Opérateurs différentiels invariants et problème de Noether

Alev, Jacques; Dumas, François

doi:10.1007/0-8176-4478-4_3

Cited by 6 publications

(14 citation statements)

References 15 publications

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“…A proof of all these results, except the last one (dealt with in the next section), can be found in [4], Theorem 2.1, Corollary 3.9. Also one can find there an explicit isomorphism D 2 (C) G ∼ = D 2 (C) when G is a finite subgroup of SL 2 (C).…”

Section: ] G Which Are Invariants Of Two Polynomial Algebras Sittinmentioning

confidence: 94%

Some aspects of noncommutative invariant theory and the Noether’s problem

Schwarz

2015

São Paulo J. Math. Sci.

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This survey discusses the classical Noether's problem and presents some important cases where it has a positive solution. With the Weyl algebras taking place of the polynomial algebras, some aspects of noncommutative invariant theory are introduced: in particular, the noncommutative version of the Noether's problem. Some known cases are discussed and some new results are presented. As an application one obtains a new proof of the Gelfand-Kirillov conjecture for gl n . There is a striking similarity between the cases with positive solution of the Noether's problem for both commutative and noncommutative versions, and this leads to many conjectures.

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Section: ] G Which Are Invariants Of Two Polynomial Algebras Sittinmentioning

confidence: 94%

Some aspects of noncommutative invariant theory and the Noether’s problem

Schwarz

2015

São Paulo J. Math. Sci.

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“…Nosso interesse será para grupos G finitos, apesar de resultados interessantes existirem para G arbitrários ( [AD06]). Neste caso, existe uma precisão importante para a resposta possível: …”

Section: Problema De Noether Não-comutativounclassified

“…Os dois últimos Teoremas e Corolário foram obtidos por J. Alev e F. Dumas, [AD06]. Neste trabalho, eles também obtiveram o resultado seguinte:…”

Section: Problema De Noether Não-comutativounclassified

“…O primeiro resultado presente na literatura semelhante ao Problema de Noether não-comutativo, do tipo D n (k) G ∼ = D n (k), se deve a J. Alev e F. Dumas [AD98], no caso k algebricamente fechado e G abeliano, com uma demonstração independente da apresentada neste capítulo. O material da seção 2 é originário, principalmente, de [AD06]. A prova do Teorema 4.7 na Seção 3 seguiu de perto a encontrada nesta referência, com algumas simplificações, pois G é finito, e no artigo original eles trabalhavam sem essa hipótese.…”

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“…Após isto, serão discutidos os principais casos onde há uma solução positiva, e serão indicados também alguns contra-exemplos. Utilizando um teorema de Miyata Teorema 1.4 ([Miy71]), serão provados alguns casos de solução positiva -casos que admitem versões análogas para a versão não-comutativa do Problema, e que admitem uma demonstração semelhante utilizando uma extensão do Teorema de Myiata para extensões de Ore ( [AD06], Teorema 1.4) -que será discutida no Capítulo 4 dessa dissertação (Teorema 4.4).…”

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Problema de Noether não-comutativo

Schwarz¹

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Noncommutative Noether’s problem vs classic Noether’s problem

Futorny

Schwarz

2019

Math. Z.

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We address the Noncommutative Noether's Problem on the invariants of Weyl fields for linear actions of finite groups. We prove that if the variety A n (k)/G is rational then the Noncommutative Noether's Problem is positively solved for G and any field k of characteristic zero. In particular, this gives positive solution for all pseudo-reflections groups, for the alternating groups (n = 3, 4, 5) and for any finite group when n = 3 and k is algebraically closed. Alternative proofs are given for the complex field and for all pseudoreflections groups. In the later case an effective algorithm of finding the Weyl generators is described. We also study birational equivalence for the rings of invariant differential operators on complex affine irreducible varieties.

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Opérateurs différentiels invariants et problème de Noether

Cited by 6 publications

References 15 publications

Some aspects of noncommutative invariant theory and the Noether’s problem

Some aspects of noncommutative invariant theory and the Noether’s problem

Problema de Noether não-comutativo

Noncommutative Noether’s problem vs classic Noether’s problem

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