Algebras and Representation Theory最新文献

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 (mge |mathscr {A}|+2), where (|mathscr {A}|) is the cardinality of (mathscr {A}). Hence Holm’s question has a positive answer when (ell =3). Finally we prove that 3-dimensional Weyl arrangements of types A and B are m-free for all (mge 0).

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 (mathcal {B}) with respect to a presilting subcategory (mathcal W) satisfying certain condition can be realized as a subfactor category of (mathcal {B}). Afterwards, we discuss the relation between silting subcategories and tilting subcategories in exact categories, which gives us a kind of important examples of our results. In particular, for a finite dimensional Gorenstein algebra, we get the relative version of the description of the singularity category due to Happel and Chen-Zhang.

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 coherent sheaves over a weighted projective line, or in its stable category of vector bundles. Moreover, we classify all the Nakayama algebras (N_n(r)) of Fuchsian type, that is, derived equivalent to the bounded derived categories of extended canonical algebras. We also provide a new way to prove the classification result on Nakayama algebras of piecewise hereditary type, which have been done by Happel–Seidel before.

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 conjecture for Verma modules.

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.

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.

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 of the generalized cluster algebra associated to the surface by Chekhov and Shapiro.

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.

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 action of the absolute Galois group.

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,R) )) (e.g., (text {type}(M)=1)). This considerably strengthens a result of Takahashi. Moreover, we show that if there is a nonzero R-module M of depth (geqslant d - 1) such that the injective dimensions of M, (text {Hom}_R(M,M)) and (text {Ext}_R^1(M,M)) are finite, then M has finite projective dimension and R is Gorenstein. In the 2nd part, we assume that R is CM with a canonical module (omega ). For CM R-modules M and N, we show that the vanishing of one of the following implies the same for others: (text {Ext}_R^{gg 0}(M,N^{+})), (text {Ext}_R^{gg 0}(N,M^{+})) and (text {Tor}_{gg 0}^R(M,N)), where (M^{+}) denotes (text {Ext}_R^{d-dim (M)}(M,omega )). This strengthens a result of Huneke and Jorgensen. Furthermore, we prove a similar result for Tate cohomologies under the additional condition that R is Gorenstein.