Papers by José Gómez-Torrecillas
arXiv (Cornell University), Jul 26, 2016
A fundamental tool of Differential Galois Theory is the assignment of an algebraic group to each ... more A fundamental tool of Differential Galois Theory is the assignment of an algebraic group to each finite-dimensional differential module over differential field in such a way that the category of differential modules it generates is equivalent, as a symmetric monoidal category, to the category of representations of the group. Its underlying set is then recognized as the group of differential automorphisms of the Picard-Vessiot field extension of the base field for this differential module. These results can be obtained by means of a Tannaka reconstruction process, applied to the abelian category of finite-dimensional differential modules. In this paper, we explore the possibility of extending this theory when the differential field is replaced by a more general differential ring A. In this case, it is reasonable to deal with differential modules which are finitely generated and projective over A. A major obstacle is that this category is not abelian, in contrast with the classical case when A is a field. To overcome this difficulty, we develop some fundamental results concerning the finite dual, hereby introduced, of a cocommutative Hopf algebroid, and a canonical monoidal functor sending (differential) modules to comodules. This functor is proved to be an equivalence of categories whenever a canonical ring homomorphism, which is introduced in this work, with domain in the aforementioned finite dual and with values in the convolutional ring, is injective. Module-theoretical sufficient conditions to get its injectivity are investigated. This machinery is applied to differential modules and their Picard-Vessiot theory.
arXiv (Cornell University), Jul 21, 2013
We develop two algorithms for computing a bound of an Ore polynomial over a skew field, under mil... more We develop two algorithms for computing a bound of an Ore polynomial over a skew field, under mild conditions. As an application, we state a criterion for deciding whether a bounded Ore polynomial is irreducible, and we discuss a factorization algorithm. The asymptotic time complexity in the degree of the given Ore polynomial is studied.
Springer eBooks, 2014
The present text surveys some relevant situations and results where basic Module Theory interacts... more The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects. Non-commutative ring, Finitely presented module, Free resolution, Ore extension, Non-commutative factorization, Eigenring, Jacobson normal form, PBW ring, PBW algebra, Gröbner basis, Filtered ring, Gelfand-Kirillov dimension, grade number Partially supported by the Spanish Ministerio de Ciencia en Innovación and the European Union-grant MTM2010-20940-C02-01. The author wishes to thank Thomas Cluzeau, Viktor Levandovskyy, Georg Regensburger, and the anonymous referees for their comments that lead to improve this paper. JOSÉ GÓMEZ-TORRECILLAS 5.3. Computation of the Gelfand-Kirillov Dimension 41 6. Appendix on Computer Algebra Systems (by V. Levandovskyy) 44 6.1. Functionality of Systems for D[x; σ, δ] 45 6.2. Functionality of Systems for Multivariate Ore Algebras 45 6.3. Functionality of Systems for PBW Algebras 46 6.4. Further Systems 48 References 48
arXiv (Cornell University), Jul 28, 2022
A class of linear codes that extends classic Goppa codes to a noncommutative context is defined. ... more A class of linear codes that extends classic Goppa codes to a noncommutative context is defined. An efficient decoding algorithm, based on the solution of a non-commutative key equation, is designed. We show how the parameters of these codes, when the alphabet is a finite field, may be adjusted to propose a McEliece-type cryptosystem.
Journal of Algebra, Nov 1, 2012
Let R be a ring with a set of local units, and a homomorphism of groups Θ : G → Pic(R) to the Pic... more Let R be a ring with a set of local units, and a homomorphism of groups Θ : G → Pic(R) to the Picard group of R. We study under which conditions Θ is determined by a factor map, and, henceforth, it defines a generalized crossed product with a same set of local units. Given a ring extension R ⊆ S with the same set of local units and assuming that Θ is induced by a homomorphism of groups G → Inv R (S) to the group of all invertible R-sub-bimodules of S, then we construct an analogue of the Chase-Harrison-Rosenberg seven terms exact sequence of groups attached to the triple (R ⊆ S, Θ), which involves the first, the second and the third cohomology groups of G with coefficients in the group of all R-bilinear automorphisms of R. Our approach generalizes the works by Kanzaki and Miyashita in the unital case.
