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We study Karoubian tensor categories which interpolate representation categories of families of so-called easy quantum groups in the same sense in which Deligne's interpolation categories $\mathrm{\underline{Rep}}(S_t)$ interpolate the…
We classify the indecomposable objects in the monoidal center of Deligne's interpolation category $Rep(S_t)$ by viewing $Rep(S_t)$ as a model-theoretic limit in rank and characteristic. We further prove that the center of $Rep(S_t)$ is…
We prove that the Grothendieck ring of the monoidal category of finitely generated graded projective modules for the nil-Brauer category is isomorphic to an integral form of the split iquantum group of rank one. Under this isomorphism, the…
The Deligne category of symmetric groups is the additive Karoubi closure of the partition category. It is semisimple for generic values of the parameter t while producing categories of representations of the symmetric group when modded out…
In this note, we compute the split Grothendieck ring of a generalized category of Soergel bimodules of type $A_2$, where we take one generator for each reflection. We give a presentation by generators and relations of it and a…
We describe a new geometric model for the Hochschild cohomology of Soergel bimodules based on the monodromic Hecke category studied earlier by the first author and Yun. Moreover, we identify the objects representing individual Hochschild…
Krull-Schmidt categories are additive categories such that each object decomposes into a finite direct sum of indecomposable objects having local endomorphism rings. We provide a self-contained introduction which is based on the concept of…
We give an easy diagrammatical description of the parabolic Kazhdan-Lusztig polynomials for the Weyl group $W_n$ of type $D_n$ with parabolic subgroup of type $A_n$ and consequently an explicit counting formula for the dimension of the…
In this article we classify indecomposable objects of the derived categories of finitely-generated modules over certain infinite-dimensional algebras. The considered class of algebras (which we call nodal algebras) contains such well-known…
We define the Hochschild complex and cohomology of a ring object in a monoidal category enriched over abelian groups. We interpret the cohomology groups and prove that the cohomology ring is graded-commutative.
The purpose of this short note is to understand the Grothendieck ring of the Deligne category $\underline{\text{Rep}}(GL_t)$. We give a formula for the class of an indecomposable object in terms of exterior powers of the tautological object…
We classify braided generalized near-group fusion categories whose global dimension is not an integer; there are exactly two up to Grothendieck equivalence and taking products with braided pointed fusion categories.
In the previous paper arxiv:math/0610552 semisimple tensor categories were constructed out of certain regular Mal'cev categories. In this paper, we calculate the tensor product multiplicities and the categorical dimensions of the simple…
The degenerate Heisenberg category $\mathcal{H}eis_k$ is a strict monoidal category which was originally introduced in the special case $k=-1$ by Khovanov in 2010. Khovanov conjectured that the Grothendieck ring of the additive Karoubi…
Let $k$ be a field. We show that locally presentable, $k$-linear categories $\mathcal{C}$ dualizable in the sense that the identity functor can be recovered as $\coprod_i x_i\otimes f_i$ for objects $x_i\in \mathcal{C}$ and left adjoints…
We categorify Lusztig's version of the quantized enveloping algebra for sl(2). Using a graphical calculus a 2-category is constructed whose split Grothendieck ring is isomorphic to Lusztig's algebra. The indecomposable morphisms of this…
Combining results from Keller and Buchweitz, we describe the 1-periodic derived category of a finite dimensional algebra $A$ of finite global dimension as the stable category of maximal Cohen-Macaulay modules over some Gorenstein algebra…
Following Krause \cite{Kr}, we prove Krull-Schmidt type decomposition theorems for thick subcategories of various triangulated categories including the derived categories of rings, Noetherian stable homotopy categories, stable module…
We introduce analogues of Soergel bimodules for complex reflection groups of rank one. We give an explicit parametrization of the indecomposable objects of the resulting category and give a presentation of its split Grothendieck ring by…
Two decades ago P. Martin and D. Woodcock made a surprising and prophetic link between statistical mechanics and representation theory. They observed that the decomposition numbers of the blob algebra (that appeared in the context of…