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Criterion for Quasi-Heredity Algebr. Represent. Theor. (IF 0.6) Pub Date : 2024-03-12 Yuichiro Goto
Dlab and Ringel showed that algebras being quasi-hereditary in all orders for indices of primitive idempotents becomes hereditary. So, we are interested in for which orders a given quasi-hereditary algebra is again quasi-hereditary. As a matter of fact, we consider permutations of indices, and if the algebra with permuted indices is quasi-hereditary, then we say that this permutation gives quasi-heredity
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Rota-Baxter Lie bialgebras, classical Yang-Baxter equations and special L-dendriform bialgebras Algebr. Represent. Theor. (IF 0.6) Pub Date : 2024-02-28
Abstract This paper extends the well-known fact that a Rota-Baxter operator of weight 0 on a Lie algebra induces a pre-Lie algebra, to the level of bialgebras. We first show that a nondegenerate symmetric bilinear form that is invariant on a Rota-Baxter Lie algebra of weight 0 gives such a form that is left-invariant on the induced pre-Lie algebra and thereby gives a special L-dendriform algebra. This
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Two $$\mathbb {Z}$$ -Graded Infinite Lie Conformal Algebras Related to the Virasoro Conformal Algebra Algebr. Represent. Theor. (IF 0.6) Pub Date : 2024-02-24 Xiaoqing Yue, Shun Zou
In this paper, we study two \(\mathbb {Z}\)-graded infinite Lie conformal algebras, which are closely related to a class of Lie algebras of the generalized Block type, and which both have a quotient algebra isomorphic to the Virasoro conformal algebra. We concretely determine their isomorphic mappings, conformal derivations, extensions by a one-dimensional center under some conditions, finite conformal
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Soergel Calculus with Patches Algebr. Represent. Theor. (IF 0.6) Pub Date : 2024-02-23 Leonardo Maltoni
We adapt the diagrammatic presentation of the Hecke category to the dg category formed by the standard representatives for the Rouquier complexes. We use this description to recover basic results about these complexes, namely the categorification of the relations of the braid group and the Rouquier formula.
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Recollements of Derived Categories from Two-Term Big Tilting Complexes Algebr. Represent. Theor. (IF 0.6) Pub Date : 2024-02-19 Huabo Xu
We introduce the notion of big tilting complexes over associative rings, which is a simultaneous generalization of good tilting modules and tilting complexes over rings. Given a two-term big tilting complex over an arbitrary associative ring, we show that the derived module category of its (derived) endomorphism ring is a recollement of the one of the given ring and the one of a universal localization
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Elliptic Quantum Toroidal Algebras, Z-algebra Structure and Representations Algebr. Represent. Theor. (IF 0.6) Pub Date : 2024-02-13 Hitoshi Konno, Kazuyuki Oshima
We introduce a new elliptic quantum toroidal algebra \(U_{q,\kappa ,p}({\mathfrak {g}}_{tor})\) associated with an arbitrary toroidal algebra \({\mathfrak {g}}_{tor}\). We show that \(U_{q,\kappa ,p}({\mathfrak {g}}_{tor})\) contains two elliptic quantum algebras associated with a corresponding affine Lie algebra \(\widehat{\mathfrak {g}}\) as subalgebras. They are analogue of the horizontal and the
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Presentations of Braid Groups of Type A Arising from $$(m+2)$$ -angulations of Regular Polygons Algebr. Represent. Theor. (IF 0.6) Pub Date : 2024-02-02 Davide Morigi
Coloured quiver mutation, introduced by Buan, A.B., Thomas, H (Adv. Math. 222(3), 971–995 2009), gives a combinatorial interpretation of tilting in higher cluster categories. In type A work of Baur, K., Marsh, B. (Trans. Am. Math. Soc. 360(11), 5789-5803 2008) shows that m-coloured quivers and m-coloured quiver mutations have a nice geometrical description, given in terms of \((m+2)\)-angulations of
