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Walter Ferrer Santos

Publications and source records attributed to Walter Ferrer Santos.

13 recordsLinked to original sources
arXiv2020

Quasi-compact group schemes, Hopf sheaves, and their representations

We explore the notion of representation of an affine extension of an abelian variety -- such an extension is a faithfully flat affine morphism of $\Bbbk$-group schemes $q:G\to A$, where $A$ is an abelian variety. We characterize the categories that arise as the category of representations of an affine extension $q:G\to A$, generalizing the classical results of Tannaka Duality established for affine $\Bbbk$-group schemes (that is, when $A=\operatorname{Spec}(\Bbbk)$). We also prove the existence of a contravariant equivalence between the category of affine extensions of a given $A$ and the category of faithful commutative Hopf sheaves on $A$, generalizing in this manner the well-known op-equivalence between affine group schemes and commutative Hopf algebras. If $\mathcal H_q$ is the Hopf sheaf on $A$ associated to $q$, the category of representations of $q$ is equivalent to the category of $\mathcal H_q$-comodules.

math.AG↗
arXiv2018

A short survey on observability

The exploration of the notion of observability exhibits transparently the rich interplay between algebraic and geometric ideas in \emph{geometric invariant theory}. The concept of \emph{observable subgroup} was introduced in the early 1960s with the purpose of studying extensions of representations from an affine algebraic subgroup to the whole group. The extent of its importance in \emph{representation and invariant theory} in particular for Hilbert's $14^{\text{th}}$ problem was noticed almost immediately. An important strenghtening appeared in the mid 1970s when the concept of \emph{strong observability} was introduced and it was shown that the notion of observability can be understood as an intermediate step in the notion of reductivity (or semisimplicity), when adequately generalized. More recently starting in 2010, the concept of observable subgroup was expanded to include the concept of \emph{observable action} of an affine algebraic group on an affine variety, launching a series of new applications. In 2006 the related concept of \emph{observable adjunction} was introduced, and its application to module categories over tensor categories was noticed. In the current survey, we follow (approximately) the historical development of the subject introducing along the way, the definitions and some of the main results including some of the proofs. For the unproven parts, precise references are mentioned.

math.AG↗
arXiv2015

Realizability in OCAs and AKSs

In the context of the $\mathcal{OCA}$ associated to an ${\mathcal{AKS}}$ we introduce a closure operator and two associated maps that replace the closure and the maps defined in \cite{kn:ocar}. We were motivated by the search of a full adjunction to the original implication map. We show that all the constructions from $\mathcal{OCA}$s to triposes developped in \cite{kn:ocar} can be also implemented in the new situation.

math.LO↗
arXiv2015

The Heisenberg product: from Hopf algebras and species to symmetric functions

Many related products and coproducts (e.g. Hadamard, Cauchy, Kronecker, induction, internal, external, Solomon, composition, Malvenuto-Reutenauer, convolution, etc.) have been defined in the following objects : species, representations of the symmetric groups, symmetric functions, endomorphisms of graded connected Hopf algebras, permutations, non-commutative symmetric functions, quasi-symmetric functions, etc. With the purpose of simplifying and unifying this diversity we introduce yet, another -non graded- product the Heisenberg product, that for the highest and lowest degrees produces the classical external and internal products (and their namesakes in different contexts). In order to define it, we start from the two opposite more general extremes: species in the "commutative context", and endomorphisms of Hopf algebras in the "non-commutative" environment. Both specialize to the space of commutative symmetric functions where the definitions coincide. We also deal with the different coproducts that these objects carry -to which we add the Heisenberg coproduct for quasi-symmetric functions-, and study their Hopf algebra compatibility particularly for symmetric and non commutative symmetric functions. We obtain combinatorial formulas for the structure constants of the new product that extend, generalize and unify results due to Garsia, Remmel, Reutenauer and Solomon. In the space of quasi- symmetric functions, we describe explicitly the new operations in terms of alphabets.

