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Register a new userHall algebras associated to triangulated categories, II: almost associativity
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Fan Xu
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By using the approach in cite{XX2006} to Hall algebras arising in homologically finite triangulated categories, we find an `almost associative multiplication structure for indecomposable objects in a 2-periodic triangulated category. As an application, we give a new proof of the theorem of Peng and Xiao in cite{PX2000} which provides a way of realizing symmetrizable Kac-Moody algebras and elliptic Lie algebras via 2-periodic triangulated categories.
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We show that a tilting module over the endomorphism algebra of a cluster-tilting object in a 2-Calabi-Yau triangulated category lifts to a cluster-tilting object in this 2-Calabi-Yau triangulated category. This generalizes a recent work of D. Smith for cluster categories.
This is an introduction to some aspects of Fomin-Zelevinskys cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.
The objective of the present paper is to give a survey of recent progress on applications of the approaches of Ringel-Hall type algebras to quantum groups and cluster algebras via various forms of Greens formula. In this paper, three forms of Greens formula are highlighted, (1) the original form of Greens formula cite{Green}cite{RingelGreen}, (2) the degeneration form of Greens formula cite{DXX} and (3) the projective form of Greens formula cite{XX2007a} i.e. Green formula with a $bbc^{*}$-action.
Building on work by Geiss-Leclerc-Schroer and by Buan-Iyama-Reiten-Scott we investigate the link between certain cluster algebras with coefficients and suitable 2-Calabi-Yau categories. These include the cluster-categories associated with acyclic quivers and certain Frobenius subcategories of module categories over preprojective algebras. Our motivation comes from the conjectures formulated by Fomin and Zelevinsky in `Cluster algebras IV: Coefficients. We provide new evidence for Conjectures 5.4, 6.10, 7.2, 7.10 and 7.12 and show by an example that the statement of Conjecture 7.17 does not always hold.
In the present paper we describe Leibniz algebras with three-dimensional Euclidean Lie algebra $mathfrak{e}(2)$ as its liezation. Moreover, it is assumed that the ideal generated by the squares of elements of an algebra (denoted by $I$) as a right $mathfrak{e}(2)$-module is associated to representations of $mathfrak{e}(2)$ in $mathfrak{sl}_2({mathbb{C}})oplus mathfrak{sl}_2({mathbb{C}}), mathfrak{sl}_3({mathbb{C}})$ and $mathfrak{sp}_4(mathbb{C})$. Furthermore, we present the classification of Leibniz algebras with general Euclidean Lie algebra ${mathfrak{e(n)}}$ as its liezation $I$ being an $(n+1)$-dimensional right ${mathfrak{e(n)}}$-module defined by transformations of matrix realization of $mathfrak{e(n)}.$ Finally, we extend the notion of a Fock module over Heisenberg Lie algebra to the case of Diamond Lie algebra $mathfrak{D}_k$ and describe the structure of Leibniz algebras with corresponding Lie algebra $mathfrak{D}_k$ and with the ideal $I$ considered as a Fock $mathfrak{D}_k$-module.
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