Indecomposable representations of graphs and algebras

Dlab, Vlastimil; Ringel, Claus Michael

doi:10.1090/memo/0173

Cited by 466 publications

(493 citation statements)

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“…It is well-known that !/ is representation-directed, see [BGP], [DR2]; this means that there exists a total ordering of the positive roots, say 8 (Mg'(ai),Mg' (a j ) 4= 0 (or equivalently, e(a i , a j ) ::f 0) implies that i~j: such an ordering will be called A-admissible. The usual visualization using the Auslander-Reiten quiver will be recalled in Section 6.…”

Section: Ordering the Positive Rootsmentioning

confidence: 99%

PBW-bases of quantum groups.

Ringel¹

1996

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Section: Ordering the Positive Rootsmentioning

confidence: 99%

PBW-bases of quantum groups.

Ringel¹

1996

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Accordingly, the graphs underlying the Gabriel quivers of the principal blocks are exactly those occurring in the classification of the tame hereditary algebras (cf. [14]). In fact, the precise relationship with these hereditary algebras is provided via the notion of a trivial extension of an algebra by its dual bimodule: The tame blocks are certain generalizations of the trivial extensions of the tame hereditary radical square zero algebras.…”

Section: Introductionmentioning

confidence: 99%

Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks

Farnsteiner

2006

Invent. math.

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Abstract. Given an algebraically closed field k of characteristic p ≥ 3, we classify the finite algebraic k-groups whose algebras of measures afford a principal block of tame representation type. The structure of such a group G is largely determined by a linearly reductive subgroup schemeĜ of SL(2), with the McKay quiver ofĜ relative to its standard module being the Gabriel quiver of the principal block B0(G). The graphs underlying these quivers are extended Dynkin diagrams of typẽ A,D orẼ, and the tame blocks are Morita equivalent to generalizations of the trivial extensions of the radical square zero tame hereditary algebras of the corresponding type.

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“…Meltzer and A. Skowroflski (1) R is a division algebra and a p-Sylow subgroup of G is cyclic or (2) p=3, a 3-SyIow subgroup of G is simple, and R is isomorphic to an algebra Re(O , 1) or (3) p=2, a 2-Sylow subgroup of G is simple, and R is isomorphic to an algebra RF (m ,n), to an algebra Rv(i), i=1, 2, 3, 4, Let R be a non-simple algebra over a fixed field K of characteristic p > 0 and let G be a cyclic group of order n=pk. From [8], for our aim, we can assume that R is isomorphic to a bounden quiver algebra FQ/I, where F is a division algebra and (2 is of the form Q: 1 ~1 2--...m-1 ~,,-1 m and I(i,j)=O if i and j are neighbours in Q.…”

Section: Theorem Let G Be a Finite Group And Let R Be A Connected Fimentioning

confidence: 98%

Correction to Group algebras of finite representation type

Meltzer

Skowroński

1984

Math Z

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Recently we have observed that Lemma 4 in [-8] is not true and hence the characterization of group algebras of finite representation type given in [-8] is not complete.Here, we will give the correct version of [-8, Theorem] and we will complete the proof given there.We use the same notation as in E8]. In order to state the main result, we need some additional. Let F be a division algebra over a fixed field K. We will denote by Rr(1 ) the quiver algebra FQ1 (in the sense of [3]) of the quiver QI" 1~2~-3 and by RF(2 ) the opposite algebra to Rv(1 ). Moreover, we will denote by RF (3 ) the bounden quiver algebra FQz/I 2 where Q2 is the quiver ~3 ~2~ =2 3+---4and 12 is the ideal in the quiver algebra FQ2 generated by c%c%, and by Re(4 ) the opposite algebra to RF(3 ). Finally, recall that for nonnegative integers m and n, Rv(m,n ) is the bounden quiver algebra FQ,,,./Im, . where Q,,,, is the quiver quiver 9 m ;, 9 I-+ )e and Ira, . is the ideal in the quiver algebra FQm,, generated by the composed arrows fiifii+ 1, i= 1, ..., m -1, and c~j+ 1 c~j, j = 1 ..... n -1.

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Indecomposable representations of graphs and algebras

Cited by 466 publications

References 0 publications

PBW-bases of quantum groups.

PBW-bases of quantum groups.

Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks

Correction to Group algebras of finite representation type

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