Algèbre associative

Résumé

vignette|Relations entre certaines structures algébriques. En mathématiques, une algèbre associative (sur un anneau commutatif A) est une des structures algébriques utilisées en algèbre générale. C'est un anneau (ou simplement un pseudo-anneau) B muni d'une structure supplémentaire de module sur A et tel que la loi de multiplication de l'anneau B soit A-bilinéaire. C'est donc un cas particulier d'algèbre sur un anneau. Définition formelle Soit A un anneau commutatif. On dit que (B , + , . , × ) est une A-algèbre associative lorsque : #(B , + , . ) est un A-module, #(B , + , × ) est un pseudo-anneau, #\forall \lambda \in A,~\forall x, y \in B,\qquad\lambda \cdot (x \times y) = x \times (\lambda \cdot y) = (\lambda \cdot x) \times y~. Les éléments de A sont appelés les scalaires. Dans le cas particulier où l'anneau A est un corps, on parle alors d'algèbre associative sur un corps. On parle d'algèbre unitaire (ou unifère) lorsque B possède un neutre pour la mul

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https://fr.wikipedia.org/wiki/Alg%C3%A8bre_associative

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In this thesis, we study interactions between algebraic and coalgebraic structures in infinity-categories (more precisely, in the quasicategorical model of (infinity, 1)-categories). We define a notion of a Hopf algebra H in an E-2-monoidal infinity-category and lift some results about ordinary Hopf algebras, such as the fundamental theorem of Hopf modules, to this setting. We also study Hopf-Galois extensions in this context. Given a candidate Hopf-Galois extension, i.e., a map f : A -> B of H-comodule algebras where H coacts on A trivially, we construct a structured version of the comparison map B (x)_A B -> H (x) B that allows us to compare the category of descent data for f with a category of "B-modules equipped with a semilinear coaction of H". We provide further insights into the case of commutative (i.e., E-infinity) comodule algebras over a commutative Hopf algebra, for instance a description of the aforementioned category of modules equipped with a semilinear coaction as the limit of a "categorified cobar construction". Moreover, we provide a simple description of comodules over a space in slice categories of the infinity-category of spaces, which enables us to realize multiplicative Thom objects as comodule algebras and thus incorporate them into the aforementioned framework.

EPFL2022

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Isomorphism Problems of Noncommutative Deformations of Type D Kleinian Singularities

We construct all possible noncommutative deformations of a Kleinian singularity C-2/Gamma of type D-n in terms of generators and relations, and solve the isomorphism problem for the associative algebras thus constructed. We prove that (in our parametrization) all isomorphisms arise from the action of the normalizer N-SL(2)(Gamma) on C/Gamma. We deduce that the moduli space of isomorphism classes of noncommutative deformations in type D-n is isomorphic to a vector space of dimension n.

2009

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Algebraic Homotopy Interleaving Distance

Nicolas Michel Berkouk

The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous difficulties, where it has been proven that no barcode-like decomposition exists. To tackle this problem, algebraic invariants have been proposed to summarize multi-parameter persistence modules, adapting classical ideas from commutative algebra and algebraic geometry to this context. Nevertheless, the crucial question of their stability has raised little attention so far, and many of the proposed invariants do not satisfy a naive form of stability. In this paper, we equip the homotopy and the derived category of multi-parameter persistence modules with an appropriate interleaving distance. We prove that resolution functors are always isometric with respect to this distance. As an application, this explains why the graded-Betti numbers of a persistence module do not satisfy a naive form of stability. This opens the door to performing homological algebra operations while keeping track of stability. We believe this approach can lead to the definition of new stable invariants for multi-parameter persistence, and to new computable lower bounds for the interleaving distance (which has been recently shown to be NP-hard to compute in [2]).

SPRINGER INTERNATIONAL PUBLISHING AG2021

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Algèbre associative

(A, +, ·) est un

Isomorphism Problems of Noncommutative Deformations of Type D Kleinian Singularities

Algebraic Homotopy Interleaving Distance

Algebraic Homotopy Interleaving Distance

Isomorphism Problems of Noncommutative Deformations of Type D Kleinian Singularities