Diagrammatic categorification of the Chebyshev polynomials of the second kind

We develop a diagrammatic categorification of the polynomial ring Z[x], based on a geometrically defined graded algebra. This construction generalizes to categorification of some special functions, such as Chebyshev polynomials. Diagrammatic algebras featured in these categorifications lead to the first topological interpretations of the Bernstein-Gelfand-Gelfand reciprocity property.     

Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

We construct an explicit isomorphism between blocks of cyclotomic Hecke algebras and (sign-modified) cyclotomic Khovanov-Lauda algebras in type A. These isomorphisms connect the categorification conjecture of Khovanov and Lauda to Ariki’s categorification theorem. The Khovanov-Lauda algebras are naturally graded, which allows us to exhibit a non-trivial ℤ-grading on blocks of cyclotomic Hecke algebras, including symmetric groups in positive characteristic.     

Categorification, term rewriting and the Knuth-Bendix procedure

An axiomatization of a finitary, equational universal algebra by a convergent term rewrite system gives rise to a finite, coherent categorification of the algebra.     

Categorification and Group Extensions

We review several known categorification procedures, and introduce a functorial categorification of group extensions (Section 4.1) with applications to non-Abelian group cohomology (Section 4.2). The obstruction to the existence of group extensions (Section 4.2.4, Equation (9)) is interpreted as a coboundary condition (Proposition 4.5).     

Categorification of (induced) cell modules and the rough structure of generalised Verma modules

This paper presents categorifications of (right) cell modules and induced cell modules for Hecke algebras of finite Weyl groups. In type A we show that these categorifications depend only on the isomorphism class of the cell module, not on the cell itself. Our main application is multiplicity formulas for parabolically induced modules over a reductive Lie algebra of type A, which finally determines the so-called rough structure of generalised Verma modules. On the way we present several categorification results and give a positive answer to Kostant's problem from [A. Joseph, Kostant's problem, Goldie rank and the Gelfand-Kirillov conjecture, Invent. Math. 56 (3) (1980) 191-213] in many cases. We also present a general setup of decategorification, precategorification and categorification.     

A Quillen model for classical Morita theory and a tensor categorification of the Brauer group

Let K$\mathbb{K}$ be a commutative ring. In this article we construct a well-behaved symmetric monoidal Quillen model structure on the category of small K$\mathbb{K}$-categories which enhances classical Morita theory. Making use of it, we then obtain the usual categorification of the Brauer group and of its functoriality. Finally, we prove that the (contravariant) corestriction map for finite Galois extensions also lifts to this categorification.     

Fiat categorification of the symmetric inverse semigroup $$\textit{IS}_n$$ and the semigroup $$F^*_n$$Fn∗

Starting from the symmetric group \(S_n\), we construct two fiat 2-categories. One of them can be viewed as the fiat “extension” of the natural 2-category associated with the symmetric inverse semigroup (considered as an ordered semigroup with respect to the natural order). This 2-category provides a fiat categorification for the integral semigroup algebra of the symmetric inverse semigroup. The other 2-category can be viewed as the fiat “extension” of the 2-category associated with the maximal factorizable subsemigroup of the dual symmetric inverse semigroup (again, considered as an ordered semigroup with respect to the natural order). This 2-category provides a fiat categorification for the integral semigroup algebra of the maximal factorizable subsemigroup of the dual symmetric inverse semigroup.     

The categorified Heisenberg algebra I: A combinatorial representation

We show how Khovanov's categorification of the Heisenberg algebra arises as a linearization of a discrete combinatorial structure in the bicategory of spans of groupoids. We also treat a categorification of $U({\mathrm{sl}}_n)$ in a similar way.     

Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras

In this paper, we prove Khovanov-Lauda??s cyclotomic categorification conjecture for all symmetrizable Kac-Moody algebras. Let $U_{q}(\mathfrak{g})$ be the quantum group associated with a symmetrizable Cartan datum and let V(??) be the irreducible highest weight $U_{q}(\mathfrak{g})$ -module with a dominant integral highest weight ??. We prove that the cyclotomic Khovanov-Lauda-Rouquier algebra R ^?? gives a categorification of V(??).     

$D_4$型李代数包络代数旋模的范畴化

本文我们定义复数域$C$上一般线性李代数${\rm gl}_n$ BGG 范畴的若干子范畴及其上的投射函子,利用这些子范畴和投射函子范畴化了$D_4$型李代数包络代数旋模的$n$-次张量积.     

