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.

This thesis contains four related papers which study different aspects of double constructions for braided Hopf algebras. The main result is a categorical action of a braided version of the Drinfeld center on a Heisenberg analogue, called the Hopf center. Moreover, an application of this action to the representation theory of rational Cherednik algebras is considered. Chapter 1 : In this chapter, the Drinfeld center of a monoidal category is generalized to a class of mixed Drinfeld centers. This gives a unified picture for the Drinfeld center and a natural Heisenberg analogue. Further, there is an action of the former on the latter. This picture is translated to a description in terms of Yetter-Drinfeld and Hopf modules over quasi-bialgebras in a braided monoidal category. Via braided reconstruction theory, intrinsic definitions of braided Drinfeld and Heisenberg doubles are obtained, together with a generalization of the result of Lu (1994) that the Heisenberg double is a 2-cocycle twist of the Drinfeld double for general braided Hopf algebras. Chapter 2 : In this chapter, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Benkart and Witherspoon (2004) who showed that two-parameter quantum groups are Drinfeld doubles. It is possible to recover a Lie algebra from these doubles in the case where the group is free and the parameters are generic. The Lie algebras arising are generated by Lie subalgebras isomorphic to sl2. Chapter 3 : The universal enveloping algebra <i>U</i>(tr_n) of a Lie algebra associated to the classical Yang-Baxter equation was introduced in 2006 by Bartholdi-Enriquez-Etingof-Rains where it was shown to be Koszul. This algebra appears as the A_<i>n</i>-1 case in a general class of braided Hopf algebras in work of Bazlov-Berenstein (2009) for any complex reection group. In this chapter, we show that the algebras corresponding to the series <i>B_n</i> and <i>D_n</i>, which are again universal enveloping algebras, are Koszul. This is done by constructing a PBW-basis for the quadratic dual. We further show how results of Bazlov-Berenstein can be used to produce pairs of adjoint functors between categories of rational Cherednik algebra representations of different rank and type for the classical series of Coxeter groups. Chapter 4 : Quantum groups can be understood as braided Drinfeld doubles over the group algebra of a lattice. The main objects of this chapter are certain braided Drinfeld doubles over the Drinfeld double of an irreducible complex reflection group. We argue that these algebras are analogues of the Drinfeld-Jimbo quantum enveloping algebras in a setting relevant for rational Cherednik algebra. This analogy manifests itself in terms of categorical actions, related to the general Drinfeld-Heisenberg double picture developed in Chapter 2, using embeddings of Bazlov and Berenstein (2009). In particular, this work provides a class of quasitriangular Hopf algebras associated to any complex reflection group which are in some cases finite-dimensional.

Cette thèse présente quelques résultats de la théorie des représentations des algèbres de Cherednikrationnelles en t=0 et traite en particulier des différents ordres construits sur la partition de Calogero-Moserdes groupes imprimitifs.On commence par généraliser au cas abélien certains résultats obtenus par M. Chlouveraki concernant lesblocs d'algèbres en système de Clifford pour un groupe cyclique, puis on construit un ordre sur les C*-pointsfixes d'une variété complexe quasi-projective normale, en utilisant la décomposition de Bialynicki-Birula.Dans la deuxième partie, on s'intéresse à la description des partitions de Calogero-Moser de deux groupesde réflexions complexes K et W quand K est un sous-groupe distingué de W et on généralise au cas abélienles résultats obtenus par G. Bellamy dans le cas d'un quotient W/K cyclique.Dans la troisième partie, on présente les différents ordres, construits par I. Gordon, sur la partition deCalogero-Moser des groupes G(l,1,n) pour certains paramètres : les ordres des a et c-fonctions, un ordrecombinatoire et l'ordre géométrique, qui est défini grâce aux C*-points fixes de certaines variétés decarquois, ces points fixes paramétrant les blocs de la partition de Calogero-Moser de G(l,1,n). On donneensuite les relations entre ces ordres, puis on étend ces constructions ainsi que ces liens à l'ensemble desparamètres.Enfin, dans la dernière partie, on tente de généraliser ces propriétés aux groupes G(l,e,n). On cherche alors,pour construire l'ordre géométrique sur la partition de Calogero-Moser de G(l,e,n), une variété dont les C*-points fixes décrivent les blocs de la partition de G(l,e,n). Dans le cas où e ne divise pas n, on construit lavariété qui nous permet de définir l'ordre géométrique et de le relier aux autres ordres. Pour le cas e divise n,on propose une variété qui pourrait décrire par ses points fixes les blocs de Calogero-Moser de G(l,e,n) etnous permettre de construire l'ordre géométrique. / This work is a contribution to the representation theory of Rational Cherednik Algebras for t=0 and deals inparticular with different orders on the Calogero-Moser partition of imprimitive reflection groups.In the first part, we generalize to the abelian case some results about blocs of algebras in Clifford systemobtained by M. Chlouveraki in the cyclic case, and then we build an order on the C*-fixed points of acomplex, quasi-projective and normal variety, using the Bialynicki-Birula decomposition.The second part deals with the Calogero-Moser partition of two groups K and W, when K is a normalsubgroup of W, and generalize to the abelian case the results that G. Bellamy obtained when the quotientW/K is cyclic.In the third part, we present the different orders that I. Gordon built in the Calogero-Moser partition of thegroups G(l,1,n) and for some parameters : the orders of the a and c-functions, a combinatorial order and thegeometric order, defined using the C*-fixed points of some quiver varieties which parametrise the blocs of theCalogero-Moser partition of G(l,1,n). Then we give some relations between these orders and we extendthese constructions and these links for all parameters.Finally, in the last part, we try to generalize these properties for the groups G(l,e,n). We are looking for avariety whose C*-fixed points describe blocs of G(l,e,n) to construct the geometric order on the Calogero-Moser partition of G(l,e,n). When n is not divided by e, we build this variety that enables us to define thegeometric order and to show all the links with the other orders. When e don't divide n, we suggest a varietywhich could describe the blocs of G(l,e,n) and allow us to build the geometric order.