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Some Remarks on Weierstrass Sections, Adapted Pairs and Polynomiality

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Book cover Lie Theory and Its Applications in Physics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 36))

Abstract

Let A be a polynomial subalgebra of the algebra of regular functions on affine n-space k n. A Weierstrass section for A is a translate x + Y of a linear subspace of k n such that the restriction of A to x + Y induces an isomorphism of A onto the algebra R[x + Y ] of regular functions on x + Y. They arise notably in describing algebras of invariants both for reductive and non-reductive actions as well as in describing maximal Poisson commutative subalgebras of R[k n] in the case that the latter has a Poisson bracket structure. A Weierstrass section need not always exist and in any case can be very difficult to construct. A review of some known results and open problems is presented in an entirely elementary fashion.

Work supported in part by Israel Science Foundation Grant, no. 710724.

AMS Classification: 17B35

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Notes

  1. 1.

    For nineteenth century mathematicians constucting invariants was a popular exercise. The modern mathematician would no doubt prefer to use the Weyl character formula which can be adjusted to compute the character of a given symmetric power of \(V\) and then to show it has a non-zero scalar product with the trivial character. Despite good intentions the author was too lazy to illustrate the recovery of the required invariants by these means.

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Acknowledgements

I would like to thank Vladimir Dobrev for the kind invitation to speak at the 9-th International Workshop “Lie Theory and Its Applications in Physics” held during 20–26 June 2011 in Varna, Bulgaria. I would also like to thank my former student Anna Melnikov who served as an advisor in preparing this manuscript.

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Authors and Affiliations

  1. Donald Frey Professional Chair, Department of Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel

    Anthony Joseph

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Correspondence to Anthony Joseph .

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  1. Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee, Sofia, 1784, Sofiya, Bulgaria

    Vladimir Dobrev

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Joseph, A. (2013). Some Remarks on Weierstrass Sections, Adapted Pairs and Polynomiality. In: Dobrev, V. (eds) Lie Theory and Its Applications in Physics. Springer Proceedings in Mathematics & Statistics, vol 36. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54270-4_4

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