Gelfand–Ponomarev and Herrmann constructions for quadruples and sextuples

Stekolshchik, Rafael

doi:10.1016/j.jpaa.2007.01.005

Cited by 2 publications

(5 citation statements)

References 36 publications

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“…Under the additional relation a + b = 1, the structure of these subdirect products has been analysed in [19] to such extent that neutrality could be proved requiring only the classification of frame generated lattices according to Proposition 9.7. The general structure of F L p (4) is still to be determined, based on the atomic elements of Stekolshchik [32].…”

Section: Results On Quadruplesmentioning

confidence: 99%

“…The dual elements s * n+1 ≥ t * n ≥ p * ni ≥ s * n are associated with preinjectives and s n ≥ s * n for all n. In defect 0 one has s n = 1 and s * n = 0 for all n. Thus, these elements are perfect and linearly equivalent to those established by Gel'fand and Ponomarev [11]. A detailed analysis of the relationship between both sets of elements has been given by Stekolshchik [32].…”

Section: Introductionmentioning

confidence: 94%

“…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%

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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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Section: Results On Quadruplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 94%

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

Herrmann¹

2009

Algebra Univers.

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“…• The question of defining invariants for isometry types of isotropic triples (in particular in relation to the perfect elements established by Stekolchshik [31]).…”

Section: Resumémentioning

confidence: 99%

“…Representations of this particular quiver have also been studied in quite some detail by Stekolchshik, see e.g. [31] and [32]. The study of poset representations in spaces equipped with an (anti-)symmetric inner product was first developed, to our knowledge, by Scharlau and collaborators; see [28] for a concise and enlightening overview.…”

Section: Introductionmentioning

confidence: 99%

(Co)isotropic triples and poset representations

Herrmann¹,

Lorand²,

Weinstein³

2019

Preprint

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We study triples of coisotropic or isotropic subspaces in symplectic vector spaces; in particular, we classify indecomposable structures of this kind. The classification depends on the ground field, which we only assume to be perfect and not of characteristic 2. Our work uses the theory of representations of partially ordered sets with (order reversing) involution; for (co)isotropic triples, the relevant poset is "2+2+2" consisting of three independent ordered pairs, with the involution exchanging the members of each pair.A key feature of the classification is that any indecomposable (co)isotropic triple is either "split" or "non-split". The latter is the case when the poset representation underlying an indecomposable (co)isotropic triple is itself indecomposable. Otherwise, in the "split" case, the underlying representation is decomposable and necessarily the direct sum of a dual pair of indecomposable poset representations; the (co)isotropic triple is a "symplectification".If one ignores the involution, representations of the poset "2 + 2 + 2" are essentially the same as representations of a certain quiver of extended Dynkin type Ẽ6 , and our work relies on the known classification of indecomposable representations of such quivers. In particular from these results it follows that the poset representations underlying (co)isotropic triples come in two types: there are families of indecomposables which depend on a parameter taking a continuum of values ("continuous-type"), and there are those indecomposables which are characterized only by the dimensions of the spaces involved ("discrete-type"). This pattern is reflected in the classification of (co)isotropic triples; in particular, there are families of triples which depend on a parameter. Also, indecomposable triples exist in every even dimension.In the course of the paper we develop the framework of "symplectic poset representations", which can be applied to a range of problems of symplectic linear algebra. The classification of linear hamiltonian vector fields, up to conjugation, is an example; we briefly explain the connection between these and (co)isotropic triples. The framework lends itself equally well to studying poset representations on spaces carrying a non-degenerate symmetric bilinear form; we mainly keep our focus, however, on the symplectic side.

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Gelfand–Ponomarev and Herrmann constructions for quadruples and sextuples

Cited by 2 publications

References 36 publications

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

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

(Co)isotropic triples and poset representations

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