Relations Between Arithmetical Binary Cubic Forms and Their Hessians

Mathews, G. B.

doi:10.1112/plms/s2-9.1.200

Cited by 6 publications

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“…These maps generalise the three classical covariants of the space of binary cubics (see [Eis44]) which is a special ǫ-orthogonal representation of the Lie algebra sl(2, k). We will prove a set of identities satisfied by them which generalise the Mathews identities for binary cubics (see [Mat11]) and their analogues for special symplectic representations of Lie algebras (see [SS15]).…”

Section: Mathews Identities For the Covariants Of A Special ǫ-Orthogo...mentioning

confidence: 99%

“…Finally in Section 5 we study geometric properties of special ǫ-orthogonal representations. It is well-known that the space of binary cubics, a special symplectic representation of sl(2, k), admits three covariants and that these covariants satisfy remarkable identities ([Eis44], [Mat11]). More generally, special symplectic representations of Lie algebras admit three covariants which are polynomial functions on the representation space and these covariants satisfy generalised Mathews identities ( [SS15]).…”

Section: Introductionmentioning

confidence: 99%

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The Kostant invariant and special $ε$-orthogonal representations for $ε$-quadratic colour Lie algebras

Meyer

2019

Preprint

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Let k be a field of characteristic not two or three, let g be a finite-dimensional colour Lie algebra and let V be a finite-dimensional representation of g. In this article we give various ways of constructing a colour Lie algebra g whose bracket in some sense extends both the bracket of g and the action of g on V . Colour Lie algebras, originally introduced by R. Ree ([Ree60]), generalise both Lie algebras and Lie superalgebras, and in those cases our results imply many known results ([Kos99], [Kos01], [CK15], [SS15]). For a class of representations arising in this context we show there are covariants satisfying identities analogous to Mathews identities for binary cubics.

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Section: Mathews Identities For the Covariants Of A Special ǫ-Orthogo...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Kostant invariant and special $ε$-orthogonal representations for $ε$-quadratic colour Lie algebras

Meyer

2019

Preprint

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“…Note that the great number of geometric publications in comparison with the others echoes the usual presentation of Grace as a geometer, [Todd 1958, p. 94], [Barrow-Green and Gray 2006, p. 328 sqq.]. also used in this paper: strikingly, they are the topics of several papers that Mathews had published at the turn of the century, such as [Mathews 1891[Mathews , 1893a[Mathews ,b, 1911. We have no direct evidence that Mordell knew these papers in the early 1910s, but their existence allows us to see that the various objects that would be used by Mordell for the equation y 2 − k = x 3 were part and parcel of Mathews' works.16 The preceding paragraphs show that even if Mordell repeatedly presented himself as a self-educated mathematician, he was not a mathematician with no resources, in particular in number theory.…”

Section: The Theory Of Numbers In Great Britain: Mordell's Viewmentioning

confidence: 61%

“…The case of George Ballard Mathews is different from that of Grace, for he was not in Cambridge but in Bangor (Wales) when Mordell was a student. The book that Mordell refers to is certainly Theory of Numbers, published in 1892, [Mathews 1892]. Since Mordell cited this book in one of his first papers [Mordell 1915], it is likely that he had become acquainted with it when he was a student in Cambridge.…”

Section: The Theory Of Numbers In Great Britain: Mordell's Viewmentioning

confidence: 99%

On the youthful writings of Louis J. Mordell on the Diophantine equation $$y^2-k=x^3$$

Gauthier

Lê

2019

Arch. Hist. Exact Sci.

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This article examines the research of Louis J. Mordell on the Diophantine equation y 2 − k = x 3 as it appeared in one of his first papers, published in 1914. After presenting a number of elements relating to Mordell's mathematical youth and his (problematic) writing, we analyze the 1914 paper by following the three approaches he developed therein, respectively based on the quadratic reciprocity law, on ideal numbers, and on binary cubic forms. This analysis allows us to describe many of the difficulties in reading and understanding Mordell's proofs, difficulties which we make explicit and comment on in depth.

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“…Later G.B. Mathews [17] proved some identities for integral binary cubics that essentially determined the projective fibre of the determinant of the Hessian (cf. Example 2.10).…”

Section: Symplectic Relations Of An Ssrmentioning

confidence: 99%

The geometry of special symplectic representations

Slupinski

Stanton

2015

Journal of Algebra

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We show there is a class of symplectic Lie algebra representations over any field of characteristic not 2 or 3 that have many of the exceptional algebraic and geometric properties of both symmetric three forms in two dimensions and alternating three forms in six dimensions. All nonzero orbits are coisotropic and the covariants satisfy relations generalising classical identities of Eisenstein and Mathews. The main algebraic result is that suitably generic elements of these representation spaces can be uniquely written as the sum of two elements of a naturally defined Lagrangian subvariety. We give universal explicit formulae for the summands and show how they lead to the existence of geometric structure on appropriate subsets of the representation space. Over the real numbers this structure reduces to either a conic, special pseudo-Kähler metric or a conic, special para-Kähler metric.

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Relations Between Arithmetical Binary Cubic Forms and Their Hessians

Cited by 6 publications

References 0 publications

The Kostant invariant and special $ε$-orthogonal representations for $ε$-quadratic colour Lie algebras

The Kostant invariant and special $ε$-orthogonal representations for $ε$-quadratic colour Lie algebras

On the youthful writings of Louis J. Mordell on the Diophantine equation $$y^2-k=x^3$$

The geometry of special symplectic representations

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