Category Of Finite-dimensional Modules Research Articles

Tensor ideals for quantum groups via minimal tilting complexes

We use minimal tilting complexes to construct an explicit bijection between the set of thick tensor ideals with the two-out-of-three property in the category of finite-dimensional modules over a quantum group at a root of unity and the set of thick tensor ideals in the subcategory of tilting modules. We also explain why the analogous construction for rational representations of a reductive algebraic group over a field of positive characteristic does not give rise to a bijection.

On linked modules over the super-Yangian of the superalgebra Q(1)

Let Q(n) be the queer Lie superalgebra. We determine conditions under which two one-dimensional modules over the super-Yangian of Q(1) can be extended nontrivially, and thus belong to the same block of the subcategory of finite-dimensional YQ(1)-modules admitting generalized central character χ = 0. We use these results to determine conditions under which two one-dimensional modules over the finite W-algebra for Q(n) can be extended nontrivially. We describe blocks in the category of finite-dimensional modules over the finite W-algebra for Q(2). In certain cases, we determine conditions under which two simple finite-dimensional YQ(1)-modules admitting central character χ ≠ 0 can be extended nontrivially and propose a conjecture in the general case.

Whittaker category for the Lie algebra of polynomial vector fields

For any positive integer n, let An=C[t1,…,tn], Wn=Der(An) and Δn=Span{∂∂t1,…,∂∂tn}. Then (Wn,Δn) is a Whittaker pair. A Wn-module M is called a Whittaker module if the action of Δn on M is locally finite. We show that each block ΩaW˜ of the category of (An,Wn)-Whittaker modules with finite dimensional Whittaker vector spaces is equivalent to the category of finite dimensional modules over Ln, where Ln is the Lie subalgebra of Wn consisting of vector fields vanishing at the origin. As a corollary, we classify all simple non-singular Whittaker Wn-modules with finite dimensional Whittaker vector spaces using gln-modules. We also obtain an analogue of Skryabin's equivalence for the non-singular block ΩaW.

Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types

For a Dynkin quiver Q Q (of type A D E \mathrm {ADE} ), we consider a central completion of the convolution algebra of the equivariant K K -group of a certain Steinberg type graded quiver variety. We observe that it is affine quasi-hereditary and prove that its category of finite-dimensional modules is identified with a block of Hernandez-Leclerc’s monoidal category C Q \mathcal {C}_{Q} of modules over the quantum loop algebra U q ( L g ) U_{q}(L\mathfrak {g}) via Nakajima’s homomorphism. As an application, we show that Kang-Kashiwara-Kim’s generalized quantum affine Schur-Weyl duality functor gives an equivalence between the category of finite-dimensional modules over the quiver Hecke algebra associated with Q Q and Hernandez-Leclerc’s category C Q \mathcal {C}_{Q} , assuming the simpleness of some poles of normalized R R -matrices for type E \mathrm {E} .

Equivalence between module categories over quiver Hecke algebras and Hernandez–Leclerc's categories in general types

We prove in full generality that the generalized quantum affine Schur–Weyl duality functor, introduced by Kang–Kashiwara–Kim, gives an equivalence between the category of finite-dimensional modules over a quiver Hecke algebra and a certain full subcategory of finite-dimensional modules over a quantum affine algebra which is a generalization of the Hernandez–Leclerc's category CQ. This was previously proved in untwisted ADE types by Fujita using the geometry of quiver varieties, which is not applicable in general. Our proof is purely algebraic, and so can be extended uniformly to general types.

WHEN ARE THERE ENOUGH PROJECTIVE PERVERSE SHEAVES?

AbstractLet X be a topologically stratified space, p be any perversity on X and k be a field. We show that the category of p-perverse sheaves on X, constructible with respect to the stratification and with coefficients in k, is equivalent to the category of finite-dimensional modules over a finite-dimensional algebra if and only if X has finitely many strata and the same holds for the category of local systems on each of these. The main component in the proof is a construction of projective covers for simple perverse sheaves.

A geometric q‐character formula for snake modules

Let $\mathscr{C}$ be the category of finite dimensional modules over the quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ of a simple complex Lie algebra ${\mathfrak{g}}$. Let $\mathscr{C}^-$ be the subcategory introduced by Hernandez and Leclerc. We prove the geometric $q$-character formula conjectured by Hernandez and Leclerc in types $\mathbb{A}$ and $\mathbb{B}$ for a class of simple modules called snake modules introduced by Mukhin and Young. Moreover, we give a combinatorial formula for the $F$-polynomial of the generic kernel associated to the snake module. As an application, we show that snake modules correspond to cluster monomials with square free denominators and we show that snake modules are real modules. We also show that the cluster algebras of the category $\mathscr{C}_1$ are factorial for Dynkin types $\mathbb{A,D,E}$.