arXiv (Cornell University), Jun 6, 2013
A non-unital generalization of weak bialgebra is proposed with a multiplier-valued comultiplicati... more A non-unital generalization of weak bialgebra is proposed with a multiplier-valued comultiplication. Certain canonical subalgebras of the multiplier algebra (named the 'base algebras') are shown to carry coseparable co-Frobenius coalgebra structures. Appropriate modules over a weak multiplier bialgebra are shown to constitute a monoidal category via the (co)module tensor product over the base (co)algebra. The relation to Van Daele and Wang's (regular and arbitrary) weak multiplier Hopf algebra is discussed.
International Journal of Mathematics and Mathematical Sciences, 2002
We develop some basic functorial techniques for the study of the categories of comodules over cor... more We develop some basic functorial techniques for the study of the categories of comodules over corings. In particular, we prove that the induction functor stemming from every morphism of corings has a left adjoint, called ad-induction functor. This construction generalizes the known adjunctions for the categories of Doi-Hopf modules and entwined modules. The separability of the induction and ad-induction functors are characterized, extending earlier results for coalgebra and ring homomorphisms, as well as for entwining structures.
Mathematische Zeitschrift, Aug 1, 2003
Simon Stevin, May 1, 2021
A fundamental tool of Differential Galois Theory is the assignment of an algebraic group to each ... more A fundamental tool of Differential Galois Theory is the assignment of an algebraic group to each finite-dimensional differential module over differential field in such a way that the category of differential modules it generates is equivalent, as a symmetric monoidal category, to the category of representations of the group. Its underlying set is then recognized as the group of differential automorphisms of the Picard-Vessiot field extension of the base field for this differential module. These results can be obtained by means of a Tannaka reconstruction process, applied to the abelian category of finite-dimensional differential modules. In this paper, we explore the possibility of extending this theory when the differential field is replaced by a more general differential ring A. In this case, it is reasonable to deal with differential modules which are finitely generated and projective over A. A major obstacle is that this category is not abelian, in contrast with the classical case when A is a field. To overcome this difficulty, we develop some fundamental results concerning the finite dual, hereby introduced, of a cocommutative Hopf algebroid, and a canonical monoidal functor sending (differential) modules to comodules. This functor is proved to be an equivalence of categories whenever a canonical ring homomorphism, which is introduced in this work, with domain in the aforementioned finite dual and with values in the convolutional ring, is injective. Module-theoretical sufficient conditions to get its injectivity are investigated. This machinery is applied to differential modules and their Picard-Vessiot theory.
IEEE Transactions on Information Theory, 2017
This paper investigates the application of the theoretical algebraic notion of a separable ring e... more This paper investigates the application of the theoretical algebraic notion of a separable ring extension, in the realm of cyclic convolutional codes or, more generally, ideal codes. We work under very mild conditions, that cover all previously known as well as new non trivial examples. It is proved that ideal codes are direct summands as left ideals of the underlying non-commutative algebra, in analogy with cyclic block codes. This implies, in particular, that they are generated by an idempotent element. Hence, by using a suitable separability element, we design an efficient algorithm for computing one of such idempotents.
Applied Categorical Structures, Nov 17, 2006
arXiv (Cornell University), Jan 29, 2003
To any bimodule which is finitely generated and projective on one side one can associate a coring... more To any bimodule which is finitely generated and projective on one side one can associate a coring, known as a comatrix coring. A new description of comatrix corings in terms of data reminiscent of a Morita context is given. It is also studied how properties of bimodules are reflected in the associated comatrix corings. In particular it is shown that separable bimodules give rise to coseparable comatrix corings, while Frobenius bimodules induce Frobenius comatrix corings.
arXiv (Cornell University), Jan 3, 2005
We extend Masuoka's Theorem [11] concerning the isomorphism between the group of invertible bimod... more We extend Masuoka's Theorem [11] concerning the isomorphism between the group of invertible bimodules in a non-commutative ring extension and the group of automorphisms of the associated Sweedler's canonical coring, to the class of finite comatrix corings introduced in [6].