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Unique Factorization for Tensor Products of Parabolic Verma Modules Algebr. Represent. Theor. (IF 0.6) Pub Date : 2024-02-01 K. N. Raghavan, V. Sathish Kumar, R. Venkatesh, Sankaran Viswanath
Let \(\mathfrak g\) be a symmetrizable Kac-Moody Lie algebra with Cartan subalgebra \(\mathfrak h\). We prove a unique factorization property for tensor products of parabolic Verma modules. More generally, we prove unique factorization for products of characters of parabolic Verma modules when restricted to certain subalgebras of \(\mathfrak h\). These include fixed point subalgebras of \(\mathfrak
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Preradicals Over Some Group Algebras Algebr. Represent. Theor. (IF 0.6) Pub Date : 2024-01-25 Rogelio Fernández-Alonso, Benigno Mercado, Silvia Gavito
For a field \(\varvec{K}\) and a finite group \(\varvec{G}\), we study the lattice of preradicals over the group algebra \(\varvec{KG}\), denoted by \(\varvec{KG}\)-\(\varvec{pr}\). We show that if \(\varvec{KG}\) is a semisimple algebra, then \(\varvec{KG}\)-\(\varvec{pr}\) is completely described, and we establish conditions for counting the number of its atoms in some specific cases. If \(\varvec{KG}\)
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A Quantization of the Loday-Ronco Hopf Algebra Algebr. Represent. Theor. (IF 0.6) Pub Date : 2024-01-20 João N. Esteves
We propose a quantization algebra of the Loday-Ronco Hopf algebra \(k[Y^\infty ]\), based on the Topological Recursion formula of Eynard and Orantin. We have shown in previous works that the Loday-Ronco Hopf algebra of planar binary trees is a space of solutions for the genus 0 version of Topological Recursion, and that an extension of the Loday Ronco Hopf algebra as to include some new graphs with
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Minimal Triangular Structures on Abelian Extensions Algebr. Represent. Theor. (IF 0.6) Pub Date : 2024-01-12
Abstract We study minimal triangular structures on abelian extensions. In particular, we construct a family of minimal triangular semisimple Hopf algebras and prove that the Hopf algebra \(H_{b:y}\) in the semisimple Hopf algebras of dimension 16 classified by Y. Kashina in 2000 is minimal triangular Hopf algebra with smallest dimension among non-trivial semisimple triangular Hopf algebras (i.e. not
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A Note on Singularity Categories and Triangular Matrix Algebras Algebr. Represent. Theor. (IF 0.6) Pub Date : 2024-01-06 Yongyun Qin
Let \(\Lambda = \left[ \begin{array}{cc} A &{} 0 \\ M &{} B \end{array}\right] \) be an Artin algebra and \(_BM_A\) a B-A-bimodule. We prove that there is a triangle equivalence \(D_{sg}(\Lambda ) \cong D_{sg}(A)\coprod D_{sg}(B)\) between the corresponding singularity categories if \(_BM\) is semi-simple and \(M_A\) is projective. As a result, we obtain a new method for describing the singularity
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Fullness of the Kuznetsov–Polishchuk Exceptional Collection for the Spinor Tenfold Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-12-28 Riccardo Moschetti, Marco Rampazzo
Kuznetsov and Polishchuk provided a general algorithm to construct exceptional collections of maximal length for homogeneous varieties of type A, B, C, D. We consider the case of the spinor tenfold and we prove that the corresponding collection is full, i.e. it generates the whole derived category of coherent sheaves. We also verify strongness and purity of such collection. As a step of the proof,
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Growth Rates of the Number of Indecomposable Summands in Tensor Powers Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-12-19 Kevin Coulembier, Victor Ostrik, Daniel Tubbenhauer
In this paper we study the asymptotic behavior of the number of summands in tensor products of finite dimensional representations of affine (semi)group (super)schemes and related objects.
