We begin by discussing motivation for our consideration of a structure called a Lie 2-algebra, in particular an important class of Lie 2-algebras are the Courant Algebroids introduced in 1990 by Courant. We wish to attach some natural definitions from operad theory, mainly the notion of a module over an algebra, to Lie 2-algebras and hence to Courant algebroids. To this end our goal is to show that Lie 2-algebras can be described as what are called \emph{homotopy algebras over an operad}. Describing Lie 2-algebras using operads also solves the problem of showing that the equations defining a Lie 2-algebra are consistent. Our technical discussion begins by introducing some notions from operad theory, which is a generalization of the theory of operations on a set and their compositions. We define the idea of a quadratic operad and a homotopy algebra over a quadratic operad. We then proceed to describe Lie 2-algebras as homotopy algebras over a given quadratic operad using a theorem of Ginzburg and Kapranov. Next we briefly discuss the structure of a braided monoidal category. Following this, motivated by our discussion of braided monoidal categories, a new structure is introduced, which we call a commutative 2-algebra. As with the Lie 2-algebra case we show how a commutative 2-algebra can be seen as a homotopy algebra over a particular quadratic operad. Finally some technical results used in previous theorems are mentioned. In discussing these technical results we apply some ideas about distributive laws and Koszul operads.

We begin by discussing motivation for our consideration of a structure called a Lie 2-algebra, in particular an important class of Lie 2-algebras are the Courant Algebroids introduced in 1990 by Courant. We wish to attach some natural definitions from operad theory, mainly the notion of a module over an algebra, to Lie 2-algebras and hence to Courant algebroids. To this end our goal is to show that Lie 2-algebras can be described as what are called \emph{homotopy algebras over an operad}. Describing Lie 2-algebras using operads also solves the problem of showing that the equations defining a Lie 2-algebra are consistent. Our technical discussion begins by introducing some notions from operad theory, which is a generalization of the theory of operations on a set and their compositions. We define the idea of a quadratic operad and a homotopy algebra over a quadratic operad. We then proceed to describe Lie 2-algebras as homotopy algebras over a given quadratic operad using a theorem of Ginzburg and Kapranov. Next we briefly discuss the structure of a braided monoidal category. Following this, motivated by our discussion of braided monoidal categories, a new structure is introduced, which we call a commutative 2-algebra. As with the Lie 2-algebra case we show how a commutative 2-algebra can be seen as a homotopy algebra over a particular quadratic operad. Finally some technical results used in previous theorems are mentioned. In discussing these technical results we apply some ideas about distributive laws and Koszul operads.

We place a differential on $\dot\UC_{\mathfrak{sl}_3}^+$ and show that $\dot\UC_{\mathfrak{sl}_3}^+$ is Fc-filtered. This gives a categorification of the positive half of quantum $\sl_3$ at a prime root of unity.

<p>Quantum sl_3 projectors are morphisms in Kuperberg's sl_3 spider, a diagrammatically defined category equivalent to the full pivotal subcategory of the category of (type 1) finite-dimensional representations of the quantum group U_q (sl_3 ) generated by the defining representation, which correspond to projection onto (and then inclusion from) the highest weight irreducible summand. These morphisms are interesting from a topological viewpoint as they allow the combinatorial formulation of the sl_3 tangle invariant (in which tangle components are labelled by the defining representation) to be extended to a combinatorial formulation of the invariant in which components are labelled by arbitrary finite-dimensional irreducible representations. They also allow for a combinatorial description of the SU(3) Witten-Reshetikhin-Turaev 3-manifold invariant. </p><p>There exists a categorification of the sl_3 spider, due to Morrison and Nieh, which is the natural setting for Khovanov's sl_3 link homology theory and its extension to tangles. An obvious question is whether there exist objects in this categorification which categorify the sl_3 projectors. </p><p>In this dissertation, we show that there indeed exist such "categorified projectors," constructing them as the stable limit of the complexes assigned to k-twist torus braids (suitably shifted). These complexes satisfy categorified versions of the defining relations of the (decategorified) sl_3 projectors and map to them upon taking the Grothendieck group. We use these categorified projectors to extend sl_3 Khovanov homology to a homology theory for framed links with components labeled by arbitrary finite-dimensional irreducible representations of sl_3 .</p> / Dissertation

This dissertation uses techniques from the theory of categorical actions of Kac-Moody algebras to study the analog of the BGG category O for the queer Lie superalgebra. Chen recently reduced many questions about this category to its so-called types A, B, and C blocks. The type A blocks were completely described in joint work with Brundan in terms of the general linear Lie superalgebra. This dissertation proves that the type C blocks admit the structure of a tensor product categorification of the n-fold tensor power of the natural sp_\infty-module. Using this result, we relate the combinatorics for these blocks to Webster’s orthodox bases for the quantum group of type C_\infty, verifying the truth of a recent conjecture of Cheng-Kwon-Wang. This dissertation contains coauthored material.

