Noncommutative Noether's problem is almost equivalent to the classical Noether's problem

2021/04/18 by Tikaradze, Akaki · 1 citation
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2104.08674

Abstract

Motivated by the classical Noether's problem, J. Alev and F. Dumas proposed the following question, commonly referred to as the noncommutative Noether's problem: Let a finite group G act linearly on ℂⁿ, inducing the action on Frac(A_n(ℂ))-the skew field of fractions of the n-th Weyl algebra A_n(ℂ), then is Frac(A_n(ℂ))^G isomorphic to Frac(A_n(ℂ))? In this note we show that if Frac(A_n(ℂ))^G≅ Frac(A_n(ℂ)), then for any algebraically closed field k of large enough characteristic, field k(x₁,⋯, x_n)^G is stably rational. This result allows us to produce counterexamples to the noncommutative Noether's problem based on well-known counterexamples to the Noether's problem for algebraically closed fields.

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