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.
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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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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