Perfect elements in the modular structure associated with the extended Dynkin diagram E6

Stekolshchik, Rafael

doi:10.1007/bf01079544

Cited by 2 publications

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“…Part of the results presented in the present paper was announced in [49] and published in papers [51,52], to which the reader is also referred for some proofs that are not presented here.…”

Section: An Outline Of the Main Results Of The Papermentioning

confidence: 89%

Gelfand–Ponomarev and Herrmann constructions for quadruples and sextuples

Stekolshchik¹

2007

Journal of Pure and Applied Algebra

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The notions of a perfect element and an admissible element of the free modular lattice D r generated by r ≥ 1 elements are introduced by Gelfand and Ponomarev in [I.M. Gelfand, V.A. Ponomarev, Free modular lattices and their representations, Collection of articles dedicated to the memory of Ivan Georgievic Petrovskii , IV. Uspehi Mat. Nauk 29 (6(180)) (1974) 3-58 (Russian); English translation: Russian Math. Surv. 29 (6) (1974) 1-56]. We recall that an element a ∈ D of a modular lattice L is perfect, if for each finite-dimension indecomposable K -linear representation ρ X : L → L(X ) over any field K , the image ρ X (a) ⊆ X of a is either zero, or ρ X (a) = X , where L(X ) is the lattice of all vector K -subspaces of X . A complete classification of such elements in the lattice D 4 , associated to the extended Dynkin diagram D 4 (and also in D r , where r > 4) is given in [I.M. Gelfand, V.A. Ponomarev, Free modular lattices and their representations, Collection of articles dedicated to the memory of Ivan Georgievic Petrovskii (1901-1973), IV. Uspehi Mat. Nauk 29 (6(180)) (1974) 3-58 (Russian); English translation: Russian Math. Surv. 29 (6) (1974) 1-56; I.M. Gelfand, V.A. Ponomarev, Lattices, representations, and their related algebras, I, Uspehi Mat. Nauk 31 (5(191)) (1976) 71-88 (Russian); English translation: Russian Math. Surv. 31 (5) (1976) 67-85; I.M. Gelfand, V.A. Ponomarev, Lattices, representations, and their related algebras, II. Uspehi Mat. Nauk 32 (1(193)) (1977) 85-106 (Russian); English translation: Russian Math. Surv. 32 (1) (1977) 91-114]. The main aim of the present paper is to classify all the admissible elements and all the perfect elements in the Dedekind lattice D 2,2,2 generated by six elements that are associated to the extended Dynkin diagram E 6 . We recall that in [I.M. Gelfand, V.A. Ponomarev, Free modular lattices and their representations, Collection of articles dedicated to the memory of Ivan Georgievic Petrovskii , IV. Uspehi Mat. Nauk 29 (6(180)) (1974) 3-58 (Russian); English translation: Russian Math. Surv. 29 (6) (1974) 1-56], Gelfand and Ponomarev construct admissible elements of the lattice D r recurrently. We suggest a direct method for creating admissible elements. Using this method we also construct admissible elements for D 4 and show that these elements coincide modulo linear equivalence with admissible elements constructed by Gelfand and Ponomarev. Admissible sequences and admissible elements for D 2,2,2 (resp. D 4 ) form 14 classes (resp. 8 classes) and possess some periodicity.Our classification of perfect elements for D 2,2,2 is based on the description of admissible elements. The constructed set H + of perfect elements is the union of 64-element distributive lattices H + (n), and H + is the distributive lattice itself. The lattice of perfect elements B + obtained by Gelfand and Ponomarev for D 4 can be imbedded into the lattice of perfect elements H + , associated with D 2,2,2 . 96 R. Stekolshchik / Journal of Pure and Applied Algebra 211 (2007) 95-202 Herrmann in...

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“…Part of the results presented in the present paper was announced in [49] and published in papers [51,52], to which the reader is also referred for some proofs that are not presented here.…”

Section: An Outline Of the Main Results Of The Papermentioning

confidence: 89%

Gelfand–Ponomarev and Herrmann constructions for quadruples and sextuples

Stekolshchik¹

2007

Journal of Pure and Applied Algebra

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“…For the union 2 + 2 + 2 of three 2-element chains, Stekolshchik [31,32] gave an explicit construction of perfect elements based on atomic elements 2 C. Herrmann Algebra univers.…”

Section: Introductionmentioning

confidence: 99%

On perfect pairs for quadruples in complemented modular lattices and concepts of perfect elements

Herrmann¹

2009

Algebra Univers.

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Gel'fand and Ponomarev [11] introduced the concept of perfect elements and constructed such in the free modular lattice on 4 generators. We present an alternative construction of such elements u (linearly equivalent to theirs) and for each u a direct decomposition u, u of the generating quadruple within the free complemented modular lattice on 4 generators; u, u are said to form a perfect pair. This builds on [17] and fills a gap left there. We also discuss various notions of perfect elements and relate them to preprojective and preinjective representations.

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Perfect elements in the modular structure associated with the extended Dynkin diagram E6

Cited by 2 publications

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Gelfand–Ponomarev and Herrmann constructions for quadruples and sextuples

Gelfand–Ponomarev and Herrmann constructions for quadruples and sextuples

On perfect pairs for quadruples in complemented modular lattices and concepts of perfect elements

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