math.RA↗
arXiv2014

Ordered combinatory algebras and realizability

We consider different classes of combinatory structures related to Krivine realizability. We show, in the precise sense that they give rise to the same class of triposes, that they are equivalent for the purpose of modeling higher-order logic. We center our attentions in the role of a special kind of Ordered Combinatory Algebras-- that we call the "Krivine ordered combinatory algebras" ($\mathcal{KOCA}$s)-- that we propose as the foundational pillars for the categorical perspective of Krivine's classical realizability as presented by Streicher. Our procedure is the following: we show that each of the considered combinatory structures gives rise to an indexed preorder, and describe a way to transform the different structures into each other that preserves the associated indexed preorders up to equivalence. Since all structures give rise to the same indexed preorders, we only prove that they are triposes once: for the class of $\mathcal{KOCA}$s. We finish showing that in $\mathcal{KOCA}$s, one can define realizability in every higher-order language and in particular in higher-order arithmetic.

math.LO↗
arXiv2013

Almost involutive Hopf algebras

We define the concept of \emph{companion automorphism} of a Hopf algebra $H$ as an automorphism $σ:H \rightarrow H$: $σ^2=S^2$ --where $S$ denotes the antipode--. A Hopf algebra is said to be \emph{almost involutive} (AI) if it admits a companion automorphism that can be viewed as a special additional symmetry. We present examples and study some of the basic properties and constructions of AI-Hopf algebras centering the attention in the finite dimensional case. In particular we show that within the family of Hopf algebras of dimension smaller or equal than 15, only in dimension eight and twelve, there are non almost involutive Hopf algebras.

math.RA↗
arXiv2013

A Report on Realizability

Besides recalling the basic definitions of Realizability Lattices, Abstract Krivine Structures, Ordered Combinatory Algebras and Tripos and reviewing its relationships, we propose a new foundational framework for realizability. Motivated by Streicher's paper "Krivine's Classical Realizability from a Categorical Perspective" [9], we define the concept of Krivine's Ordered Combinatory Algebras (kOKA) as a common platform that is strong enough to do both: categorical and computational semantics. The OCAs produced by Streicher from AKSs in [9] are particular cases of kOKAs.

math.LO↗
arXiv2013

Some constructions of compact quantum groups

The purpose of this paper is to consider some basic constructions in the category of compact quantum groups --for example de case of extensions, of Drinfeld twists, of matched pairs, of extensions, of linked pairs and of cocycle Singer pairs -- with special emphasis in the finite dimensional situation. We give conditions, in some cases necessary and sufficient, to extend to the new objects the original compact structure. We illustrate the results in the case of matched pairs of groups.

math.QA↗
arXiv2009

The beginnings of the theory of Hopf algebras

We consider issues related to the origins, sources and initial motivations of the theory of Hopf algebras. We consider the two main sources of primeval development: algebraic topology and algebraic group theory. Hopf algebras are named from the work of Heinz Hopf in the 1940's. In this note we trace the infancy of the subject back to papers from the 40's, 50's and 60's in the two areas mentioned above. Many times we just describe -- and/or transcribe parts of -- some of the relevant original papers on the subject.

math.HO↗
arXiv2008

Monoidal categories of comodules for coquasi Hopf algebras and Radford's formula

We study the basic monoidal properties of the category of Hopf modules for a coquasi Hopf algebra. In particular we discuss the so called fundamental theorem that establishes a monoidal equivalence between the category of comodules and the category of Hopf modules. We present a categorical proof of Radford's $S^4$ formula for the case of a finite dimensional coquasi Hopf algebra, by establishing a monoidal isomorphism between certain double dual functors.

math.QA↗
arXiv2006

Radford's formula for biFrobenius algebras and applications

In a biFrobenius algebra H, in particular in the case that H is a finite dimensional Hopf algebra, the antipode S can be decomposed as S= cf where c and f are the Frobenius and coFrobenius isomorphisms. We use this decomposition to present an easy proof of Radford's formula for the fourth composition power of S. Then, in the case that the map S is the convolution inverse of the identity, we prove the trace formula for the trace of the square of S. We finish by applying the above results to study the semisimplicity and cosemisimplicity of H.

math.RA↗
arXiv2005

Generalized Cayley's $Ω$-processes

In this paper we generalize some constructions and results due to Cayley and Hilbert. We define the concept of $Ω$--process for an arbitrary algebraic monoid with zero and unit group $G$. Then we show how to produce from the process and for a linear rational representation of $G$, a number of elements of the ring of $G$-invariants, that is large enough as to guarantee its finite generation. Moreover, we give an explicit construction of all $Ω$-processes for general reductive monoids and, in the case of the monoid of all the $n^2$ matrices, compare our construction with Cayley's definition.

math.AG↗