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A Gelfand–MacPherson Correspondence for Quiver Moduli Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-12-16 Hans Franzen
We show that a semi-stable moduli space of representations of an acyclic quiver can be identified with two GIT quotients by reductive groups. One of a quiver Grassmannian of a projective representation, the other of a quiver Grassmannian of an injective representation. This recovers as special cases the classical Gelfand–MacPherson correspondence and its generalization by Hu and Kim to bipartite quivers
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Classification of Orbit Closures in the Variety of 4-Dimensional Symplectic Lie Algebras Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-12-14
Abstract The aim of this paper is to study the natural action of the real symplectic group, \({\text {Sp}}(4, \mathbb {R})\) , on the algebraic set of 4-dimensional Lie algebras admitting symplectic structures and to give a complete classification of orbit closures. We present some applications of such classification to the study of the Ricci curvature of left-invariant almost Kähler structures on
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Quipu Quivers and Nakayama Algebras with Almost Separate Relations Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-12-12 Didrik Fosse
A Nakayama algebra with almost separate relations is one where the overlap between any pair of relations is at most one arrow. In this paper we give a derived equivalence between such Nakayama algebras and path algebras of quivers of a special form known as quipu quivers. Furthermore, we show how this derived equivalence can be used to produce a complete classification of linear Nakayama algebras with
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Reflection Representations of Coxeter Groups and Homology of Coxeter Graphs Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-12-07 Hongsheng Hu
We study and classify a class of representations (called generalized geometric representations) of a Coxeter group of finite rank. These representations can be viewed as a natural generalization of the geometric representation. The classification is achieved by using characters of the integral homology group of certain graphs closely related to the Coxeter graph. On this basis, we also provide an explicit
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Frobenius Kernels of Algebraic Supergroups and Steinberg’s Tensor Product Theorem Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-12-04 Taiki Shibata
For a split quasireductive supergroup \(\mathbbm {G}\) defined over a field, we study structure and representation of Frobenius kernels \(\mathbbm {G}_r\) of \(\mathbbm {G}\) and we give a necessary and sufficient condition for \(\mathbbm {G}_r\) to be unimodular in terms of the root system of \(\mathbbm {G}\). We also establish Steinberg’s tensor product theorem for \(\mathbbm {G}\) under some natural
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Middle Terms of AR-sequences of Graded Kronecker Modules Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-11-29 Jie Liu
Let \((T(n),\Omega )\) be the covering of the generalized Kronecker quiver K(n), where \(\Omega \) is a bipartite orientation. Then there exists a reflection functor \(\sigma \) on the category \({{\,\textrm{mod}\,}}(T(n),\Omega )\). Suppose that \(0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0\) is an AR-sequence in the regular component \(\mathcal {D}\) of \({{\,\textrm{mod}\,}}(T(n),\Omega
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On the Braid Group Action on Exceptional Sequences for Weighted Projective Lines Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-11-23 Edson Ribeiro Alvares, Eduardo Nascimento Marcos, Hagen Meltzer
We give a new and intrinsic proof of the transitivity of the braid group action on the set of full exceptional sequences of coherent sheaves on a weighted projective line. We do not use the corresponding result of Crawley-Boevey for modules over hereditary algebras. As an application we prove that the strongest global dimension of the category of coherent sheaves on a weighted projective line \(\mathbb
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High Order Free Hyperplane Arrangements in 3-Dimensional Vector Spaces Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-11-04 Norihiro Nakashima
Holm introduced m-free \(\ell \)-arrangements which is a generalization of free arrangements, while he asked whether all \(\ell \)-arrangements are m-free for m large enough. Recently Abe and the author gave a negative answer to this question when \(\ell \ge 4\). In this paper we verify that 3-arrangements \(\mathscr {A}\) are m-free and compute the m-exponents for all \(m\ge |\mathscr {A}|+2\), where
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Silting Reduction in Exact Categories Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-10-28 Yu Liu, Panyue Zhou, Yu Zhou, Bin Zhu
Presilting and silting subcategories in extriangulated categories were introduced by Adachi and Tsukamoto recently, which are generalizations of those concepts in triangulated categories. Exact categories and triangulated categories are extriangulated categories. In this paper, we prove that the Gabriel-Zisman localization \(\mathcal {B}/(\textsf{thick}\hspace{.01in}\mathcal W)\) of an exact category
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Nakayama Algebras and Fuchsian Singularities Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-10-20 Helmut Lenzing, Hagen Meltzer, Shiquan Ruan
This present paper is devoted to the study of a class of Nakayama algebras \(N_n(r)\) given by the path algebra of the equioriented quiver \(\mathbb {A}_n\) subject to the nilpotency degree r for each sequence of r consecutive arrows. We show that the Nakayama algebras \(N_n(r)\) for certain pairs (n, r) can be realized as endomorphism algebras of tilting objects in the bounded derived category of
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Kazhdan-Lusztig Algorithm for Whittaker Modules with Arbitrary Infinitesimal Characters Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-10-16 Qixian Zhao
Let \(\mathfrak {g}\) be a complex semisimple Lie algebra. We give a description of characters of irreducible Whittaker modules for \(\mathfrak {g}\) with any infinitesimal character, along with a Kazhdan-Lusztig algorithm for computing them. This generalizes Miličić-Soergel’s and Romanov’s results for integral infinitesimal characters. As a special case, we recover the non-integral Kazhdan-Lusztig
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On the Properties of Acyclic Sign-Skew-Symmetric Cluster Algebras Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-10-14 Siyang Liu
We study the tropical dualities and properties of exchange graphs for the totally sign-skew-symmetric cluster algebra under a condition. We prove that the condition always holds for acyclic cluster algebras, then all results hold for the acyclic case.
