Essen, University, Diss., 2001.

Cette thèse s’inscrit dans l’étude des catégories d’algèbres associées aux opérades. On développe des outils d’algèbre homologique et une méthode générale de classification (à homotopie près) des morphismes entre algèbres sur une opérade.La dualité de Koszul des opérades, introduite par V. Ginzburg et M. Kapranov, permet de construire des théories homologiques appropriées pour des catégories d’algèbres associées à certaines bonnes opérades – les opérades de Koszul. On donne dans la première partie de cette thèse un critère effectif pour qu’une opérade soit de Koszul : on montre qu’une opérade, linéairement engendrée par une base, est de Koszul dès lors que l’on peut ordonner sa base de façon compatible avec la structure de composition opéradique – on parle alors d’opérade de Poincaré-Birkhoff-Witt.La théorie originale de Ginzburg-Kapranov s’applique en caractéristique nulle seulement. On construit une théorie homologique adaptée - la Gamma-homologie - pour l’étude des catégories d’algèbres différentielles graduées associées à une opérade de Koszul en toute caractéristique. Cette théorie généralise la Gamma-homologie définie par A. Robinson et S. Whitehouse pour la catégorie des algèbres commutatives.On montre que la Gamma-homologie opéradique contient l’obstruction à la réalisation de morphismes entre algèbres sur une opérade, ainsi que l’obstruction à la réalisation d’homotopies entre morphismes, et donne de la sorte un outil général pour classifier les morphismes entre algèbres sur une opérade. / This thesis is concerned with the study of categories of algebras associated to operads. We develop tools of homological algebra and a general method to classify morphisms in the homotopy category of algebras over an operad.The Koszul duality of operads, introduced by V. Ginzburg and M. Kapranov, allows us to construct suitable homology theories for categories of algebras associated to some good operads – the Koszul operads. We give in the first part of this thesis an effective criterion to prove that an operad is Kozul : we show that an operad, linearly generated by a basis, is Koszul as soon as we can order its basis compatibly with the operadic composition structure – we call such operads Poincaré-Birkhoff-Witt operads.The original theory of Ginzburg and Kapranov works in characteristic zero only. We construct a homology theory - the Gamma-homology - for the study of the categories of the differential graded algebras associated to a Koszul operad in any characteristic. This theory generalizes the Gamma-homology introduced by A. Robinson and S. Whitehouse for the category of commutative algebras.We show that our Gamma-homology contains the obstruction to the realization of morphisms between algebras over an operad, and also the obstruction to the realization of homotopies between morphisms. We obtain in this way a general tool to classify morphisms between algebras over an operad.

In this thesis, we present a detailed exposition of Koszul algebras and Koszul duality. We begin with an overview of the required concepts of graded algebras and homological algebra. We then give a precise treatment of Koszul and quadratic algebras, together with their dualities. We fill in some arguments that are omitted in the literature and work out a number of examples in full detail to illustrate the abstract concepts.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, Jun. 2003. / Includes bibliographical references (p. 25-26). Also available on the Internet.

An algebra is a typical object of study in pure mathematics. Take a collection of numbers (for example, all whole numbers or all decimal numbers). Inside, you can add and multiply, but with respect to these operations different collections can behave differently. Here is an example of what I mean by this. The collection of whole numbers is called Z. Starting anywhere in Z you can get to anywhere else by adding other members of the collection: 9 + (-3) + (-6) = 0. This is not true with multiplication; to get from 5 to 1 you would need to multiply by 1/5 and 1/5 doesn’t exist in the restricted universe of Z. Enter R, the collection of all numbers that can be written as decimals. Now, if you start anywhere—apart from 0—you can get to anywhere else by multiplying by members of R—if you start at zero you’re stuck there. By adjusting what you mean by ‘add’ and ‘multiply’, you can add and multiply other things too, like polynomials, transformations or even symmetries. Some of these collections look different, but behave in similar ways and some look the same but are subtly different. By defining an algebra to be any collection of things with a rule to add and multiply in a sensible way, all of these examples (and many more you can’t imagine) can be treated in general. This is the power of abstraction: proving that an arbitrary algebra, A, has some property implies that every conceivable algebra (including Z and R) has that property too. In order to start navigating this universe of algebras it is useful to group them together by their behaviour or by how they are constructed. For example, R belongs to a class called simple algebras. There are mental laboratories full of machinery used to construct new and interesting algebras from old ones. One recipe, invented by Ivan Cherednik in 1993, produces Cherednik algebras. Attached to each algebra is a collection of modules (also called representations). As shadows are to a sculpture, each module is a simplified version of the algebra, with a taste of its internal structure. They are not algebras in their own right: they have no sense of multiplication, only addition. Being individually simple, modules are often much easier to study than the algebra itself. However, everything that is interesting about an algebra is captured by the collective behaviour of its modules. The analogy fails here: for example, shadows encode no information about colour. Sometimes the interplay between its modules leads to subtle and unexpected insights about the algebra itself. Nobody understands what the modules for Cherednik algebras look like. One first step is to simplify the problem by only considering modules which behave ‘nicely’. This is what is referred to as category O. Being Koszul is a rare property of an algebra that greatly helps to describe its behaviour. Also, each Koszul algebra is mysteriously linked with another called its Koszul dual. One of the main results of the thesis is that, in some cases, the modules in category O behave as if they were the modules for some Koszul algebra. It is an interesting question to ask, what the Koszul dual might be and what this has to do with Cherednik’s recipe. Geometers study tangled, many-dimensional spaces with holes. In analogy with the algebraic world, just as algebraists use modules to study algebras, geometers use sheaves to study their spaces. Suppose one could construct sheaves on some space whose behaviour is precisely the same as Cherednik algebra modules. Then, for example, theorems from geometry about sheaves could be used to say something about Cherednik algebra modules. One way of setting up this analogy is called localisation. This doesn’t always work in general. The last part of the thesis provides a rule for checking when it does.

