Produit tensoriel

Résumé

En mathématiques, le produit tensoriel est un moyen commode de coder les objets multilinéaires. Il est utilisé en algèbre, en géométrie différentielle, en géométrie riemannienne, en analyse fonctionnelle et en physique (mécanique des solides, relativité générale et mécanique quantique). Produit tensoriel d'espaces vectoriels Définition Théorème et définition. Soient E et F deux espaces vectoriels sur un corps commutatif K. Il existe un espace vectoriel, noté E\otimes F , et une application bilinéaire \phi :E\times F \rightarrow E\otimes F (on pose \phi(x,y)=x\otimes y) ayant la propriété suivante (dite universelle) : pour tout espace vectoriel G sur le même corps K, et pour toute application bilinéaire g de E \times F dans G, il existe une et une seule application linéaire \tilde g de E\otimes F

Source officielle

https://fr.wikipedia.org/wiki/Produit_tensoriel

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Publications associées (45)

Publications associées

Chargement

Personnes associées (5)

Personnes associées

Chargement

Unités associées (4)

Unités associées

Chargement

Concepts associés (85)

Concepts associés

Chargement

Cours associés (58)

Cours associés

Chargement

Séances de cours associées (105)

Séances de cours associées

Chargement

Generic direct summands of tensor products for simple algebraic groups and quantum groups at roots of unity

Jonathan Gruber

Let G be either a simple linear algebraic group over an algebraically closed field of characteristic l>0 or a quantum group at an l-th root of unity. The category Rep(G) of finite-dimensional G-modules is non-semisimple. In this thesis, we develop new techniques for studying Krull-Schmidt decompositions of tensor products of G-modules.More specifically, we use minimal complexes of tilting modules to define a tensor ideal of singular G-modules, and we show that, up to singular direct summands, taking tensor products of G-modules respects the decomposition of Rep(G) into linkage classes. In analogy with the classical translation principle, this allows us to reduce questions about tensor products of G-modules in arbitrary l-regular linkage classes to questions about tensor products of G-modules in the principal block of G. We then identify a particular non-singular indecomposable direct summand of the tensor product of two simple G-modules in the principal block (with highest weights in two given l-alcoves), which we call the generic direct summand because it appears generically in Krull-Schmidt decompositions of tensor products of simple G-modules (with highest weights in the given l-alcoves). We initiate the study of generic direct summands, and we use them to prove a necessary condition for the complete reducibility of tensor products of simple G-modules, when G is a simple algebraic group of type A_n.

EPFL2022

Chargement

A language is said to be homogeneous when all its words have the same length. Homogeneous languages thus form a monoid under concatenation. It becomes freely commutative under the simultaneous actions of every permutation group G(n) on the collection of homogeneous languages of length n is an element of N. One recovers the isothetic regions from (Haucourt 2017, to appear (online since October 2017)) by considering the alphabet of connected subsets of the space vertical bar G vertical bar, viz the geometric realization of a finite graph G. Factoring the geometric model of a conservative program amounts to parallelize it, and there exists an efficient factoring algorithm for isothetic regions. Yet, from the theoretical point of view, one wishes to go beyond the class of conservative programs, which implies relaxing the finiteness hypothesis on the graph G. Provided that the collections of n-dimensional isothetic regions over G (denoted by R-n vertical bar G vertical bar) are co -unital distributive lattices, the prime decomposition of isothetic regions is given by an algorithm which is, unfortunately, very inefficient. Nevertheless, if the collections R-n vertical bar G vertical bar satisfy the stronger property of being Boolean algebras, then the efficient factoring algorithm is available again. We relate the algebraic properties of the collections R-n vertical bar G vertical bar to the geometric properties of the space I GI. On the way, the algebraic structure R-n vertical bar G vertical bar is proven to be the universal tensor product, in the category of semilattices with zero, of n copies of the algebraic structure R-1 vertical bar G vertical bar.

2019

Chargement

The Chu Construction For Complete Atomistic Coatomistic Lattices

Boris Ischi, Gavin Jay Seal

The Chu construction is used to define a *-autonomous structure on a category of complete atomistic coatomistic lattices. This construction leads to a new tensor product that is compared with a certain number of other existing tensor products.

Univ Houston2013

Chargement