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Dipper Donkin Quantized Matrix Algebra and Reflection Equation Algebra at Root of Unity Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-10-10 Sanu Bera, Snehashis Mukherjee
In this article the quantized matrix algebras as in the title have been studied at a root of unity. A full classification of simple modules over such quantized matrix algebras of degree 2 along with some finite-dimensional indecomposable modules are explicitly presented.
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Gentle Algebras Arising from Surfaces with Orbifold Points of Order 3, Part I: Scattering Diagrams Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-09-29 Daniel Labardini-Fragoso, Lang Mou
To each triangulation of any surface with marked points on the boundary and orbifold points of order three, we associate a quiver (with loops) with potential whose Jacobian algebra is finite dimensional and gentle. We study the stability scattering diagrams of such gentle algebras and use them to prove that the Caldero–Chapoton map defines a bijection between \(\tau \)-rigid pairs and cluster monomials
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A Note on the Global Dimension of Shifted Orders Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-09-22 Özgür Esentepe
We consider the dominant dimension of an order over a Cohen-Macaulay ring in the category of centrally Cohen-Macaulay modules. There is a canonical tilting module in the case of positive dominant dimension and we give an upper bound on the global dimension of its endomorphism ring.
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Dessins D’Enfants, Brauer Graph Algebras and Galois Invariants Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-09-18 Goran Malić, Sibylle Schroll
In this paper, we associate a finite dimensional algebra, called a Brauer graph algebra, to every clean dessin d’enfant by constructing a quiver based on the monodromy of the dessin. We show that Galois conjugate dessins d’enfants give rise to derived equivalent Brauer graph algebras and that the stable Auslander-Reiten quiver and the dimension of the Brauer graph algebra are invariant under the induced
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Gorenstein Rings via Homological Dimensions, and Symmetry in Vanishing of Ext and Tate Cohomology Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-09-15 Dipankar Ghosh, Tony J. Puthenpurakal
The aim of this article is to consider the spectral sequences induced by tensor-hom adjunction, and provide a number of new results. Let R be a commutative Noetherian local ring of dimension d. In the 1st part, it is proved that R is Gorenstein if and only if it admits a nonzero CM (Cohen-Macaulay) module M of finite Gorenstein dimension g such that \(\text {type}(M) \leqslant \mu ( \text {Ext}_R^g(M
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Onset of Regularity for FI-modules Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-09-12 Cihan Bahran
In terms of local cohomology, we give an explicit range as to when the FI-homology of an FI-module attains the degree predicted by its regularity.
