Category Of Finite-dimensional Representations Research Articles

On unimodular module categories

Let C be a finite tensor category and M an exact left C-module category. We call M unimodular if the finite multitensor category RexC(M) of right exact C-module endofunctors of M is unimodular. In this article, we provide various characterizations, properties, and examples of unimodular module categories. As our first application, we employ unimodular module categories to construct (commutative) Frobenius algebra objects in the Drinfeld center of any finite tensor category. When C is a pivotal category, and M is a unimodular, pivotal left C-module category, the Frobenius algebra objects are symmetric as well. Our second application is a classification of unimodular module categories over the category of finite dimensional representations of a finite dimensional Hopf algebra; this answers a question of Shimizu [40]. Using this, we provide an example of a finite tensor category whose categorical Morita equivalence class does not contain any unimodular tensor category.

Representations of the Laurent series Rota-Baxter algebras and regular-singular decompositions

There is a Rota-Baxter algebra structure on the field A=k((t)) with P being the projection map from A=k[[t]]⊕t−1k[t−1] onto k[[t]]. We study the representation theory and regular-singular decompositions of any finite dimensional A-vector space. The main result shows that the category of finite dimensional representations is semisimple and consists of exactly three isomorphism classes of irreducible representations which are all one-dimensional. As a consequence, the number of GLA(V)-orbits in the set of all regular-singular decompositions of an n-dimensional A-vector space V is (n+1)(n+2)/2. We also use the result to compute the generalized class number, i.e., the number of the GLn(A)-isomorphism classes of finitely generated k[[t]]-submodules of An.

Invariants of Weyl Group Action and q-characters of Quantum Affine Algebras

Let W be the Weyl group corresponding to a finite dimensional simple Lie algebra $$\mathfrak {g}$$ of rank $$\ell $$ and let $$m>1$$ be an integer. In Inoue (Lett. Math. Phys. 111(1):32, 2021), by applying cluster mutations, a W-action on $$\mathcal {Y}_m$$ was constructed. Here $$\mathcal {Y}_m$$ is the rational function field on $$cm\ell $$ commuting variables, where $$c \in \{ 1, 2, 3 \}$$ depends on $$\mathfrak {g}$$ . This was motivated by the q-character map $$\chi _q$$ of the category of finite dimensional representations of quantum affine algebra $$U_q(\hat{\mathfrak {g}})$$ . We showed in Inoue (Lett. Math. Phys. 111(1):32, 2021) that when q is a root of unity, $$\textrm{Im} \chi _q$$ is a subring of the W-invariant subfield $$\mathcal {Y}_m^W$$ of $$\mathcal {Y}_m$$ . In this paper, we give more detailed study on $$\mathcal {Y}_m^W$$ ; for each reflection $$r_i \in W$$ associated to the ith simple root, we describe the $$r_i$$ -invariant subfield $$\mathcal {Y}_m^{r_i}$$ of $$\mathcal {Y}_m$$ .

Affine Super Schur Duality

Schur duality is an equivalence, for $d\leq n$, between the category of finite-dimensional representations over $\mathbb{C}$ of the symmetric group $S\_d$ on $d$ letters, and the category of finite-dimensional representations over $\mathbb{C}$ of $\operatorname{GL}(n,\mathbb{C})$ whose irreducible subquotients are subquotients of $\smash{\overline{\mathbb{E}}^{\otimes d}}$, $\overline{\mathbb{E}}=\mathbb{C}^n$. The latter are called polynomial representations homogeneous of degree $d$. It is based on decomposing $\smash{\overline{\mathbb{E}}^{\otimes d}}$ as a $\mathbb{C}\[S\_d]\times\operatorname{GL}(n,\mathbb{C})$-bimodule. It was used by Schur to conclude the semisimplicity of the category of finite-dimensional complex $\operatorname{GL}(n,\mathbb{C})$-modules from the corresponding result for $S\_d$ that had been obtained by Young. Here we extend this duality to the affine super case by constructing a functor $\mathcal{F}\colon M\mapsto M\otimes\_{\mathbb{C}\[S\_d]}\mathbb{E}^{\otimes d}$, $\mathbb{E}$ now being the super vector space $\mathbb{C}^{m|n}$, from the category of finite-dimensional $\mathbb{C}\[S\_d\ltimes\mathbb{Z}^d]$-modules, or representations of the affine Weyl, or symmetric, group $S\_d^a=S\_d\ltimes\mathbb{Z}^d$, \emph{to} the category of finite-dimensional representations of the universal enveloping algebra of the affine Lie superalgebra $\mathfrak{U}(\widehat{\operatorname{sl}}(m|n))$ that are $\mathbb{E}^{\otimes d}$-compatible, namely the subquotients of whose restriction to $\mathfrak{U}(\operatorname{sl}]\(m|n))$ are constituents of $\mathbb{E}^{\otimes d}$. Both categories are not semisimple. When $d\<m+n$ the functor defines an equivalence of categories. As an application we conclude that the irreducible finite-dimensional $\mathbb{E}^{\otimes d}$-compatible representations of the affine superalgebra $\widehat{\operatorname{sl}}(m|n)$ are tensor products of evaluation representations at distinct points of $\mathbb{C}^\times$.

