scholarly journals ORIENTED FLIP GRAPHS, NONCROSSING TREE PARTITIONS, AND REPRESENTATION THEORY OF TILING ALGEBRAS

2019 ◽  
Vol 62 (1) ◽  
pp. 147-182 ◽  
Author(s):  
ALEXANDER GARVER ◽  
THOMAS MCCONVILLE
Keyword(s):  
Representation Theory ◽  
Derived Categories ◽  
Module Category ◽  
Indecomposable Modules ◽  
Quiver Mutation ◽  
Extension Spaces ◽  
Dynkin Quiver ◽  
Torsion Pairs ◽  
Gentle Algebras

AbstractThe purpose of this paper is to understand lattices of certain subcategories in module categories of representation-finite gentle algebras called tiling algebras, as introduced by Coelho Simões and Parsons. We present combinatorial models for torsion pairs and wide subcategories in the module category of tiling algebras. Our models use the oriented flip graphs and noncrossing tree partitions, previously introduced by the authors, and a description of the extension spaces between indecomposable modules over tiling algebras. In addition, we classify two-term simple-minded collections in bounded derived categories of tiling algebras. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver.

scholarly journals On Extensions for Gentle Algebras

2020 ◽  
pp. 1-44
Author(s):  
İlke Çanakçı ◽  
David Pauksztello ◽  
Sibylle Schroll
Keyword(s):  
Module Category ◽  
Extension Spaces ◽  
Gentle Algebras

Abstract We give a complete description of a basis of the extension spaces between indecomposable string and quasi-simple band modules in the module category of a gentle algebra.

scholarly journals Automorphisms of Regular Wreath Product -Groups

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Jeffrey M. Riedl
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Representation Theory ◽  
Wreath Product ◽  
Cyclic Group ◽  
Product Group ◽  
Finite Cyclic Group ◽  
Arbitrary Finite Group

We present a useful new characterization of the automorphisms of the regular wreath product group of a finite cyclic -group by a finite cyclic -group, for any prime , and we discuss an application. We also present a short new proof, based on representation theory, for determining the order of the automorphism group Aut(), where is the regular wreath product of a finite cyclic -group by an arbitrary finite -group.

scholarly journals The Tilting–Cotilting Correspondence

2019 ◽  
Author(s):  
Leonid Positselski ◽  
Jan Šťovíček
Keyword(s):  
Abelian Category ◽  
Derived Categories ◽  
Module Category ◽  
Topological Ring ◽  
Projective Generator ◽  
Derived Equivalences ◽  
Wide Range ◽  
Injective Cogenerator ◽  
Tilting Object

Abstract To a big $n$-tilting object in a complete, cocomplete abelian category ${\textsf{A}}$ with an injective cogenerator we assign a big $n$-cotilting object in a complete, cocomplete abelian category ${\textsf{B}}$ with a projective generator and vice versa. Then we construct an equivalence between the (conventional or absolute) derived categories of ${\textsf{A}}$ and ${\textsf{B}}$. Under various assumptions on ${\textsf{A}}$, which cover a wide range of examples (for instance, if ${\textsf{A}}$ is a module category or, more generally, a locally finitely presentable Grothendieck abelian category), we show that ${\textsf{B}}$ is the abelian category of contramodules over a topological ring and that the derived equivalences are realized by a contramodule-valued variant of the usual derived Hom functor.

scholarly journals Complete slices and homological properties of tilted algebras

1994 ◽  
Vol 36 (3) ◽  
pp. 347-354 ◽  
Author(s):  
Ibrahim Assem ◽  
Flávio Ulhoa Coelho
Keyword(s):  
Representation Theory ◽  
Global Dimension ◽  
Indecomposable Modules ◽  
Tilted Algebra ◽  
Finite Dimensional ◽  
Tilted Algebras ◽  
Homological Dimensions ◽  
The Subject ◽  
Left And Right ◽  
Almost All