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Auto-Correlation Functions for Unitary Groups Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-09-11 Kyu-Hwan Lee, Se-Jin Oh
We compute the auto-correlations functions of order \(m\ge 1\) for the characteristic polynomials of random matrices from certain subgroups of the unitary groups \({\text {U}}(2)\) and \({\text {U}}(3)\) by establishing new branching rules. These subgroups can be understood as certain analogues of Sato–Tate groups of \({\text {USp}}(4)\) in our previous paper. Our computation yields symmetric polynomial
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Extremal Tensor Products of Demazure Crystals Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-09-12 Sami Assaf, Anne Dranowski, Nicolle González
Demazure crystals are subcrystals of highest weight irreducible \(\mathfrak {g}\)-crystals. In this article, we study tensor products of a larger class of subcrystals, called extremal, and give a local characterization for exactly when the tensor product of Demazure crystals is extremal. We then show that tensor products of Demazure crystals decompose into direct sums of Demazure crystals if and only
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An Indicator Formula for the Hopf Algebra $$k^{S_{n-1}}\#kC_n$$ Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-09-06 Kayla Orlinsky
The semisimple bismash product Hopf algebra \(J_n=k^{S_{n-1}}\#kC_n\) for an algebraically closed field k is constructed using the matched pair actions of \(C_n\) and \(S_{n-1}\) on each other. In this work, we reinterpret these actions and use an understanding of the involutions of \(S_{n-1}\) to derive a new Froebnius-Schur indicator formula for irreps of \(J_n\) and show that for n odd, all indicators
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Dimer Algebras, Ghor Algebras, and Cyclic Contractions Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-09-07 Charlie Beil
A ghor algebra is the path algebra of a dimer quiver on a surface, modulo relations that come from the perfect matchings of its quiver. Such algebras arise from abelian quiver gauge theories in physics. We show that a ghor algebra \(\Lambda \) on a torus is a dimer algebra (a quiver with potential) if and only if it is noetherian, and otherwise \(\Lambda \) is the quotient of a dimer algebra by homotopy
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A Drinfeld-Type Presentation of the Orthosymplectic Yangians Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-08-18 A. I. Molev
We use the Gauss decomposition of the generator matrix in the R-matrix presentation of the Yangian for the orthosymplectic Lie superalgebra \(\mathfrak {osp}_{N|2m}\) to produce its Drinfeld-type presentation. The results rely on a super-version of the embedding theorem which allows one to identify a subalgebra in the R-matrix presentation which is isomorphic to the Yangian associated with \(\mathfrak
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A Uniqueness Property of $$\tau $$ -Exceptional Sequences Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-08-16 Eric J. Hanson, Hugh Thomas
Recently, Buan and Marsh showed that if two complete \(\tau \)-exceptional sequences agree in all but at most one term, then they must agree everywhere, provided the algebra is \(\tau \)-tilting finite. They conjectured that the result holds without that assumption. We prove their conjecture. Along the way, we also show that the dimension vectors of the modules in a \(\tau \)-exceptional sequence are
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Identities of Inverse Chevalley Type for the Graded Characters of Level-Zero Demazure Submodules over Quantum Affine Algebras of Type C Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-08-15 Takafumi Kouno, Satoshi Naito, Daniel Orr
We provide identities of inverse Chevalley type for the graded characters of level-zero Demazure submodules of extremal weight modules over a quantum affine algebra of type C. These identities express the product \(e^{\mu } \text {gch} ~V_{x}^{-}(\lambda )\) of the (one-dimensional) character \(e^{\mu }\), where \(\mu \) is a (not necessarily dominant) minuscule weight, with the graded character g
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Quantization of Deformed Cluster Poisson Varieties Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-08-09 Man-Wai Mandy Cheung, Juan Bosco Frías-Medina, Timothy Magee
Fock and Goncharov described a quantization of cluster \(\mathcal {X}\)-varieties (also known as cluster Poisson varieties) in Fock and Goncharov (Ann. Sci. Éc. Norm. Supér. 42(6), 865–930 2009). Meanwhile, families of deformations of cluster \(\mathcal {X}\)-varieties were introduced in Bossinger et al. (Compos. Math. 156(10), 2149–2206, 2020). In this paper we show that the two constructions are
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Module Categories of Small Radical Nilpotency Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-07-24 Shiping Liu, Youqi Yin
This paper aims to initiate a study of the representation theory of representation-finite artin algebras in terms of the nilpotency of the radical of their module category. Firstly, we shall calculate this nilpotency explicitly for hereditary algebras of type \(\mathbb {A}_n\) and for Nakayama algebras. Surprisingly, this nilpotency for a given algebra coincides with its Loewy length if and only if
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A Note on Constructing Quasi Modules for Quantum Vertex Algebras from Twisted Yangians Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-07-24 Slaven Kožić, Marina Sertić
In this note, we consider the twisted Yangians \(\text {Y}(\mathfrak {g}_N)\) associated with the orthogonal and symplectic Lie algebras \(\mathfrak {g}_N=\mathfrak {o}_N,\mathfrak {sp}_N\). First, we introduce a certain subalgebra \(\text {A}_c(\mathfrak {g}_N)\) of the double Yangian for \(\mathfrak {gl}_N\) at the level \(c\in \mathbb {C}\), which contains the centrally extended \(\text {Y}(\mathfrak
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Wild Local Structures of Automorphic Lie Algebras Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-07-20 Drew Damien Duffield, Vincent Knibbeler, Sara Lombardo
We study automorphic Lie algebras using a family of evaluation maps parametrised by the representations of the associative algebra of functions. This provides a descending chain of ideals for the automorphic Lie algebra which is used to prove that it is of wild representation type. We show that the associated quotients of the automorphic Lie algebra are isomorphic to twisted truncated polynomial current
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Classifying Recollements of Derived Module Categories for Derived Discrete Algebras Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-07-12 Xiuli Bian
We study a class of derived discrete Nakayama algebras. All indecomposable compact objects in the derived module category are determined and all recollements generated by the indecomposable compact exceptional objects are classified. It reveals that all such recollements are derived equivalent to stratifying recollements. As a byproduct, this confirms a question due to Xi for these recollements.
