共查询到20条相似文献,搜索用时 15 毫秒
1.
Bin Zhu 《Journal of Algebraic Combinatorics》2008,27(1):35-54
We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. Using
d-cluster categories defined by Keller as triangulated orbit categories of (bounded) derived categories of representations
of valued quivers, we define a d-compatibility degree (−∥−) on any pair of “colored” almost positive real Schur roots which generalizes previous definitions on the noncolored case
and call two such roots compatible, provided that their d-compatibility degree is zero. Associated to the root system Φ corresponding to the valued quiver, using this compatibility relation, we define a simplicial complex which has colored almost
positive real Schur roots as vertices and d-compatible subsets as simplices. If the valued quiver is an alternating quiver of a Dynkin diagram, then this complex is
the generalized cluster complex defined by Fomin and Reading.
Supported by the NSF of China (Grants 10471071) and by the Leverhulme Trust through the network ‘Algebras, Representations
and Applications’. 相似文献
2.
A notion of a mixed representation of a quiver can be derived from ordinary quiver representation by considering the dual
action of groups on "vertex" vector spaces together with their usual action. A generating system for the algebra of semi-invariants
of mixed representations of a quiver is determined. This is done by reducing the problem to the case of bipartite quivers
of a special form and using a function DP on three matrices, which is a mixture of the determinant and two pfaffians. 相似文献
3.
For a given quiver and dimension vector, Kac has shown that there is exactly one indecomposable representation up to isomorphism if and only if this dimension vector is a positive real root. However, it is not clear how to compute these indecomposable representations in an explicit and minimal way, and the properties of these representations are mostly unknown. In this note we study representations of a particular wild quiver. We define operations which act on representations of this quiver, and using these operations we construct indecomposable representations for positive real roots, compute their endomorphism rings and show that these representations are tree representations. The operations correspond to the fundamental reflections in the Weyl group of the quiver. Our results are independent of the characteristic of the field. 相似文献
4.
5.
As is known, finitely presented quivers correspond to Dynkin graphs (Gabriel, 1972) and tame quivers correspond to extended
Dynkin graphs (Donovan and Freislich, Nazarova, 1973). In the article “Locally scalar representations of graphs in the category
of Hilberts spaces” (Func. Anal. Apps., 2005), the authors showed a way for carrying over these results to Hilbert spaces,
constructed Coxeter functors, and proved an analog of the Gabriel theorem for locally scalar representations (up to unitary
equivalence).
The category of locally scalar representations of a quiver can be regarded as a subcategory in the category of all representations
(over the field ℂ). In the present paper, we study the relationship between the indecomposability of locally scalar representations
in the subcategory and in the category of all representations (it is proved that for a class of quivers wide enough indecomposability
in the subcategory implies indecomposability in the category).
For a quiver corresponding to the extended Dynkin graph
, locally scalar representations that cannot be obtained from the simplest ones by Coxeter functors (regular representations)
are classified. Bibliography: 21 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 180–201. 相似文献
6.
7.
Marcel Wiedemann 《Journal of Algebra》2009,321(6):1711-1718
In this paper we consider the following question: Is it possible to construct all real root representations of a given quiver Q using universal extension functors, starting with a real Schur representation? We give a concrete example answering this question negatively. 相似文献
8.
S. A. Kruglyak L. A. Nazarova A. V. Roiter 《Functional Analysis and Its Applications》2010,44(2):125-138
It is known that finitely representable quivers correspond to Dynkin graphs and tame quivers correspond to extended Dynkin
graphs. In an earlier paper, the authors generalized some of these results to locally scalar (later renamed to orthoscalar)
quiver representations in Hilbert spaces; in particular, an analog of the Gabriel theorem was proved. In this paper, we study
the relationships between indecomposable representations in the category of orthoscalar representations and indecomposable
representations in the category of all quiver representations. For the quivers corresponding to extended Dynkin graphs, the
indecomposable orthoscalar representations are classified up to unitary equivalence. 相似文献
9.
On the Representation Ring of a Quiver 总被引:1,自引:0,他引:1
Martin Herschend 《Algebras and Representation Theory》2009,12(6):513-541
10.