arXiv (Cornell University), Nov 22, 2007
We develop a Galois (descent) theory for comonads within the framework of bicategories. We give g... more We develop a Galois (descent) theory for comonads within the framework of bicategories. We give generalizations of Beck's theorem and the Joyal-Tierney theorem. Many examples are provided, including classical descent theory, Hopf-Galois theory over Hopf algebras and Hopf algebroids, Galois theory for corings and group-corings, and Morita-Takeuchi theory for corings. As an application we construct a new type of comatrix corings based on (dual) quasi bialgebras.
arXiv (Cornell University), Jul 26, 2016
We construct a contravariant functor (the finite dual functor) from the category of co-commutativ... more We construct a contravariant functor (the finite dual functor) from the category of co-commutative Hopf algebroids to the category of commutative Hopf algebroids. Using this functor, we then show that the representation theory of a given Lie-Rinehart algebra (A, L) with A is a Dedekind domain, is the 'same' as the representation theory of an affine groupoid over the spectrum of A. In particular, this applies to the global sections of any Lie algebroid over an irreducible smooth curve over an algebraically closed field. To illustrate our methods, we give a detailed application to the transitive Lie algebroid of vector fields over the complex affine line A 1 C , which involves the theory of differential modules.
Applied Categorical Structures, Aug 24, 2013
We show that the functor from bialgebras to vector spaces sending a bialgebra to its subspace of ... more We show that the functor from bialgebras to vector spaces sending a bialgebra to its subspace of primitives has monadic length at most 2. 1. Monadic decompositions Consider categories A and B. Let (L : B → A, R : A → B) be an adjunction with unit η and counit ǫ, and consider the monad (RL, RǫL, η) generated on B. By B 1 we denote its Eilenberg-Moore category of algebras. Hence we can consider the so-called comparison functor of the adjunction (L, R) i.e. the functor K : A → B 1 , KX := (RX, RǫX) , Kf := Rf.
arXiv (Cornell University), Mar 1, 2018
arXiv (Cornell University), Apr 28, 2015
We recognise Harada's generalized categories of diagrams as a particular case of modules over a m... more We recognise Harada's generalized categories of diagrams as a particular case of modules over a monad defined on a finite direct product of additive categories. We work in the dual (albeit formally equivalent) situation, that is, with comodules over comonads. With this conceptual tool at hand, we obtain several of the Harada results with simpler proofs, some of them under more general hypothesis, besides with a characterization of the normal triangular matrix comonads that are hereditary, that is, of homological dimension less or equal than 1. Our methods rest on a matrix representation of additive functors and natural transformations, which allows us to adapt typical algebraic manipulations from Linear Algebra to the additive categorical setting.
arXiv (Cornell University), Jun 6, 2013
Weak (Hopf) bialgebras are described as (Hopf) bimonoids in appropriate duoidal (also known as 2-... more Weak (Hopf) bialgebras are described as (Hopf) bimonoids in appropriate duoidal (also known as 2-monoidal) categories. This interpretation is used to define a category wba of weak bialgebras over a given field. As an application, the "free vector space" functor from the category of small categories with finitely many objects to wba is shown to possess a right adjoint, given by taking (certain) group-like elements. This adjunction is proven to restrict to the full subcategories of groupoids and of weak Hopf algebras, respectively. As a corollary, we obtain equivalences between the category of small categories with finitely many objects and the category of pointed cosemisimple weak bialgebras; and between the category of small groupoids with finitely many objects and the category of pointed cosemisimple weak Hopf algebras.
arXiv (Cornell University), May 10, 2012
Given a weak distributive law between algebras underlying two weak bialgebras, we present suffici... more Given a weak distributive law between algebras underlying two weak bialgebras, we present sufficient conditions under which the corresponding weak wreath product algebra becomes a weak bialgebra with respect to the tensor product coalgebra structure. When the weak bialgebras are weak Hopf algebras, then the same conditions are shown to imply that the weak wreath product becomes a weak Hopf algebra, too. Our sufficient conditions are capable to describe most known examples, (in particular the Drinfel'd double of a weak Hopf algebra) .
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Papers by José Gómez-Torrecillas