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Topologically Semiperfect Topological Rings Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-07-10 Leonid Positselski, Jan Šťovíček
We define topologically semiperfect (complete, separated, right linear) topological rings and characterize them by equivalent conditions. We show that the endomorphism ring of a module, endowed with the finite topology, is topologically semiperfect if and only if the module is decomposable as an (infinite) direct sum of modules with local endomorphism rings. Then we study structural properties of topologically
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Epimorphic Quantum Subgroups and Coalgebra Codominions Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-07-08 Alexandru Chirvasitu
We prove a number of results concerning monomorphisms, epimorphisms, dominions and codominions in categories of coalgebras. Examples include: (a) representation-theoretic characterizations of monomorphisms in all of these categories that when the Hopf algebras in question are commutative specialize back to the familiar necessary and sufficient conditions (due to Bien-Borel) that a linear algebraic
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A Generalization of the Correspondences Between Quasi-Hereditary Algebras and Directed Bocses Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-07-03 Yuichiro Goto
Quasi-hereditary algebras were introduced by Cline, Parshall and Scott to study the highest weight categories in Lie theory. On the other hand, bocses were introduced in the context of Drozd’s tame and wild dichotomy theorem. Koenig, Külshammer and Ovsienko connected the two areas by giving equivalences between the categories of \(\Delta \)-filtered modules over quasi-hereditary algebras and those
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Stable Unital Bases, Hyperfocal Subalgebras and Basic Morita Equivalences Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-07-03 Tiberiu Coconeţ, Constantin-Cosmin Todea
We investigate Conjecture 1.5 introduced by Barker and Gelvin (J. Gr. Theory 25, 973–995 2022), which says that any source algebra of a p-block (p is a prime) of a finite group has the unit group containing a basis stabilized by the left and right actions of the defect group. We will reduce this conjecture to a similar statement about the bases of the hyperfocal subalgebras in the source algebras.
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Finite GK-Dimensional Nichols Algebras Over the Infinite Dihedral Group Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-06-29 Yongliang Zhang
We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GK-dimension for short, through the study of Nichols algebras over \(\mathbb {D}_{\infty }\), the infinite dihedral group.We find all the irreducible Yetter-Drinfeld modules V over \(\mathbb {D}_{\infty }\), and determine which Nichols algebras \(\mathcal {B}(V)\) of V are finite GK-dimensional.