Dans cette thèse, nous présentons des applications de la réécriture à l'étude de problèmes issus de la catégorification et de la théorie des représentations. En particulier, nous appliquons les méthodes de réécriture aux problèmes de cohérence dans les catégories linéaires et au calcul de décatégorifications. Des méthodes de réécriture ont été développées pour obtenir des résultats de cohérence dans les monoïdes et les catégories monoïdales présentés par des systèmes de réécriture nommés polygraphes. Ces constructions basées sur des résultats de Squier permettent en particulier de calculer des présentations cohérentes de catégories de dimension supérieure à partir des diagrammes de confluence de polygraphes convergents. Dans ce mémoire, nous étendons ces constructions pour obtenir des résultats de cohérence dans les catégories linéaires de dimension supérieure. Nous introduisons les polygraphes linéaires afin de présenter les catégories linéaires de dimension supérieure par des systèmes de réécriture. Nous étudions ensuite les propriétés de réécriture de ces systèmes. Nous donnons une description polygraphique du calcul de décatégorification de Grothendieck. Nous généralisons également la procédure de Knuth-Bendix appliquée aux polygraphes de dimension supérieure. Cette procédure permet de compléter des présentations de catégories de dimension supérieure n'admettant pas nécessairement d'ordre de terminaison induit par une orientation des règles. De plus, nous étudions des problèmes de cohérence dans les catégories de dimension supérieure. Etant donné un polygraphe confluent et quasi-terminant, nous introduisons une notion de complétion de Squier de ce polygraphe composée de diagrammes de décroissance. Nous prouvons que cette complétion rend asphérique la catégorie de dimension supérieure libre sur ce polygraphe. Ce résultat généralise un résultat de Squier au cas des présentations quasi-terminantes. Nous présentons enfin les applications des propriétés des polygraphes linéaire à l'étude de la catégorie AOB définie par Brundan, Comes, Nash et Reynolds. Nous retrouvons par des méthodes de réécriture les bases des espaces de morphismes de AOB exhibées par Brundan, Comes, Nash and Reynolds / In this thesis, we study applications of rewriting theory to categorification problems and representation theory. We apply rewriting methods to coherence problems in linear categories and computation of decategorifications.Proofs of coherence results for monoids and monoidal categories by rewriting methods are well known. In particular, several constructions based on Squier's results lead to the computation of coherent presentations of higher-dimensional categories from the confluence diagrams of convergent rewriting systems. In this memoir, we extend those constructions to coherence results for higher-dimensional linear categories.We introduce linear polygraphs to present higher-dimensional linear categories by rewriting systems. We then develop the main rewriting properties of these systems. We focus next on the applications of those properties to the study of categorification problems such that the computation of Grothendieck decategorification by rewriting methods. Another result we obtain on higher-dimensional polygraphs is a generalization of the Knuth-Bendix procedure to higher-dimensional polygraphs. This new procedure allows us to complete presentations of higher-dimensional categories which do not necessarily admit a termination order induced by any orientation of rules.We also study general coherence problems. Given a confluent and quasi-terminating polygraph, we define a globular extension of this polygraph called decreasing Squier's completion. We prove that this extension makes aspherical the free higher-dimensional category over the given polygraph. This result generalizes a result of Squier to the case of non terminating presentations.Finally, we focus on the applications of those properties to higher-dimensional linear categories such that the category AOB defined by Brundan, Comes, Nash and Reynolds. We find by rewriting methods the bases of the morphisms spaces of AOB that Brundan, Comes, Nash and Reynolds exhibited

No description available.

We define a faithful linear monoidal functor from the partition category, and hence from Deligne's category Rep(S_t), to the additive Karoubi envelope of the Heisenberg category. We show that the induced map on Grothendieck rings is injective and corresponds to the Kronecker coproduct on symmetric functions. We then generalize the above results to any group G, the case where G is the trivial group corresponding to the case mentioned above. Thus, to every group G we associate a linear monoidal category Par(G) that we call a group partition category. We give explicit bases for the morphism spaces and also an efficient presentation of the category in terms of generators and relations. We then define an embedding of Par(G) into the group Heisenberg category associated to G. This embedding intertwines the natural actions of both categories on modules for wreath products of G. Finally, we prove that the additive Karoubi envelope of Par(G) is equivalent to a wreath product interpolating category introduced by Knop, thereby giving a simple concrete description of that category.

We define the affinization of an arbitrary monoidal category C, corresponding to the category of C-diagrams on the cylinder. We also give an alternative characterization in terms of adjoining dot generators to C. The affinization formalizes and unifies many constructions appearing in the literature. In particular, we describe a large number of examples coming from Hecke-type algebras, braids, tangles, and knot invariants. When C is rigid, its affinization is isomorphic to its horizontal trace, although the two definitions look quite different. In general, the affinization and the horizontal trace are not isomorphic. We then use the affinization to show our main result, which is an explicit isomorphism between the central charge k reduction of the universal central extension of the elliptic Hall algebra and the trace, or zeroth Hochschild homology, of the quantum Heisenberg category of central charge k. We use this isomorphism to construct large families of representations of the universal extension of the elliptic Hall algebra.

By extending Husserl’s own historico-critical study to include the conceptual mathematics of more contemporary times – specifically category theory and its emphatic development since the second half of the 20th century – this paper claims that the delineation between mathematics and philosophy must be completely revisited. It will be contended that Husserl’s phenomenological work was very much influenced by the discoveries and limitations of the formal mathematics being developed at Göttingen during his tenure there and that, subsequently, the rôle he envisaged for his material a priori science is heavily dependent upon his conception of the definite manifold. Motivating these contentions is the idea of a mathematics which would go beyond the constraints of formal ontology and subsequently achieve coherence with the full sense of transcendental phenomenology. While this final point will be by no means proven within the confines of this paper it is hoped that the very fact of opening up for the possibility of such an idea will act as a supporting argument to the overriding thesis that the relationship between mathematics and phenomenology must be problematised.