars functors ϕ * attached to a ring homomorphism ϕ : R → S. In Proposition 3.1, we prove that ϕ * is h-separable if and only if there is a ring homomorphism E : S → R such that E • ϕ = Id. Characterizing whether ϕ * is h-separable (in this case we say that S/R is h-separable) is more laborious. In Proposition 3.4, we prove that S/R is h-separable if and only if it is endowed with what we call a h-separability idempotent, a stronger version of a separability idempotent. In Lemma 3.7 we show that the ring epimorphisms (by this we mean epimorphisms in the category of rings) provide particular examples of h-separability. In Lemma 3.11 we show that the ring of matrices is never h-separable over the base ring except in trivial cases. In the rest of the present section we investigate the particular case when S is an R-algebra i.e. Im(ϕ) ⊆ Z(S). In Theorem...

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Section: Introductionmentioning

confidence: 88%

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confidence: 99%

Heavily separable functors

Ardizzoni

Menini

2020

Journal of Algebra

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“…If this process stops exactly after N steps, meaning that N is the smallest positive integer such that U N,N +1 is a category isomorphism, then R is said to have a monadic decomposition of monadic length N . For relevant outcomes of this notion we refer to [4,6,10]. We just mention here how our interest in this construction stems from the case when (L, R) is the adjunction ( T , P ), where P is the functor that associates to any bialgebra, over a base field k, its space of primitive elements and its left adjoint T associates to a vector space V its tensor algebra T V endowed with the usual bialgebra structure in which the elements of V are primitive.…”

Section: Introductionmentioning

confidence: 99%

Monadic vs Adjoint Decomposition

Ardizzoni¹,

Menini²

2019

Preprint

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It is known that the so-called monadic decomposition, applied to the adjunction connecting the category of bialgebras to the category of vector spaces via the tensor and the primitive functors, returns the usual adjunction between bialgebras and (restricted) Lie algebras. Moreover, in this framework, the notions of heavily separable functor and combinatorial rank play a central rule. In order to set these results into a wider context, we are led to substitute the monadic decomposition by what we call the adjoint decomposition. This construction has the advantage of reducing the computational complexity when compared to the first one. We connect the two decompositions by means of an embedding and we investigate its properties by using a relative version of Grothendieck fibration. As an application, in this wider setting, by using the notion of heavily separable functor, we introduce a notion of combinatorial rank that, among other things, is expected to give some hints on the length of the monadic decomposition.

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Semiseparable Functors and Conditions up to Retracts

Ardizzoni,

Bottegoni

2024

Appl Categor Struct

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In a previous paper we introduced the concept of semiseparable functor. Here we continue our study of these functors in connection with idempotent (Cauchy) completion. To this aim, we introduce and investigate the notions of (co)reflection and bireflection up to retracts. We show that the (co)comparison functor attached to an adjunction whose associated (co)monad is separable is a coreflection (reflection) up to retracts. This fact allows us to prove that a right (left) adjoint functor is semiseparable if and only if the associated (co)monad is separable and the (co)comparison functor is a bireflection up to retracts, extending a characterization pursued by X.-W. Chen in the separable case. Finally, we provide a semi-analogue of a result obtained by P. Balmer in the framework of pre-triangulated categories.

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Restricted Lie Algebras via Monadic Decomposition

Abstract: Abstract. We give a description of the category of restricted Lie algebras over a field k of prime characteristic by means of monadic decomposition of the functor that computes the kvector space of primitive elements of a k-bialgebra.

Cited by 4 publications

References 13 publications

Heavily separable functors

Heavily separable functors

Monadic vs Adjoint Decomposition

Semiseparable Functors and Conditions up to Retracts

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