Axiomatic framework for the BGG category O

The classification of blocks in BGG category O

We classify all equivalences between the indecomposable abelian categories which appear as blocks in BGG category O for reductive Lie algebras. Our classification implies that a block in category O only depends on the Bruhat order of the relevant parabolic quotient of the Weyl group. As part of the proof, we observe that any finite dimensional algebra with simple preserving duality admits at most one quasi-hereditary structure.

Characters for Projective Modules in the BGG Category O for General Linear Lie Superalgebras

We determine the Verma multiplicities and the characters of projective modules for atypical blocks in the BGG Category O for the general linear Lie superalgebras $\frak{gl}(2|2)$ and $\frak{gl}(3|1)$. We then explicitly determine the composition factor multiplcities of Verma modules in the atypicality 2 block of $\frak{gl}(2|2)$.

Functoriality of the BGG Category O

This article aims to contribute to the study of algebras with triangular decomposition over a Hopf algebra, as well as the BGG Category O. We study functorial properties of O across various setups. The first setup is over a skew group ring, involving a finite group $Γ$ acting on a regular triangular algebra $A$. We develop Clifford theory for $A \rtimes Γ$, and obtain results on block decomposition, complete reducibility, and enough projectives. O is shown to be a highest weight category when $A$ satisfies one of the "Conditions (S)"; the BGG Reciprocity formula is slightly different because t…

The BGG Category for Generalised Reductive Lie Algebras

A Lie algebra is said to be generalised reductive if it is a direct sum of a semisimple Lie algebra and a commutative radical. In this paper we extend the BGG category $\mathcal{O}$ over complex semisimple Lie algebras to the category $\mathcal {O'}$ over complex generalised reductive Lie algebras. Then we make a preliminary research on the highest weight modules and the projective modules in this new category $\mathcal {O'}$, and generalize some conclusions in the classical case. As a critical difference from the complex semisimple Lie algebra case, we prove that there is no projective module…

BGG reciprocity for current algebras

It was conjectured by Bennett, Chari, and Manning that a BGG-type reciprocity holds for the category of graded representations with finite-dimensional graded components for the current algebra associated to a simple Lie algebra. We associate a current algebra to any indecomposable affine Lie algebra and show that, in this generality, the BGG reciprocity is true for the corresponding category of representations.

On an infinite limit of BGG categories O

We study a version of the BGG category O for Dynkin Borel subalgebras of root-reductive Lie algebras g, such as gl(\infty). We prove results about extension fullness and compute the higher extensions of simple modules by Verma modules. In addition, we show that our category O is Ringel self-dual and initiate the study of Koszul duality. An important tool in obtaining these results is an equivalence we establish between appropriate Serre subquotients of category O for g and category O for finite dimensional reductive subalgebras of g.

重访Jacobi-Trudi恒等式：通过BGG范畴 $\mathcal{O}$

Revisiting Jacobi-Trudi identities via the BGG category $\mathcal{O}$

本文通过BGG范畴O中的张量积重数重新证明斜Schur多项式的Jacobi-Trudi恒等式，并确立截断展开的Schur正性，同时解释系数为张量积重数。

Macdonald Polynomials and BGG reciprocity for current algebras

We study the category of graded representations with finite--dimensional graded pieces for the current algebra associated to a simple Lie algebra. This category has many similarities with the category $\cal O$ of modules for $\lie g$ and in this paper, we use the combinatorics of Macdonald polynomials to prove an analogue of the famous BGG duality in the case of $\lie{sl}_{n+1}$.

单位根处展开受限量子群的广义权模的BGG互反律

BGG Reciprocity for Generalized Weight Modules of Unrolled Restricted Quantum Groups at Roots of Unity

本文研究单位根处展开受限量子群的广义权模范畴，通过子范畴过滤获得投射模，证明其满足BGG互反律变体，并给出$\mathfrak{sl}_2$情形下投射覆盖的生成元与关系描述。

Jantzen coefficients and radical filtrations for generalized Verma modules

In this paper we give a sum formula for the radical filtration of generalized Verma modules in any (possibly singular) blocks of parabolic BGG category which can be viewed as a generalization of Jantzen sum formula for Verma modules in the usual BGG category $\mathcal{O}$. Combined with Jantzen coefficients, we determine the radical filtrations for all basic generalized Verma modules. The proof makes use of the graded version of parabolic BGG category. Explicit formulae for the graded decomposition numbers and inverse graded decomposition numbers of generalized Verma modules in any (possibly s…

Projective modules over classical Lie algebras of infinite rank in the parabolic category

We study the truncation functors and show the existence of projective cover of each irreducible module in parabolic BGG category $\mathcal O$ over infinite rank Lie algebra of types $\mathfrak{a,b,c,d}$. Moreover, $\mathcal O$ is a Koszul category. As a consequence, the corresponding parabolic BGG category $\overline{\mathcal O}$ over infinite rank Lie superalgebra of types $\mathfrak{a,b,c,d}$ through the super duality is also a Koszul category.

