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Search: MSC category 16G20 ( Representations of quivers and partially ordered sets )

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1. CJM Online first

Du, Jie; Zhao, Zhonghua
Multiplication formulas and canonical bases for quantum affine gln
We will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra $\mathfrak{H}_\Delta(n)$ of a cyclic quiver $\Delta(n)$. As a first application, we see immediately the existence of Hall polynomials for cyclic quivers, a fact established by J. Y. Guo and C. M. Ringel, and derive a recursive formula to compute them. We will further use the formula and the construction of a certain monomial base for $\mathfrak{H}_\Delta(n)$ given by Deng, Du, and Xiao together with the double Ringel--Hall algebra realisation of the quantum loop algebra $\mathbf{U}_v(\widehat{\mathfrak{g}\mathfrak{l}}_n)$ given by Deng, Du, and Fu to develop some algorithms and to compute the canonical basis for $\mathbf{U}_v^+(\widehat{\mathfrak{g}\mathfrak{l}}_n)$. As examples, we will show explicitly the part of the canonical basis associated with modules of Lowey length at most $2$ for the quantum group $\mathbf{U}_v(\widehat{\mathfrak{g}\mathfrak{l}}_2)$.

Keywords:Ringel-Hall algebra, quantum group, cyclic quiver, monomial basis, canonical basis
Categories:16G20, 20G42

2. CJM 2014 (vol 67 pp. 28)

Asadollahi, Javad; Hafezi, Rasool; Vahed, Razieh
Bounded Derived Categories of Infinite Quivers: Grothendieck Duality, Reflection Functor
We study bounded derived categories of the category of representations of infinite quivers over a ring $R$. In case $R$ is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left, resp. right, rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.

Keywords:derived category, Grothendieck duality, representation of quivers, reflection functor
Categories:18E30, 16G20, 18E40, 16D90, 18A40

3. CJM 2012 (vol 64 pp. 1222)

Bobiński, Grzegorz
Normality of Maximal Orbit Closures for Euclidean Quivers
Let $\Delta$ be an Euclidean quiver. We prove that the closures of the maximal orbits in the varieties of representations of $\Delta$ are normal and Cohen--Macaulay (even complete intersections). Moreover, we give a generalization of this result for the tame concealed-canonical algebras.

Keywords:normal variety, complete intersection, Euclidean quiver, concealed-canonical algebra
Categories:16G20, 14L30

4. CJM 2009 (vol 61 pp. 315)

Enochs, E.; Estrada, S.; Rozas, J. R. Garc\'{\i}a
Injective Representations of Infinite Quivers. Applications
In this article we study injective representations of infinite quivers. We classify the indecomposable injective representations of trees and describe Gorenstein injective and projective representations of barren trees.

Categories:16G20, 18A40

5. CJM 2007 (vol 59 pp. 1260)

Deng, Bangming; Du, Jie; Xiao, Jie
Generic Extensions and Canonical Bases for Cyclic Quivers
We use the monomial basis theory developed by Deng and Du to present an elementary algebraic construction of the canonical bases for both the Ringel--Hall algebra of a cyclic quiver and the positive part $\bU^+$ of the quantum affine $\frak{sl}_n$. This construction relies on analysis of quiver representations and the introduction of a new integral PBW-like basis for the Lusztig $\mathbb Z[v,v^{-1}]$-form of~$\bU^+$.

Categories:17B37, 16G20

6. CJM 2006 (vol 58 pp. 180)

Reiten, Idun; Ringel, Claus Michael
Infinite Dimensional Representations of Canonical Algebras
The aim of this paper is to extend the structure theory for infinitely generated modules over tame hereditary algebras to the more general case of modules over concealed canonical algebras. Using tilting, we may assume that we deal with canonical algebras. The investigation is centered around the generic and the Pr\"{u}fer modules, and how other modules are determined by these modules.

Categories:16D70, 16D90, 16G20, 16G60, 16G70

7. CJM 1999 (vol 51 pp. 488)

Burgess, W. D.; Saorín, Manuel
Homological Aspects of Semigroup Gradings on Rings and Algebras
This article studies algebras $R$ over a simple artinian ring $A$, presented by a quiver and relations and graded by a semigroup $\Sigma$. Suitable semigroups often arise from a presentation of $R$. Throughout, the algebras need not be finite dimensional. The graded $K_0$, along with the $\Sigma$-graded Cartan endomorphisms and Cartan matrices, is examined. It is used to study homological properties. A test is found for finiteness of the global dimension of a monomial algebra in terms of the invertibility of the Hilbert $\Sigma$-series in the associated path incidence ring. The rationality of the $\Sigma$-Euler characteristic, the Hilbert $\Sigma$-series and the Poincar\'e-Betti $\Sigma$-series is studied when $\Sigma$ is torsion-free commutative and $A$ is a division ring. These results are then applied to the classical series. Finally, we find new finite dimensional algebras for which the strong no loops conjecture holds.

Categories:16W50, 16E20, 16G20

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