Related papers: Categories of $D$-modules on a quantized flag mani…

Following Mitchell's philosophy, in this paper we define the analogous of the triangular matrix algebra to the context of rings with several objects. Given two additive categories $\mathcal{U}$ and $\mathcal{T}$ and $M\in…

We give a Quillen equivalence between May and Sigurdsson's model category of parametrized spectra over BG, and Mandell, May, Schwede, and Shipley's model category of modules over the orthogonal ring spectrum \Sigma^\infty_+ G, for each…

In [arXiv:1509.02937], the notion of a module tensor category was introduced as a braided monoidal central functor $F\colon \mathcal{V}\longrightarrow \mathcal{T}$ from a braided monoidal category $\mathcal{V}$ to a monoidal category…

We classify group gradings on the simple Lie algebra $L$ of type $D_4$ over an algebraically closed field of characteristic different from 2: fine gradings up to equivalence and $G$-gradings, with a fixed group $G$, up to isomorphism. For…

Higher cluster categories were recently introduced as a generalization of cluster categories. This paper shows that in Dynkin types A and D, half of all higher cluster categories are actually just quotients of cluster categories. The other…

Given a chiral algebra, we study modules over an arbitrary power of a curve. We describe this category in three different ways: in terms of factorization, in terms of certain chiral operations and as modules for a lie algebra in a certain…

Let $R$ be an associative ring with identity. This paper investigates the structure of the monomorphism category of large $R$-modules and establishes connections with the category of contravariant functors defined on finitely presented…

We use group homology to define invariants in algebraic K-theory and in an analogue of the Bloch group for Q-rank one lattices and for some other geometric structures. We also show that the Bloch invariants of CR structures and of flag…

We compute the quantum cohomology rings of the partial flag manifolds F_{n_1\cdots n_k}=U(n)/(U(n_1)\times \cdots \times U(n_k)). The inductive computation uses the idea of Givental and Kim. Also we define a notion of the vertical quantum…

The quantum double Schubert polynomials studied by Kirillov and Maeno, and by Ciocan-Fontanine and Fulton, are shown to represent Schubert classes in Kim's presentation of the equivariant quantum cohomology of the flag variety. We define…

Arinkin and Gaitsgory defined a category of tempered D-modules on Bun_G that is conjecturally equivalent to the category of quasi-coherent (not ind-coherent!) sheaves on LocSys_{\check{G}}. However, their definition depends on the auxiliary…

We give a geometric perspective on the algebra of Drinfeld modular forms for congruence subgroups $\Gamma\leq \GL_2(\bbF_q[T]).$ In particular, we describe an isomorphism between the section ring of a line bundle on the stacky modular curve…

Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found,…

Rietsch constructed a candidate $T$-equivariant mirror LG model for any flag variety $G/P$. In this paper, we prove the following mirror symmetry prediction: the small $T\times\mathbb{G}_m$-equivariant quantum cohomology of $G/P$ equipped…

We study finitary 2-categories associated to dual projection functors for finite dimensional associative algebras. In the case of path algebras of admissible tree quivers (which includes all Dynkin quivers of type A) we show that the monoid…

We generalize the monomorphism category from quiver (with monomial relations) to arbitrary finite dimensional algebras by a homological definition. Given two finite dimension algebras $A$ and $B$, we use the special monomorphism category…

We point out similarities between two partitions of a flag manifold with pieces indexed by Weyl group elements. One partition is defined using the action of a Frobenius map, the other partition is defined using conjugation by a regular…

We propose a new approach to the multiplication of Schubert classes in the K-theory of the flag variety. This extends the work of Fomin and Kirillov in the cohomology case, and is based on the quadratic algebra defined by them. More…

We introduce $C^*$-algebras associated to the foliation structure of a quantum flag manifold. We use these to construct $SL_q(3,\mathbb{C})$-equivariant Fredholm modules for the full quantum flag manifold $X_q = SU_q(3)/T$ of $SU_q(3)$,…

This is a review article on modular categories, extending an invited talk given at the workshop "Categorical (co)algebraic methods in quantum informatics and linguistics", Oxford, October 29-31, 2010. To appear in C. Heunen, M. Sadrzadeh,…