It is reasonable to expect that the representation theory of an algebra (finite dimensional over a field, basic and connected) can be used to study its homological properties. In particular, much is known about the structure of the Auslander-Reiten quiver of an algebra, which records most of the information we have on its module category. We ask whether one can predict the homological dimensions of a module from its position in the Auslander-Reiten quiver. We are particularly interested in the case where the algebra is a tilted algebra. This class of algebras of global dimension two, introduced by Happel and Ringel in [7], has since then been the subject of many investigations, and its representation theory is well understood by now (see, for instance, [1], [7], [8], [9], [11], [13]).In this case, the most striking feature of the Auslander-Reiten quiver is the existence of complete slices, which reproduce the quiver of the hereditary algebra from which the tilted algebra arises. It follows from well-known results that any indecomposable successor (or predecessor) of a complete slice has injective (or projective, respectively) dimension at most one, from which one deduces that a tilted algebra is representation-finite if and only if both the projective and the injective dimensions of almost all (that is, all but at most finitely many non-isomorphic) indecomposable modules equal two (see (3.1) and (3.2)). On the other hand, the authors have shown in [2, (3.4)] that a representation-infinite algebra is concealed if and only if both the projective and the injective dimensions of almost all indecomposable modules equal one (see also [14]). This leads us to consider, for tilted algebras which are not concealed, the case when the projective (or injective) dimension of almost all indecomposable successors (or predecessors, respectively) of a complete slice equal two. In order to answer this question, we define the notions of left and right type of a tilted algebra, then those of reduced left and right types (see (2.2) and (3.4) for the definitions).

scholarly journals Strongly simply connected Auslander algebras

1997 ◽  
Vol 39 (1) ◽  
pp. 21-27 ◽  
Author(s):  
Ibrahim Assem ◽  
Peter Brown
Keyword(s):  
Representation Theory ◽  
Closed Field ◽  
Interesting Problem ◽  
Indecomposable Modules ◽  
Finite Algebras ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Infinite Case ◽  
Simple Connectedness ◽  
Simply Connected

Letkbe an algebraically closed field. By an algebra is meant an associative finite dimensionalk-algebra A with an identity. We are interested in studying the representation theory of Λ, that is, in describing the category mod Λ of finitely generated right Λ-modules. Thus we may, without loss of generality, assume that Λ is basic and connected. For our purpose, one strategy consists in using covering techniques to reduce the problem to the case where the algebra is simply connected, then in solving the problem in this latter case. This strategy was proved efficient for representation-finite algebras (that is, algebras having only finitely many isomorphism classes of indecomposable modules) and representation-finite simply connected algebras are by now well-understood: see, for instance [5], [7],[8]. While little is known about covering techniques in the representation-infinite case, it is clearly an interesting problem to describe the representation-infinite simply connected algebras. The objective of this paper is to give a criterion for the simple connectedness of a class of (mostly representationinfinite) algebras.

JAPAN’S EFFORTS TO DILUTE ITS DARK HISTORY THROUGH ANIME “HETALIA AXIS POWERS”

2018 ◽  
Vol 3 (2) ◽  
Author(s):  
Citra Rindu Prameswari
Keyword(s):  
World War Ii ◽  
Representation Theory ◽  
Image Representation ◽  
World War ◽  
Positive Image

“Hetalia” is one of many animes with history, specifically World War II, as its theme, but what makes it different from other animes is the fact that it gained many controversies even before it was broadcasted. The characters in “Hetalia” are world countries‟ personifications with stereotypes as their characteristics. Because of that, it can be seen easily that these characters automatically have ties with each respective country. Japan is Japan‟s character in it, but his scenes mainly only contain cultural related things rather than war-related activities. This article aims to find out the agenda behind Japan‟s image representation via “Hetalia’s” narrative plots and the characterization of Japan. Using cinema concept and representation theory to analyze the characterization through dialogues and narrations, I argue that there is a certain agenda to feminize to soften Japan‟s image and to dilute Japan‟s dark history so that Japan will have more positive image via “Hetalia”. It automatically highlights Japan‟s position as not one of the „main villains‟ in World War II by making one party look positively good (Japan) and another party look not so good in relation with geopolitics strategy.