Modified Turaev-Viro invariants from quantum 𝔰𝔩(2|1)

The category of finite dimensional modules over the quantum superalgebra [Formula: see text] is not semi-simple and the quantum dimension of a generic [Formula: see text]-module vanishes. This vanishing happens for any value of [Formula: see text] (even when [Formula: see text] is not a root of unity). These properties make it difficult to create a fusion or modular category. Loosely speaking, the standard way to obtain such a category from a quantum group is: (1) specialize [Formula: see text] to a root of unity; this forces some modules to have zero quantum dimension, (2) quotient by morphisms of modules with zero quantum dimension, (3) show the resulting category is finite and semi-simple. In this paper, we show an analogous construction works in the context of [Formula: see text] by replacing the vanishing quantum dimension with a modified quantum dimension. In particular, we specialize [Formula: see text] to a root of unity, quotient by morphisms of modules with zero modified quantum dimension and show the resulting category is generically finite semi-simple. Moreover, we show the categories of this paper are relative [Formula: see text]-spherical categories. As a consequence, we obtain invariants of 3-manifold with additional structures.

A Category of Wide Subcategories

Abstract An algebra is said to be $\tau$-tilting finite provided it has only a finite number of $\tau$-rigid objects up to isomorphism. To each such algebra, we associate a category whose objects are the wide subcategories of its category of finite dimensional modules, and whose morphisms are indexed by support $\tau$-tilting pairs.

Equivariant perverse sheaves on Coxeter arrangements and buildings

When $W$ is a finite Coxeter group acting by its reflection representation on $E$, we describe the category ${\mathsf{Perv}}_W(E_{\mathbb C}, {\mathcal{H}}_{\mathbb C})$ of $W$-equivariant perverse sheaves on $E_{\mathbb C}$, smooth with respect to the stratification by reflection hyperplanes. By using Kapranov and Schechtman's recent analysis of perverse sheaves on hyperplane arrangements, we find an equivalence of categories from ${\mathsf{Perv}}_W(E_{\mathbb C}, {\mathcal{H}}_{\mathbb C})$ to a category of finite-dimensional modules over an algebra given by explicit generators and relations. We also define categories of equivariant perverse sheaves on affine buildings, e.g., $G$-equivariant perverse sheaves on the Bruhat--Tits building of a $p$-adic group $G$. In this setting, we find that a construction of Schneider and Stuhler gives equivariant perverse sheaves associated to depth zero representations.

Exceptional sequences and Drinfeld double Hall algebras

Let A be a finitary hereditary abelian category and D(A) be its reduced Drinfeld double Hall algebra. By giving explicit formulas in D(A) for left and right mutations, we show that the subalgebras of D(A) generated by exceptional sequences are invariant under mutation equivalences. As an application, we obtain that if A is the category of finite dimensional modules over a finite dimensional hereditary algebra, or the category of coherent sheaves on a weighted projective line, the double composition algebra of A is generated by any complete exceptional sequence. Moreover, for the Lie algebra case, we also have paralleled results.

Whittaker coinvariants for GL(m|n)

Let Wm|n be the (finite) W-algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra glm|n(C). In this paper we study the Whittaker coinvariants functor, which is an exact functor from category O for glm|n(C) to a certain category of finite-dimensional modules over Wm|n. We show that this functor has properties similar to Soergel's functor V in the setting of category O for a semisimple Lie algebra. We also use it to compute the center of Wm|n explicitly, and deduce consequences for the classification of blocks of O up to Morita/derived equivalence.

Grothendieck rings for Lie superalgebras and the Duflo–Serganova functor

We show that the Duflo-Serganova functor on the category of finite-dimensional modules over a finite-dimensional contragredient Lie superalgebra induces a ring homomorphism on a natural quotient of the Grothendieck ring, which is isomorphic to the ring of characters. We realize this homomorphism as a certain evaluation of functions related to the supersymmetry property. We use this realization to describe the kernel and image of the homomorphism induced by the Duflo-Serganova functor.

Stable Grothendieck rings of wreath product categories

Let k be an algebraically closed field of characteristic zero, and let $${\mathcal {C}} = {\mathcal {R}} -\hbox {mod}$$ be the category of finite-dimensional modules over a fixed Hopf algebra over k. One may form the wreath product categories $$ {\mathcal {W}}_{n}({\mathcal {C}}) = ( {\mathcal {R}} \wr S_n)-\hbox {mod}$$ whose Grothendieck groups inherit the structure of a ring. Fixing distinguished generating sets (called basic hooks) of the Grothendieck rings, the classification of the simple objects in $$ {\mathcal {W}}_{n}({\mathcal {C}}) $$ allows one to demonstrate stability of structure constants in the Grothendieck rings (appropriately understood), and hence define a limiting Grothendieck ring. This ring is the Grothendieck ring of the wreath product Deligne category $$S_t({\mathcal {C}})$$ . We give a presentation of the ring and an expression for the distinguished basis arising from simple objects in the wreath product categories as polynomials in basic hooks. We discuss some applications when $$ {\mathcal {R}} $$ is the group algebra of a finite group, and some results about stable Kronecker coefficients. Finally, we explain how to generalise to the setting where $${\mathcal {C}}$$ is a tensor category.