BGG 范畴 $\mathcal{O}$ 的 Dirac 上同调

Dirac cohomology for the BGG category $\mathcal{O}$

本文研究有限维复半单李代数范畴 $\mathcal{O}$ 中模的 Dirac 上同调，证明 Vogan 猜想，揭示其与幂零李代数上同调的联系，并验证高阶 Dirac 上同调的同调性质。

Hypertoric category O

We study the representation theory of the invariant subalgebra of the Weyl algebra under a torus action, which we call a "hypertoric enveloping algebra." We define an analogue of BGG category O for this algebra, and identify it with a certain category of sheaves on a hypertoric variety. We prove that a regular block of this category is highest weight and Koszul, identify its Koszul dual, compute its center, and study its cell structure. We also consider a collection of derived auto-equivalences analogous to the shuffling and twisting functors for BGG category O.

Kazhdan-Lusztig Theory of super type D and quantum symmetric pairs

We reformulate the Kazhdan-Lusztig theory for the BGG category $\mathcal{O}$ of Lie algebras of type D via the theory of canonical bases arising from quantum symmetric pairs initiated by Weiqiang Wang and the author. This is further applied to formulate and establish for the first time the Kazhdan-Lusztig theory for the BGG category $\mathcal{O}$ of the ortho-symplectic Lie superalgebra $\mathfrak{osp}(2m|2n)$.

BGG reciprocity for current algebras

We study the category $\cal I_{\gr}$ of graded representations with finite--dimensional graded pieces for the current algebra $\lie g\otimes\bc[t]$ where $\lie g$ is a simple Lie algebra. This category has many similarities with the category $\cal O$ of modules for $\lie g$ and in this paper, we formulate and study an analogue of the famous BGG duality. We recall the definition of the projective and simple objects in $\cal I_{\gr}$ which are indexed by dominant integral weights. The role of the Verma modules is played by a family of modules called the global Weyl modules. We show that in the c…

Reduction method for representations of queer Lie superalgebras

We develop a reduction procedure which provides an equivalence from an arbitrary block of the BGG category for the queer Lie superalgebra $\mathfrak{q}(n)$ to a "$\mathbb{Z}\pm s$-weights" ($s\in \mathbb{C}$) block of a BGG category for finite direct sum of queer Lie superalgebras. We give descriptions of blocks. We also establish equivalences between certain maximal parabolic subcategories for $\mathfrak{q}(n)$ and blocks of atypicality-one of the category of finite-dimensional modules for $\mathfrak{gl}(\ell|n-\ell)$.

Category O for quantum groups

In this paper we study of the BGG-categories $\mathcal O_q$ associated to quantum groups. We prove that many properties of the ordinary BGG-category $\mathcal O$ for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when $q$ is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for $\mathcal O$ and for finite dimensional $U_q$-modules we are able to determine all irreducible characters as well as the characters…

An axiomatic approach for degenerations in triangulated categories

We generalise Yoshino's definition of a degeneration of two Cohen Macaulay modules to a definition of degeneration between two objects in a triangulated category. We derive some natural properties for the triangulated category and the degeneration under which the Yoshino-style degeneration is equivalent to the degeneration defined by a specific distinguished triangle analogous to Zwara's characterisation of degeneration in module varieties.

通过弯曲 BGG 对应对 Koszul 复形上的厚子范畴进行分类

Classifying thick subcategories over a Koszul complex via the curved BGG correspondence

本文利用弯曲 BGG 对应及完美弯曲 dg 模的厚张量子模分类，刻画了正则环中任意元素列表的 Koszul 复形上有界 dg 模导出范畴的厚子范畴，并恢复 Stevenson、Hopkins–Neeman 等结果。

A note on Hecke patterns in Category O

We study the family of derived auto-equivalences of the BGG-category O that correspond to the action of the standard generators of the Hecke algebra. Both the non-graded and graded situations are considered.