BGG CATEGORY FOR THE QUANTUM SCHRÖDINGER ALGEBRA

AbstractIn 1996, aq-deformation of the universal enveloping algebra of the Schrödinger Lie algebra was introduced in Dobrevet al. [J. Phys.A29(1996) 5909–5918.]. This algebra is called the quantum Schrödinger algebra. In this paper, we study the Bernstein-Gelfand-Gelfand (BGG) category$\mathcal{O}$for the quantum Schrödinger algebra$U_q(\mathfrak{s})$, whereqis a nonzero complex number which is not a root of unity. If the central charge$\dot z\neq 0$, using the module$B_{\dot z}$over the quantum Weyl algebra$H_q$, we show that there is an equivalence between the full subcategory$\mathcal{O}[\dot Z]$consisting of modules with the central charge$\dot z$and the BGG category$\mathcal{O}^{(\mathfrak{sl}_2)}$for the quantum group$U_q(\mathfrak{sl}_2)$. In the case that$\dot z = 0$, we study the subcategory$\mathcal{A}$consisting of finite dimensional$U_q(\mathfrak{s})$-modules of type 1 with zero action ofZ. We directly construct an equivalence functor from$\mathcal{A}$to the category of finite dimensional representations of an infinite quiver with some quadratic relations. As a corollary, we show that the category of finite dimensional$U_q(\mathfrak{s})$-modules is wild.

Braided module categories via quantum symmetric pairs

Let ${\mathfrak g}$ be a finite dimensional complex semisimple Lie algebra. The finite dimensional representations of the quantized enveloping algebra $U_q({\mathfrak g})$ form a braided monoidal category $O_{int}$. We show that the category of finite dimensional representations of a quantum symmetric pair coideal subalgebra $B_{c,s}$ of $U_q({\mathfrak g})$ is a braided module category over an equivariantization of $O_{int}$. The braiding for $B_{c,s}$ is realized by a universal K-matrix which lies in a completion of $B_{c,s}\otimes U_q({\mathfrak g})$. We apply these results to describe a distinguished basis of the center of $B_{c,s}$.

First fundamental theorems of invariant theory for quantum supergroups

Let \(\mathrm{U}_q({{\mathfrak {g}}})\) be the quantum supergroup of \({\mathfrak {gl}}_{m|n}\) or the modified quantum supergroup of \({\mathfrak {osp}}_{m|2n}\) over the field of rational functions in q, and let \(V_q\) be the natural module for \(\mathrm{U}_q({{\mathfrak {g}}})\). There exists a unique tensor functor associated with \(V_q\), from the category of ribbon graphs to the category of finite dimensional representations of \(\mathrm{U}_q({{\mathfrak {g}}})\), which preserves ribbon category structures. We show that this functor is full in the cases \({{\mathfrak {g}}}={\mathfrak {gl}}_{m|n}\) or \({\mathfrak {osp}}_{2\ell +1|2n}\). For \({{\mathfrak {g}}}={\mathfrak {osp}}_{2\ell |2n}\), we show that the space is spanned by images of ribbon graphs if \(r+s< 2\ell (2n+1)\). The proofs involve an equivalence of module categories for two versions of the quantisation of \(\mathrm{U}({{\mathfrak {g}}})\).

Modular Representation Theory of Affine and Cyclotomic Yokonuma-Hecke Algebras

We explore the modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras. We provide an equivalence between the category of finite dimensional representations of the affine (resp. cyclotomic) Yokonuma-Hecke algebra and that of an algebra which is a direct sum of tensor products of affine Hecke algebras of type A (resp. Ariki-Koike algebras). As one of the applications, the irreducible representations of affine and cyclotomic Yokonuma-Hecke algebras are classified over an algebraically closed field of characteristic p. Secondly, the modular branching rules for these algebras are obtained; moreover, the resulting modular branching graphs for cyclotomic Yokonuma-Hecke algebras are identified with crystal graphs of irreducible integrable representations of affine Lie algebras of type A.