Homological invariants in category $\mathcal {O}$ for the general linear superalgebra

We study three related homological properties of modules in the BGG category $ \mathcal {O}$ for basic classical Lie superalgebras, with specific focus on the general linear superalgebra. These are the projective dimension, associated variety and complexity. We demonstrate connections between projective dimension and singularity of modules and blocks. Similarly we investigate the connection between complexity and atypicality. This creates concrete tools to describe singularity and atypicality as homological, and hence categorical, properties of a block. However, we also demonstrate how two integral blocks in category $ \mathcal {O}$ with identical global categorical characteristics of singularity and atypicality will generally still be inequivalent. This principle implies that category $ \mathcal {O}$ for $ \mathfrak{gl}(m\vert n)$ can contain infinitely many non-equivalent blocks, which we work out explicitly for $ \mathfrak{gl}(3\vert 1)$. All of this is in sharp contrast with category $ \mathcal {O}$ for Lie algebras, but also with the category of finite dimensional modules for superalgebras. Furthermore we characterise modules with finite projective dimension to be those with trivial associated variety. We also study the associated variety of Verma modules. To do this, we also classify the orbits in the cone of self-commuting odd elements under the action of an even Borel subgroup.

Indecomposable decomposition of tensor products of modules over Drinfeld doubles of Taft algebras

In this paper, we study the tensor product structure of the category of finite dimensional modules over Drinfeld doubles of Taft Hopf algebras. Tensor product decomposition rules for all finite dimensional indecomposable modules are explicitly given.

Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras IV

Let $U'_q(\mathfrak{g})$ be a twisted affine quantum group of type $A_{N}^{(2)}$ or $D_{N}^{(2)}$ and let $\mathfrak{g}_{0}$ be the finite-dimensional simple Lie algebra of type $A_{N}$ or $D_{N}$. For a Dynkin quiver of type $\mathfrak{g}_{0}$, we define a full subcategory ${\mathcal C}_{Q}^{(2)}$ of the category of finite-dimensional integrable $U'_q(\mathfrak{g})$-modules, a twisted version of the category ${\mathcal C}_{Q}$ introduced by Hernandez and Leclerc. Applying the general scheme of affine Schur-Weyl duality, we construct an exact faithful KLR-type duality functor ${\mathcal F}_{Q}^{(2)}: Rep(R) \rightarrow {\mathcal C}_{Q}^{(2)}$, where $Rep(R)$ is the category of finite-dimensional modules over the quiver Hecke algebra $R$ of type $\mathfrak{g}_{0}$ with nilpotent actions of the generators $x_k$. We show that ${\mathcal F}_{Q}^{(2)}$ sends any simple object to a simple object and induces a ring isomorphism $K(Rep(R)) \simeq K({\mathcal C}_{Q}^{(2)})$.

Derived equivalences for hereditary Artin algebras

We study the role of the Serre functor in the theory of derived equivalences. Let A be an abelian category and let (U,V) be a t-structure on the bounded derived category DbA with heart H. We investigate when the natural embedding H→DbA can be extended to a triangle equivalence DbH→DbA. Our focus of study is the case where A is the category of finite-dimensional modules over a finite-dimensional hereditary algebra. In this case, we prove that such an extension exists if and only if the t-structure is bounded and the aisle U of the t-structure is closed under the Serre functor.

Categorification of Virasoro–Magri Poisson vertex algebra

Let Σ be the direct sum of algebra of symmetric groups CΣn, n∈Z≥0. We show that the Grothendieck group K0(Σ) of the category of finite dimensional modules of Σ is isomorphic to the differential algebra of polynomials Z[∂nx|n∈Z≥0]. Moreover, we define m-th products (m∈Z≥0) on K0(Σ) which make the algebra K0(Σ) isomorphic to an integral form of the Virasoro–Magri Poisson vertex algebra. Also, we investigate relations between K0(Σ) and K0(N) where K0(N) is the direct sum of Grothendieck groups K0(Nn), n≥0, of finitely generated projective Nn-modules. Here Nn is the nil-Coxeter algebra generated by n−1 elements.

Covering theory for linear categories with application to derived categories

We extend the Galois covering theory introduced by Bongartz–Gabriel for skeletal linear categories to general linear categories. We show that a Galois covering between Krull–Schmidt categories preserves irreducible morphisms and almost splits sequences. Specializing to derived categories, we study when a Galois covering between locally bounded linear categories induces a Galois covering between the bounded derived categories of finite dimensional modules. As an application, we show that each locally bounded linear category with radical squared zero admits a gradable Galois covering, which induces a Galois covering between the bounded derived categories of finite dimensional modules, and a Galois covering between the Auslander–Reiten quivers of these bounded derived categories. In a future paper, this will enable us to obtain a complete description of the bounded derived category of finite dimensional modules over a finite dimensional algebra with radical squared zero.