On Category $\mathcal{O}$ over triangular Generalized Weyl Algebras

We analyze the BGG Category $\mathcal{O}$ over a large class of generalized Weyl algebras (henceforth termed GWAs). Given such a "triangular" GWA for which Category $\mathcal{O}$ decomposes into a direct sum of subcategories, we study in detail the homological properties of blocks with finitely many simples. As consequences, we show that the endomorphism algebra of a projective generator of such a block is quasi-hereditary, finite-dimensional, and graded Koszul. We also classify all tilting modules in the block, as well as all submodules of all projective and tilting modules. Finally, we prese…

Applications of the category of linear complexes of tilting modules associated with the category O

We use the category of linear complexes of tilting modules for the BGG category O, associated with a semi-simple complex finite-dimensional Lie algebra g, to reprove in purely algebraic way several known results about O obtained earlier by different authors using geometric methods. We also obtain several new results about the parabolic category O.

Support varieties - an axiomatic approach

We provide an axiomatic approach for studying support varieties of objects in a triangulated category via the action of a tensor triangulated category, where the tensor product is not necessarily symmetric. This is illustrated by examples, taken in particular from representation theory of finite dimensional algebras.

Janzten coefficients and Blocks of category $\mathcal{O}^\mathfrak{p}$

The BGG category $\mathcal{O}$ and its generalization $\mathcal{O}^\mathfrak{p}$ play essential roles in representation theory and have led to far-reaching work. Some elementary problems remain open for several decades, such as the block decomposition of category $\mathcal{O}^\mathfrak{p}$. In this paper, we solve the problem of blocks by applying the theory of Jantzen coefficients.

Super duality for general linear Lie superalgebras and applications

We apply the super duality formalism recently developed by the authors to obtain new equivalences of various module categories of general linear Lie superalgebras. We establish the correspondence of standard, tilting, and simple modules, as well as the identification of the u-homology groups, under these category equivalences. As an application, we obtain a complete solution of the irreducible character problem for some new parabolic BGG categories of gl(m|n)-modules, including the full BGG category of gl(m|2)-modules, in terms of type A Kazhdan-Lusztig polynomials.

权重模与旗帜簇上sheaves的粘合

Weight modules and gluing of sheaves on the flag variety

本文研究了半单李代数权重模的几何实现，并通过导出版本D与Kazhdan-Laumon Category O的Koszul对偶关系，揭示了权重模的新解释及与量子群和仿射Springer纤维的联系。

Generalized Weyl algebras: category O and graded Morita equivalence

We study the structural and homological properties of graded Artinian modules over generalized Weyl algebras (GWAs), and this leads to a decomposition result for the category of graded Artinian modules. Then we define and examine a category of graded modules analogous to the BGG category O. We discover a condition on the data defining the GWA that ensures O has a system of projective generators. Under this condition, O has nice representation-theoretic properties. There is also a decomposition result for O. Next, we give a necessary condition for there to be a strongly graded Morita equivalenc…

范畴 $\mathcal{O}$ 的若干同调性质，VII

Some homological properties of category $\mathcal{O}$, VII

本文刻画半单有限维复 Lie 代数对应的（抛物）BGG 范畴 $\mathcal{O}$ 正则块中的 Calabi–Yau 对象，并给出其与 Serre 函子及移位恒等函子自然变换的等价条件。

Serre猜想的de Rham重量部分及广义的mod p BGG分解

A de Rham weight part of Serre's conjecture and generalized mod $p$ BGG decompositions

本文通过de Rham上同调与BGG分解提出Serre猜想的几何版本，并证明其等价性，同时改进了相关结果。

范畴 $\mathcal{O}$ 上的投射函子与 Whittaker 函子

Projective and Whittaker functors on category $\mathcal{O}$

本文证明了半单复李代数 BGG 范畴 $\mathcal{O}$ 正则块上的 Whittaker 函子可由平移至墙函子与 Soergel-Miličić 等价复合得到，并证明它是与所有投射函子及其自同态交换的商函子。

An axiomatic characterization of the Gabriel-Roiter measure

Given an abelian length category A, the Gabriel-Roiter measure with respect to a length function l is characterized as a universal morphism ind(A)-->P of partially ordered sets. The map is defined on the isomorphism classes of indecomposable objects of A and is a suitable refinement of the length function l.