We present a particular connection between classical partition combinatorics and the theory of quiver representations. Specifically, we give a bijective proof of an analogue of A. L. Cauchy’s Durfee square identity to multipartitions. We then use this result to give a new proof of M. Reineke’s identity in the case of quivers \({\mathcal {Q}}\) of Dynkin type A. Our identity is stated in terms of the lacing diagrams of S. Abeasis–A. Del Fra, which parameterize orbits of the representation space of \({\mathcal {Q}}\) for a fixed dimension vector. 相似文献
11.
12.
Xintian Wang 《代数通讯》2017,45(11):4809-4816
Stability conditions play an important role in the study of representations of a quiver. In the present paper, we study semistable representations of quivers. In particular, we describe the slopes of semistable representations of a tame quiver for a fixed stability condition. 相似文献
13.
Martin Herschend 《Journal of Pure and Applied Algebra》2008,212(2):452-469
The aim of this article is to translate the well-known tensor product of representations of a group given by diagonal action to the case of representations of a quiver. We provide three different approaches and exhibit their close relationship to the point-wise tensor product, which is considered in [M. Herschend, Solution to the Clebsch-Gordan problem for Kronecker representations. U.U.D.M Project Report 2003:P1, Uppsala University, 2003; M. Herschend, Solution to the Clebsch-Gordan problem for representations of quivers of type , J. Algebra Appl. 4 (5) (2005) 481-488; M. Herschend, On the Clebsch-Gordan problem for quiver representations. U.U.D.M Report 2005:43, Uppsala University, 2005]. 相似文献
14.
Magnus Engenhorst 《Algebras and Representation Theory》2017,20(1):163-174
Maximal green sequences were introduced as combinatorical counterpart for Donaldson-Thomas invariants for 2-acyclic quivers with potential by B. Keller. We take the categorical notion and introduce maximal green sequences for hearts of bounded t-structures of triangulated categories that can be tilted indefinitely. We study the case where the heart is the category of modules over the preprojective algebra of a quiver without loops. The combinatorical counterpart of maximal green sequences for Dynkin quivers are maximal chains in the Hasse quiver of basic support τ-tilting modules. We show that a quiver has a maximal green sequence if and only if it is of Dynkin type. More generally, we study module categories for finite-dimensional algebras with finitely many bricks. 相似文献
15.
《Journal of Pure and Applied Algebra》2019,223(12):5251-5278
The purpose of this paper is to study stable representations of partially ordered sets (posets) and compare it to the well known theory for quivers. In particular, we prove that every indecomposable representation of a poset of finite type is stable with respect to some weight and construct that weight explicitly in terms of the dimension vector. We show that if a poset is primitive then Coxeter transformations preserve stable representations. When the base field is the field of complex numbers we establish the connection between the polystable representations and the unitary χ-representations of posets. This connection explains the similarity of the results obtained in the series of papers. 相似文献
16.
17.
William Chin 《Journal of Algebra》2012,353(1):1-21
We develop the theory of special biserial and string coalgebras and other concepts from the representation theory of quivers. These tools are then used to describe the finite-dimensional comodules and Auslander–Reiten quiver for the coordinate Hopf algebra of quantum at a root of unity. We also compute quantum dimensions and the stable Green ring. 相似文献
18.
Kay Großblotekamp 《代数通讯》2017,45(12):5103-5110
A set valued representation of the Kronecker quiver is nothing but a quiver. We apply the forgetful functor from vector spaces to sets and compare linear with set valued representations of the Kronecker quiver. 相似文献
19.
We study quivers with relations given by noncommutative analogs of Jacobian ideals in the complete path algebra. This framework
allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching
generalization of Bernstein–Gelfand–Ponomarev reflection functors. The motivations for this work come from several sources:
superpotentials in physics, Calabi–Yau algebras, cluster algebras.
相似文献
20.
Silvia Montarani 《Selecta Mathematica, New Series》2010,16(3):631-671
Given a quiver, a fixed dimension vector, and a positive integer n, we construct a functor from the category of D-modules on the space of representations of the quiver to the category of modules over a corresponding Gan–Ginzburg algebra
of rank n. When the quiver is affine Dynkin, we obtain an explicit construction of representations of the corresponding wreath product
symplectic reflection algebra of rank n. When the quiver is star-shaped, but not finite Dynkin, we use this functor to obtain a Lie-theoretic construction of representations
of a “spherical” subalgebra of the Gan–Ginzburg algebra isomorphic to a rational generalized double affine Hecke algebra of
rank n. Our functors are a generalization of the type A and type BC functors from [1] and [4], respectively. 相似文献