On braided almost square-free fusion categories with odd dimension

Let q be a odd prime number, d be an odd square-free natural number. We prove that a braided fusion category of Frobenius–Perron dimension dqn is solvable. If then we further prove that is tensor equivalent to the category of finite-dimensional representations of a semisimple quasitriangular Hopf algebra which is kG for some abelian group G, or a crossed product for some σ.

The Morita Theory of Quantum Graph Isomorphisms

We classify instances of quantum pseudo-telepathy in the graph isomorphism game, exploiting the recently discovered connection between quantum information and the theory of quantum automorphism groups. Specifically, we show that graphs quantum isomorphic to a given graph are in bijective correspondence with Morita equivalence classes of certain Frobenius algebras in the category of finite-dimensional representations of the quantum automorphism algebra of that graph. We show that such a Frobenius algebra may be constructed from a central type subgroup of the classical automorphism group, whose action on the graph has coisotropic vertex stabilisers. In particular, if the original graph has no quantum symmetries, quantum isomorphic graphs are classified by such subgroups. We show that all quantum isomorphic graph pairs corresponding to a well-known family of binary constraint systems arise from this group-theoretical construction. We use our classification to show that, of the small order vertex-transitive graphs with no quantum symmetry, none is quantum isomorphic to a non-isomorphic graph. We show that this is in fact asymptotically almost surely true of all graphs.

Supports of representations of the rational Cherednik algebra of type A

We first consider the rational Cherednik algebra corresponding to the action of a finite group on a complex variety, as defined by Etingof. We define a category of representations of this algebra which is analogous to “category O” for the rational Cherednik algebra of a vector space. We generalise to this setting Bezrukavnikov and Etingof's results about the possible support sets of such representations. Then we focus on the case of Sn acting on Cn, determining which irreducible modules in this category have which support sets. We also show that the category of representations with a given support, modulo those with smaller support, is equivalent to the category of finite dimensional representations of a certain Hecke algebra.

On a cluster category of type D infinity

We study the canonical orbit category of the bounded derived category of finite dimensional representations of the quiver of type D∞. We prove that this orbit category is a cluster category, that is, its cluster-tilting subcategories form a cluster structure as defined in [2].

Cluster categories of type $${\mathbb {A}}_\infty ^\infty $$ A ∞ ∞ and triangulations of the infinite strip

We first study the (canonical) orbit category of the bounded derived category of finite dimensional representations of a quiver with no infinite path, and we pay more attention on the case where the quiver is of infinite Dynkin type. In particular, its Auslander-Reiten components are explicitly described. When the quiver is of type $\mathbb{A}_\infty$ or $\mathbb{A}_\infty^\infty$, we show that this orbit category is a cluster category, that is, its cluster-tilting subcategories form a cluster structure. When the quiver is of type $\mathbb{A}_\infty^\infty$, we shall give a geometrical description of the cluster structure of the cluster category by using triangulations of the infinite strip in the plane. In particular, we shall show that the cluster-tilting subcategories are precisely given by compact triangulations.

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Let $U_q(\mathfrak{g})$ be the quantum supergroup of $\mathfrak{gl}_{m|n}$ or the modified quantum supergroup of $osp_{m|2n}$ over the field of rational functions in $q$, and let $V_q$ be the natural module for $U_q(\mathfrak{g})$. There exists a unique tensor functor, associated with $V_q$, from the category of ribbon graphs to the category of finite dimensional representations of $U_q(\mathfrak{g}$, which preserves ribbon category structures. We show that this functor is full in the cases $\mathfrak{g}=\mathfrak{gl}_{m|n}$ or $osp_{2\ell+1|2n}$. For $\mathfrak{g}=osp_{2\ell|2n}$, we show that the space $Hom_{U_q(\mathfrak{g}}(V_q^{\otimes r}, V_q^{\otimes s})$ is spanned by images of ribbon graphs if $r+s< 2\ell(2n+1)$. The proofs involve an equivalence of module categories for two versions of the quantisation of $U(\mathfrak{g})$.

On Weyl groups and Artin groups associated to orbifold projective lines

We associate a generalized root system in the sense of Kyoji Saito to an orbifold projective line via the derived category of finite dimensional representations of a certain bound quiver algebra. We generalize results by Saito and Takebayashi and Yamada for elliptic Weyl groups and elliptic Artin groups to the Weyl groups and the fundamental groups of the regular orbit spaces associated to the generalized root systems. Moreover we study the relation between this fundamental group and a certain subgroup of the autoequivalence group of a triangulated subcategory of the derived category of 2-Calabi–Yau completion of the bound quiver algebra.