Infinitesimal 2-braidings and differential crossed modules

We categorify the notion of an infinitesimal braiding in a linear strict symmetric monoidal category, leading to the notion of a (strict) infinitesimal 2-braiding in a linear symmetric strict monoidal 2-category. We describe the associated categorification of the 4-term relations, leading to six categorified relations. We prove that any infinitesimal 2-braiding gives rise to a flat and fake flat 2-connection in the configuration space of n particles in the complex plane, hence to a categorification of the Knizhnik–Zamolodchikov connection. We discuss infinitesimal 2-braidings in a certain monoidal 2-category naturally assigned to every differential crossed module, leading to the notion of a symmetric quasi-invariant tensor in a differential crossed module. Finally, we prove that symmetric quasi-invariant tensors exist in the differential crossed module associated to Wagemann's version of the String Lie-2-algebra. As a corollary, we obtain a more conceptual proof of the flatness of a previously constructed categorified Knizhnik–Zamolodchikov connection with values in the String Lie-2-algebra.     

Uno's Invariant Conjecture for the Finite Symplectic Group Sp4(q) in the Defining Characteristic

The purpose of this article is to study a categorification of the n-th tensor power of the spin representation of U(𝔰𝔬(7, ?)) by using certain subcategories and projective functors of the Bernstein–Gelfand–Gelfand (BGG) category of the complex Lie algebra 𝔤𝔩 _n .     

A categorification of finite-dimensional irreducible representations of quantum \mathfraksl2{\mathfrak{sl}_2} and their tensor products

The purpose of this paper is to study categorifications of tensor products of finite-dimensional modules for the quantum group for . The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra . For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed.We also give a categorical version of the quantised Schur–Weyl duality and an interpretation of the (dual) canonical bases and the (dual) standard bases in terms of projective, tilting, standard and simple Harish-Chandra bimodules.     

Categorification of invariants in gauge theory and symplectic geometry

This is a mixture of survey article and research announcement. We discuss instanton Floer homology for 3 manifolds with boundary. We also discuss a categorification of the Lagrangian Floer theory using the unobstructed immersed Lagrangian correspondence as a morphism in the category of symplectic manifolds. During the year 1998–2012, those problems have been studied emphasizing the ideas from analysis such as degeneration and adiabatic limit (instanton Floer homology) and strip shrinking (Lagrangian correspondence). Recently we found that replacing those analytic approach by a combination of cobordism type argument and homological algebra, we can resolve various difficulties in the analytic approach. It thus solves various problems and also simplify many of the proofs.     

The degenerate analogue of Ariki’s categorification theorem

We explain how to deduce the degenerate analogue of Ariki’s categorification theorem over the ground field \mathbbC{\mathbb{C}} as an application of Schur–Weyl duality for higher levels and the Kazhdan–Lusztig conjecture in finite type A. We also discuss some supplementary topics, including Young modules, tensoring with sign, tilting modules and Ringel duality.     

On structure of cluster algebras of geometric type Ⅰ:In view of sub-seeds and seed homomorphisms

Our motivation is to build a systematic method in order to investigate the structure of cluster algebras of geometric type. The method is given through the notion of mixing-type sub-seeds, the theory of seed homomorphisms and the view-point of gluing of seeds. As an application, for(rooted) cluster algebras, we completely classify rooted cluster subalgebras and characterize rooted cluster quotient algebras in detail. Also,we build the relationship between the categorification of a rooted cluster algebra and that of its rooted cluster subalgebras. Note that cluster algebras of geometric type studied here are of the sign-skew-symmetric case.     

Even More Spectra: Tensor Triangular Comparison Maps via Graded Commutative 2-rings

We initiate the theory of graded commutative 2-rings, a categorification of graded commutative rings. The goal is to provide a systematic generalization of Paul Balmer’s comparison maps between the spectrum of tensor-triangulated categories and the Zariski spectra of their central rings. By applying our constructions, we compute the spectrum of the derived category of perfect complexes over any graded commutative ring, and we associate to every scheme with an ample family of line bundles an embedding into the spectrum of an associated graded commutative 2-ring.     

On structure of cluster algebras of geometric type I: In view of sub-seeds and seed homomorphisms

Our motivation is to build a systematic method in order to investigate the structure of cluster algebras of geometric type. The method is given through the notion of mixing-type sub-seeds, the theory of seed homomorphisms and the view-point of gluing of seeds. As an application, for (rooted) cluster algebras, we completely classify rooted cluster subalgebras and characterize rooted cluster quotient algebras in detail. Also, we build the relationship between the categorification of a rooted cluster algebra and that of its rooted cluster subalgebras. Note that cluster algebras of geometric type studied here are of the sign-skew-symmetric case.     

Categorified quantum sl2 is an inverse limit of flag 2-categories

We prove that categorified quantum sl₂ is an inverse limit of Flag 2-categories defined using cohomology rings of iterated flag varieties. This inverse limit is an instance of a 2-limit in a bicategory giving rise to a universal property that characterizes the categorification of quantum sl₂ uniquely up to equivalence. As an application we characterize all bimodule homomorphisms in the Flag 2-category and prove that on the homological level the categorified quantum Casimir of sl₂ acts appropriately on these 2-representations.