Stone Lattices. I: Construction Theorems

1969 ◽  
Vol 21 ◽  
pp. 884-894 ◽  
Author(s):  
C. C. Chen ◽  
G. Grätzer
Keyword(s):  
Representation Theory ◽  
Boolean Algebras ◽  
Prime Ideals ◽  
Satisfactory Theory

Stone lattices were (named and) first studied in 1957 (5). Since then, a great number of papers have been written on Stone lattices and a very satisfactory theory evolved. Despite the fact that all chains with 0, 1 as well as all Boolean algebras are Stone lattices, it turns out that many of the nice theorems on Boolean algebras have analogues, in fact, generalizations for Stone lattices. To give just two examples: the characterization of Boolean algebras in terms of prime ideals (Nachbin (6)) is generalized in (5) (see also (9)); Stone's representation theory (8) is generalized in (4); see also (2).

A REPRESENTATION THEORY FOR POLYNOMIAL COFRACTIONALITY IN VECTOR AUTOREGRESSIVE MODELS

2009 ◽  
Vol 26 (4) ◽  
pp. 1201-1217 ◽  
Author(s):  
Massimo Franchi
Keyword(s):  
Representation Theory ◽  
Autoregressive Model ◽  
Moving Average ◽  
Recursive Algorithm ◽  
Autoregressive Models ◽  
Vector Autoregressive Models ◽  
Vector Autoregressive ◽  
Moving Average Representation ◽  
Average Representation

We extend the representation theory of the autoregressive model in the fractional lag operator of Johansen (2008, Econometric Theory 24, 651–676). A recursive algorithm for the characterization of cofractional relations and the corresponding adjustment coefficients is given, and it is shown under which condition the solution of the model is fractional of order d and displays cofractional relations of order d − b and polynomial cofractional relations of order d − 2b,…, d − cb ≥ 0 for integer c; the cofractional relations and the corresponding moving average representation are characterized in terms of the autoregressive coefficients by the same algorithm. For c = 1 and c = 2 we find the results of Johansen (2008).

scholarly journals EXPLICIT CONSTRUCTION OF COMPANION BASES

2015 ◽  
Vol 58 (2) ◽  
pp. 357-384
Author(s):  
MARK JAMES PARSONS
Keyword(s):  
Root System ◽  
Explicit Construction ◽  
Finitely Generated ◽  
Root Lattice ◽  
Indecomposable Modules ◽  
Tilted Algebra ◽  
Underlying Graph ◽  
Inner Products ◽  
Dynkin Quiver ◽  
Mutation Type

AbstractA companion basis for a quiver Γ mutation equivalent to a simply-laced Dynkin quiver is a subset of the associated root system which is a$\mathbb{Z}$-basis for the integral root lattice with the property that the non-zero inner products of pairs of its elements correspond to the edges in the underlying graph of Γ. It is known in typeA(and conjectured for all simply-laced Dynkin cases) that any companion basis can be used to compute the dimension vectors of the finitely generated indecomposable modules over the associated cluster-tilted algebra. Here, we present a procedure for explicitly constructing a companion basis for any quiver of mutation typeAorD.

scholarly journals Split t-structures and torsion pairs in hereditary categories

2018 ◽  
Vol 17 (11) ◽  
pp. 1850218
Author(s):  
Ibrahim Assem ◽  
María José Souto-Salorio ◽  
Sonia Trepode
Keyword(s):  
Derived Category ◽  
Module Category ◽  
Hereditary Algebra ◽  
Tilted Algebra ◽  
Text Structures ◽  
Torsion Pairs

We construct a bijection between split torsion pairs in the module category of a tilted algebra having a complete slice in the preinjective component with corresponding [Formula: see text]-structures. We also classify split [Formula: see text]-structures in the derived category of a hereditary algebra.

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