Grading the Translation Functors in type A

Kazhdan-Lusztig tensoring and Harish-Chandra categories

We use the Kazhdan-Lusztig tensoring to define affine translation functors, describe annihilating ideals of highest weight modules over an affine Lie algebra in terms of the corresponding VOA, and to sketch a functorial approach to ``affine Harish-Chandra bimodules''.

Characters of irreducible modules with non-critical highest weights over affine Lie algebras

We shall derive Kazhdan-Lusztig type character formula for the irreducible modules with arbitrary non-critical highest weights over affine Lie algebras from the rational case by using the translation functor, the Enright functor and Jantzen's deformation argument.

General representation theory in relatively closed monoidal categories

We apply the notion of relative adjoint functor to generalise closed monoidal categories. We define representations in such categories and give their relation with left actions of monoids. The translation of these representations under lax monoidal functors is investigated. We introduce tensor product of representations of bimonoids as a functorial binary operation and show how symmetric lax monoidal functors act on this product. Finally we apply the general theory to classical and quantum representations.

受限量子群作为分层霍普夫代数

Restricted quantum groups as graded Hopf algebras

本文引入π²-分层霍普夫代数概念，探讨其有限维表示构成的刚性单向范畴，并以受限量子群为例，展示其在统计力学模型中的应用。

半单张量范畴的扭曲德利涅积

Twisted Deligne products of semisimple tensor categories

探讨半单张量范畴$\mathcal C$、$\mathcal D$的扭曲德利涅积分类，给出特定条件下唯一扭曲德利涅积是普通积的结论，借助相关工作实现群论分类及融合范畴精确分解，附录引入范畴$n$-上链概念并回答相关问题。

Fiber 2-函子与 Tambara-Yamagami 融合 2-范畴

Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories

本文定义并刻画群论融合 2-范畴，分析其结构、融合规则与分次，并分类带 fiber 2-函子及 Tambara-Yamagami 缺陷的融合 2-范畴。

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…

On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology

We relate decategorifications of Ozsváth-Szabó's new bordered theory for knot Floer homology to representations of $\mathcal{U}_q(\mathfrak{gl}(1|1))$. Specifically, we consider two subalgebras $\mathcal{C}_r(n,\mathcal{S})$ and $\mathcal{C}_l(n,\mathcal{S})$ of Ozsváth- Szabó's algebra $\mathcal{B}(n,\mathcal{S})$, and identify their Grothendieck groups with tensor products of representations $V$ and $V^*$ of $\mathcal{U}_q(\mathfrak{gl}(1|1))$, where $V$ is the vector representation. We identify the decategorifications of Ozsváth-Szabó's DA bimodules for elementary tangles with corresponding…

Regularity of quantum tau-functions generated by quantum birational Weyl group actions

We canonically quantize the tau-functions for the birational Weyl group action arising from a nilpotent Poisson algebra proposed by Noumi and Yamada. We also construct the q-difference deformation of the canonical quantization of the tau-functions. Using the translation functors for the symmetrizable Kac-Moody algebras, we prove the regularity of the quantum tau-functions, namely, we show that the quantum tau-functions are polynomials in dependent variables.

Grade zero part of forced graded algebras

The paper concerns a certain subcategory of the category of representations for a semisimple algebraic group $G$ in characteristic $p$, which arise from the semisimple modules for the corresponding quantum group at a $p$-th root of unity. The subcategory, thus, records the cohomological difference between quantum groups and algebraic groups. We define translation functors in this category and use them to obtain information on the irreducible characters for $G$ when the Lusztig character formula does not hold.

Second quantized Frobenius algebras

We show that given a Frobenius algebra there is a unique notion of its second quantization, which is the sum over all symmetric group quotients of n--th tensor powers, where the quotients are given by symmetric group twisted Frobenius algebras. To this end, we consider the setting of Frobenius algebras given by functors from geometric categories whose objects are endowed with geometric group actions and prove structural results, which in turn yield a constructive realization in the case of n--th tensor powers and the natural permutation action. We also show that naturally graded symmetric grou…

Quantum Hamiltonian reduction of W-algebras and category O

W-algebras are a class of non-commutative algebras related to the classical universal enveloping algebras. They can be defined as a subquotient of U(g) related to a choice of nilpotent element e and compatible nilpotent subalgebra m. The definition is a quantum analogue of the classical construction of Hamiltonian reduction. We define a quantum version of Hamiltonian reduction by stages and use it to construct intermediate reductions between different W-algebras U(g,e) in type A.This allows us to express the W-algebra U(g,e') as a subquotient of U(g,e) for nilpotent elements e' covering e. It …

泊松代数的通用构造及其应用

Universal constructions for Poisson algebras. Applications

本文定义了泊松代数的通用代数，证明其对有限维泊松代数存在，构造了相关伴随函子，刻画了其代数结构，并给出了自同构群描述与阿贝尔群分次分类两项应用。