The Jordan-Hölder property and Grothendieck monoids of exact categories

Monoidal categories of modules over quantum affine algebras of type A and B

We construct an exact tensor functor from the category $\mathcal{A}$ of finite-dimensional graded modules over the quiver Hecke algebra of type $A_\infty$ to the category $\mathscr C_{B^{(1)}_n}$ of finite-dimensional integrable modules over the quantum affine algebra of type $B^{(1)}_n$. It factors through the category $\mathcal T_{2n}$, which is a localization of $\mathcal{A}$. As a result, this functor induces a ring isomorphism from the Grothendieck ring of $\mathcal T_{2n}$ (ignoring the gradings) to the Grothendieck ring of a subcategory $\mathscr C^{0}_{B^{(1)}_n}$ of $\mathscr C_{B^{(1…

Duality structures for module categories of vertex operator algebras and the Feigin Fuchs boson

Huang, Lepowsky and Zhang have developed a module theory for vertex operator algebras that endows suitably chosen module categories with the structure of braided monoidal categories. Included in the theory is a functor which assigns to discretely strongly graded modules a contragredient module, obtained as a gradewise dual. In this paper, we show that this gradewise dual endows the module category with the structure of a ribbon Grothendieck-Verdier category. This duality structure is more general than that of a rigid monoidal category; in contrast to rigidity, it naturally accommodates the fac…

Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm

We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories of finite-dimensional representations of quantum affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in part on the corresponding quantum cluster algebra structures. Moreover, we prove that our isomorphisms specialize at $t = 1$ to the isomorphisms of (classical) Grothendieck rings obtained recently by Kashiwara, Kim and Oh by other methods. As a consequence, we prove a conjecture formulated by the first author in 2002 : the multiplicities of simple modules in sta…

Gluing derived equivalences together

The Grothendieck construction of a diagram $X$ of categories can be seen as a process to construct a single category $\Gr(X)$ by gluing categories in the diagram together. Here we formulate diagrams of categories as colax functors from a small category $I$ to the 2-category $\kCat$ of small $\k$-categories for a fixed commutative ring $\k$. In our previous paper we defined derived equivalences of those colax functors. Roughly speaking two colax functors $X, X' \colon I \to \kCat$ are derived equivalent if there is a derived equivalence from $X(i)$ to $X'(i)$ for all objects $i$ in $I$ satisfyi…

Plethysm 积、元素构造与 Plus 构造

Plethysm Products, Element and Plus Constructions

本文在双模、相对双模和可因子化双模三个层面构造 plethysm 积幺半结构，建立其与 Grothendieck 元素构造等的统一理论，并证明 plus 构造可实现为与幺半结构相容的元素构造，从而得到元素构造与 plus 构造的交换性。

2-Dimensional Categories

This book is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, adjunctions and monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double cate…

Grothendieck-Verdier范畴中的交换代数、刚性与顶点算子代数

Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras

研究Grothendieck-Verdier范畴中交换代数的模范畴及其局部子范畴的刚性继承问题，给出刚性判定条件，应用于顶点算子代数扩张的强有理性证明，降低相关证明难度。