A new connection between combinatorics and noncommutative algebra is established by relating a certain class of directed graphs to noncommutative Koszul algebras. The directed graphs in this class are called full graphs and are defined by a set of criteria on the edges. The structural properties of full graphs are studied as they relate to the edge criteria. A method is introduced for generating a Koszul algebra Lambda from a full graph G. The properties of Lambda are examined as they relate to the structure of G, with special attention being given to the construction of a projective resolution of certain semisimple Lambda-modules based on the structural properties of G. The characteristics of the Koszul algebra Lambda that is derived from the product of two full graphs G' and G' are studied as they relate to the properties of the Koszul algebras Lambda' and Lambda' derived from G' and G'. / Ph. D.

In this paper, for a select group of quadratic algebras, we investigate restrictions necessary on the generators of the ideal for the resulting algebra to be Koszul. Techniques include the use of Gröbner bases and development of Koszul resolutions. When the quadratic algebra is Koszul, we provide the associated linear resolution of the field. When not Koszul, we describe the maps of the resolution up to the instance of nonlinearity. / Ph. D.

Orientador: Adriano Adrega de Moura / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T09:20:36Z (GMT). No. of bitstreams: 1 Ferreira_GilmardeSousa_M.pdf: 480832 bytes, checksum: 42db64084fe99183a4fb95f90cd9a833 (MD5) Previous issue date: 2012 / Resumo: Nessa dissertação estudamos certas categorias de módulos graduados para uma classe de álgebras de Lie que inclui as álgebras de correntes. Em particular, estudamos diversas propriedades homológicas dessas categorias tais como resoluções projetivas e o espaço de extensões entre seus objetos simples. Em certas situações, os resultados levam ao estabelecimento de um relacionamento com álgebras de Koszul. O estudo é baseado em artigos recentes de Vyjayanthi Chari e seus co-autores / Abstract: In this dissertation, we study certain categories of graded modules for a class of Lie algebras which include current algebras. In particular, we study several homological properties of these categories such as projective resolutions and the space of extensions between two given simple objects. Under certain conditions, these results establish a relationship with Koszul algebras. The study is based on recent papers by Vyjayanthi Chari and her co-authors / Mestrado / Matematica / Mestre em Matemática

Recent work of Bambozzi, Ben-Bassat, and Kremnitzer suggests that derived analytic geometry over a valued field k can be modelled as geometry relative to the quasi-abelian category of Banach spaces, or rather its completion Ind(Ban_k). In this thesis we develop a robust theory of homotopical algebra in Ch(E) for E any sufficiently 'nice' quasi-abelian, or even exact, category. Firstly we provide sufficient conditions on weakly idempotent complete exact categories E such that various categories of chain complexes in E are equipped with projective model structures. In particular we show that as soon as E has enough projectives, the category Ch₊(E) of bounded below complexes is equipped with a projective model structure. In the case that E also admits all kernels we show that it is also true of Ch≥0(E), and that a generalisation of the Dold-Kan correspondence holds. Supplementing the existence of kernels with a condition on the existence and exactness of certain direct limit functors guarantees that the category of unbounded chain complexes Ch(E) also admits a projective model structure. When E is monoidal we also examine when these model structures are monoidal. We then develop the homotopy theory of algebras in Ch(E). In particular we show, under very general conditions, that categories of operadic algebras in Ch(E) can be equipped with transferred model structures. Specialising to quasi-abelian categories we prove our main theorem, which is a vast generalisation of Koszul duality. We conclude by defining analytic extensions of the Koszul dual of a Lie algebra in Ind(Ban_k).

A prop profile of a differential geometric structure is a minimal resolution of an algebraic prop such that representations of this resolution are in one-to-one correspondence with structures of the given type. We begin this thesis with a detailed account of the algebraic tools necessary to construct prop profiles; we treat operads and props, and resolutions of these through Koszul duality. Our main results can be summarized as follows. Firstly, we contribute to the work of S.A. Merkulov on the prop profiles of Poisson and Nijenhuis structures. We prove that the operad of the latter prop profile is Koszul by showing that it has a PBW-basis, and we provide a geometrical interpretation of the former in terms of an L-infinity structure on the structure sheaf of a manifold. Secondly, we construct prop profiles of compatible Poisson and Nijenhuis structures. Representations of minimal resolutions of props correspond to Maurer-Cartan elements of certain Lie algebras associated to the resolved props. Also the differential geometric structures are defined as solutions of Maurer-Cartan equations. We show the correspondence between props and differential geometry by providing explicit isomorphisms between these Lie algebras. Thirdly, in order to construct the prop profiles of compatible Poisson and Nijenhuis structures we study operads of compatible algebraic structures. By studying Cohen-Macaulay properties of posets associated to such operads we prove the Koszulness of a large class of operads of compatible structures.