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Modular Products and Modules for Finite Groups Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-06-26 John F. R. Duncan, Jeffrey A. Harvey, Brandon C. Rayhaun
Motivated by the appearance of penumbral moonshine, and by evidence that penumbral moonshine enjoys an extensive relationship to generalized monstrous moonshine via infinite products, we establish a general construction in this work which uses singular theta lifts and a concrete construction at the level of modules for a finite group to translate between moonshine in weight one-half and moonshine in
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On the Ideals of Ultragraph Leavitt Path Algebras Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-06-24 T. T. H. Duyen, D. Gonçalves, T. G. Nam
In this article, we provide an explicit description of a set of generators for any ideal of an ultragraph Leavitt path algebra. We provide several additional consequences of this description, including information about generating sets for graded ideals, the graded uniqueness and Cuntz-Krieger theorems, the semiprimeness, and the semiprimitivity of ultragraph Leavitt path algebras, a complete characterization
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Crystal Bases of Modified $$\imath $$ quantum Groups of Certain Quasi-Split Types Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-06-14 Hideya Watanabe
In order to see the behavior of \(\imath \)canonical bases at \(q = \infty \), we introduce the notion of \(\imath \)crystals associated to an \(\imath \)quantum group of certain quasi-split type. The theory of \(\imath \)crystals clarifies why \(\imath \)canonical basis elements are not always preserved under natural homomorphisms. Also, we construct a projective system of \(\imath \)crystals whose
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The Igusa-Todorov $$\phi $$ -Dimension on Morita Context Algebras Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-06-13 Marcos Barrios, Gustavo Mata
In this article we prove that, under certain hypotheses, Morita context algebras with zero bimodule morphisms have finite \(\phi \)-dimension. For these algebras we also study the behaviour of the \(\phi \)-dimension for an algebra and its opposite. In particular we show that the \(\phi \)-dimension of an Artin algebra is not symmetric, i.e. there exists an Artin algebra A such that \(\phi \dim (A)
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From the Lattice of Torsion Classes to the Posets of Wide Subcategories and ICE-closed Subcategories Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-06-13 Haruhisa Enomoto
In this paper, we compute the posets of wide subcategories and ICE-closed subcategories from the lattice of torsion classes in an abelian length category in a purely lattice-theoretical way, by using the kappa map in a completely semidistributive lattice. As for the poset of wide subcategories, we give two more simple constructions via a bijection between wide subcategories and torsion classes with
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Entwined Modules Over Representations of Categories Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-05-22 Abhishek Banerjee
We introduce a theory of modules over a representation of a small category taking values in entwining structures over a semiperfect coalgebra. This takes forward the aim of developing categories of entwined modules to the same extent as that of module categories as well as the philosophy of Mitchell of working with rings with several objects. The representations are motivated by work of Estrada and
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Invariants of Weyl Group Action and q-characters of Quantum Affine Algebras Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-05-09 Rei Inoue, Takao Yamazaki
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
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Littlewood-Richardson rule for generalized Schur Q-functions Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-04-03 Fang Huang, Yanjun Chu, Chuanzhong Li
Littlewood-Richardson rule gives the expansion formula for decomposing a product of two Schur functions as a linear sum of Schur functions, while the decomposition formula for the multiplication of two Schur Q-functions is also given as the combinatorial model by using the shifted tableaux. In this paper, we firstly use the shifted Littlewood-Richardson coefficients to give the coefficients of generalized
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Representation Stability and Finite Orthogonal Groups Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-03-29 Arun S. Kannan, Zifan Wang
In this paper, we prove homological stability results about orthogonal groups over finite commutative rings where 2 is a unit. Inspired by Putman and Sam (2017), we construct a category OrI(R) and prove a local Noetherianity theorem for the category of OrI(R)-modules. This implies an asymptotic structure theorem for orthogonal groups. In addition, we show general homological stability theorems for
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On Quasi Steinberg Characters of Complex Reflection Groups Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-02-27 Ashish Mishra, Digjoy Paul, Pooja Singla
Let G be a finite group and p be a prime number dividing the order of G. An irreducible character χ of G is called a quasi p-Steinberg character if χ(g) is nonzero for every p-regular element g in G. In this paper, we classify the quasi p-Steinberg characters of complex reflection groups G(r,q,n) and exceptional complex reflection groups. In particular, we obtain this classification for Weyl groups
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Local Characterizations for Decomposability of 2-Parameter Persistence Modules Algebr. Represent. Theor. (IF 0.6) Pub Date : 2023-02-14 Magnus B. Botnan, Vadim Lebovici, Steve Oudot
We investigate the existence of sufficient local conditions under which poset representations decompose as direct sums of indecomposables from a given class. In our work, the indexing poset is the product of two totally ordered sets, corresponding to the setting of 2-parameter persistence in topological data analysis. Our indecomposables of interest belong to the so-called interval modules, which by