scholarly journals Graded simple modules and loop modules

2017 ◽  
pp. 53-85 ◽  
Author(s):  
Alberto Elduque ◽  
Mikhail Kochetov
Keyword(s):  
Simple Modules ◽  
Loop Modules

scholarly journals Laurent phenomenon and simple modules of quiver Hecke algebras

2019 ◽  
Vol 155 (12) ◽  
pp. 2263-2295 ◽  
Author(s):  
Masaki Kashiwara ◽  
Myungho Kim
Keyword(s):  
Simple Module ◽  
Hecke Algebras ◽  
Duke Math ◽  
Cluster Algebras ◽  
Coordinate Ring ◽  
Simple Modules ◽  
Laurent Phenomenon ◽  
Cluster Variable ◽  
Quantum Cluster

In this paper we study consequences of the results of Kang et al. [Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), 349–426] on a monoidal categorification of the unipotent quantum coordinate ring $A_{q}(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category ${\mathcal{C}}_{w}$ strongly commutes with all the cluster variables in a cluster $[\mathscr{C}]$, then $[S]$ is a cluster monomial in $[\mathscr{C}]$. If $S$ strongly commutes with cluster variables except for exactly one cluster variable $[M_{k}]$, then $[S]$ is either a cluster monomial in $[\mathscr{C}]$ or a cluster monomial in $\unicode[STIX]{x1D707}_{k}([\mathscr{C}])$. We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Qin [Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. 166 (2017), 2337–2442]) of the localization $\widetilde{A}_{q}(\mathfrak{n}(w))$ of $A_{q}(\mathfrak{n}(w))$ at the frozen variables. A characterization on the commutativity of a simple module $S$ with cluster variables in a cluster $[\mathscr{C}]$ is given in terms of the denominator vector of $[S]$ with respect to the cluster $[\mathscr{C}]$.

CLOSE NEIGHBORS IN THE QUIVER OF TILTING MODULES

2008 ◽  
Vol 07 (03) ◽  
pp. 379-392
Author(s):  
DIETER HAPPEL
Keyword(s):  
Representation Type ◽  
Tilting Module ◽  
Tilting Modules ◽  
Hereditary Algebra ◽  
Simple Modules ◽  
Finite Dimensional ◽  
Wild Representation Type ◽  
Local Properties

For a finite dimensional hereditary algebra Λ local properties of the quiver [Formula: see text] of tilting modules are investigated. The existence of special neighbors of a given tilting module is shown. If Λ has more than 3 simple modules it is shown as an application that Λ is of wild representation type if and only if [Formula: see text] is a subquiver of [Formula: see text].

scholarly journals Primitive ideals, twisting functors and star actions for classical Lie superalgebras

2016 ◽  
Vol 2016 (718) ◽  
Author(s):  
Kevin Coulembier ◽  
Volodymyr Mazorchuk
Keyword(s):  
Representation Theory ◽  
Lie Superalgebras ◽  
Simple Modules ◽  
Primitive Ideals

AbstractWe study three related topics in representation theory of classical Lie superalgebras. The first one is classification of primitive ideals, i.e. annihilator ideals of simple modules, and inclusions between them. The second topic concerns Arkhipov’s twisting functors on the BGG category

scholarly journals INJECTIVE MODULES OVER DOWN-UP ALGEBRAS

2010 ◽  
Vol 52 (A) ◽  
pp. 53-59 ◽  
Author(s):  
PAULA A. A. B. CARVALHO ◽  
CHRISTIAN LOMP ◽  
DILEK PUSAT-YILMAZ
Keyword(s):  
Roots Of Unity ◽  
Finiteness Conditions ◽  
Simple Modules ◽  
Injective Modules ◽  
Injective Hulls

AbstractThe purpose of this paper is to study finiteness conditions on injective hulls of simple modules over Noetherian down-up algebras. We will show that the Noetherian down-up algebras A(α, β, γ) which are fully bounded are precisely those which are module-finite over a central subalgebra. We show that injective hulls of simple A(α, β, γ)-modules are locally Artinian provided the roots of X2 − αX − β are distinct roots of unity or both equal to 1.

Simple Modules for Grace

2000 ◽  
pp. 383-395 ◽  
Author(s):  
Reiko Heckel ◽  
Berthold Hoffmann ◽  
Peter Knirsch ◽  
Sabine Kuske
Keyword(s):  
Simple Modules

scholarly journals Asymptotic representations and Drinfeld rational fractions

2012 ◽  
Vol 148 (5) ◽  
pp. 1593-1623 ◽  
Author(s):  
David Hernandez ◽  
Michio Jimbo
Keyword(s):  
Rational Functions ◽  
Loop Algebra ◽  
Explicit Formulas ◽  
Simple Modules ◽  
Asymptotic Representations

AbstractWe introduce and study a category of representations of the Borel algebra associated with a quantum loop algebra of non-twisted type. We construct fundamental representations for this category as a limit of the Kirillov–Reshetikhin modules over the quantum loop algebra and establish explicit formulas for their characters. We prove that general simple modules in this category are classified by n-tuples of rational functions in one variable which are regular and non-zero at the origin but may have a zero or a pole at infinity.

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