F-zips with additional structure

An $F$-zip over a scheme $S$ over a finite field is a certain object of semi-linear algebra consisting of a locally free module with a descending filtration and an ascending filtration and a $\Frob_q$-twisted isomorphism between the respective graded sheaves. In this article we define and systematically investigate what might be called "$F$-zips with a $G$-structure", for an arbitrary reductive linear algebraic group $G$. These objects come in two incarnations. One incarnation is an exact linear tensor functor from the category of finite dimensional representations of $G$ to the category of $F$-zips over $S$. Locally any such functor has a type $\chi$, which is a cocharacter of $G$. The other incarnation is a certain $G$-torsor analogue of the notion of $F$-zips. We prove that both incarnations define stacks that are naturally equivalent to a quotient stack of the form $[E_{G,\chi}\backslash G_k]$ that was studied in an earlier paper. By the results obtained there they are therefore smooth algebraic stacks of dimension 0 over $k$. Using our earlier results we can also classify the isomorphism classes of such objects over an algebraically closed field, describe their automorphism groups, and determine which isomorphism classes can degenerate into which others. For classical groups we can deduce the corresponding results for twisted or untwisted symplectic, orthogonal, or unitary $F$-zips. The results can be applied to the algebraic de Rham cohomology of smooth projective varieties (or generalizations thereof such as smooth proper Deligne-Mumford stacks) and to truncated Barsotti-Tate groups of level 1. In addition, we hope that our systematic group theoretical approach will help to understand the analogue of the Ekedahl-Oort stratification of the special fibers of arbitrary Shimura varieties.

On a symmetry of Müger’s centralizer for the Drinfeld double of a semisimple Hopf algebra

In this paper we prove a formula that relates Muger’s centralizer in the category of representations of a factorizable Hopf algebra to the notion of Hopf kernel of a representation of the dual Hopf algebra. Using this relation we obtain a complete description for Muger’s centralizer of some fusion subcategories of the fusion category of finite dimensional representations of a Drinfeld double of a semisimple Hopf algebra.

Reduced representations of rooted trees

Let k be a field and Q a quiver. The category of finite dimensional representations of Q will be denoted by repk(Q). We can define the pointwise tensor product on the category repk(Q), i.e. for two representations N, M the representation N⊗M is defined at a vertex x∈Q0 as (N⊗M)x=Nx⊗Mx and for an arrow α∈Q1 we define (N⊗M)α=Nα⊗Mα. This tensor product has been investigated in recent work of Herschend and Kinser [1–3]. In [3] Kinser investigated the representation ring of rooted trees. A rooted tree (Q,σ) is a quiver Q, whose underlying graph is tree and which has a unique sink σ. The representation ring of Q is the free abelian group generated by the isomorphism classes of indecomposable objects in repk(Q). It is endowed with a multiplication induced by the pointwise tensor product. See [3] for details. For these rooted trees, Kinser constructed reduced representations, which are defined inductively. These reduced representations then lead to a set of orthogonal primitive idempotents in the representation ring of Q. On the other hand, there is an inductive construction of so-called radiation modules for a tree with arbitrary orientation in [5]. In the special case of a rooted tree quiver, this construction results in the reduced representations introduced by Kinser. The idea used in the construction of the radiation modules then allows us to prove a new characterization of reduced representations which isTheoremLet Q be a rooted tree quiver. An indecomposable representationV∈repk(Q)is reduced if and only if V is a direct summand ofV⊗V.This theorem then allows us to prove further results for the special case, where Q is a rooted tree quiver, whose underlying graph is of Dynkin type. In that case V being reduced is also equivalent to the condition that the vector space of V at the sink σ is one dimensional.

The Gabriel–Roiter Measure for $\widetilde{\mathbb{A}}_{\boldsymbol n}$ II

Let Q be a tame quiver of type \(\widetilde{\mathbb{A}}_n\) and Rep(Q) the category of finite dimensional representations over an algebraically closed field. A representation is simply called a module. It will be shown that a regular string module has, up to isomorphism, at most two Gabriel–Roiter submodules. The quivers Q with sink-source orientations will be characterized as those, whose central parts do not contain preinjective modules. It will also be shown that there are only finitely many (central) Gabriel–Roiter measures admitting no direct predecessors. This fact will be generalized for all tame quivers.

Simple tensor products

Let F be the category of finite dimensional representations of an arbitrary quantum affine algebra. We prove that a tensor product $S_1\otimes ... \otimes S_N$ of simple objects of F is simple if and only if for any $i < j$, $S_i\otimes S_j$ is simple.