Non-commutative amoebas

Article Article scientifique Article Non-commutative amoebas eng 2022-03-17 2022-04 author Mikhalkin Grigory 681622 Section de mathématiques (MATH) author Shkolnikov Mikhail 752442 335-368 10.1112/blms.12622 0024-6093 The group of isometries of the hyperbolic space ℍ3 is the 3-dimensional group PSL2(ℂ), which is one of the simplest non-commutative complex Lie groups. Its quotient by the subgroup SO(3) ⊂ PSL2(ℂ) naturally maps it back to ℍ3. Each fiber of this map is diffeomorphic to the real projective 3-space ℝℙ3. The resulting map PSL2(ℂ) → ℍ3 can be viewed as the simplest non-commutative counterpart of the map Log ∶ (ℂ×)𝑛 → ℝ𝑛 from the commutative complex Lie group (ℂ×)𝑛 with the Lagrangian torus fibers that can be considered as a Liouville–Arnold type integrable system. Gelfand, Kapranov and Zelevinsky have introduced amoebas of algebraic varieties 𝑉 ⊂ (ℂ×)𝑛 as images Log(𝑉) ⊂ ℝ𝑛. We define the amoeba of an algebraic subvariety of PSL2(ℂ) as its image inℍ3. The paper surveys basic properties of the resulting hyperbolic amoebas and compares them against the commutative amoebas ℝ𝑛. Dewey 510 Swiss National Science Foundation Topics in real algebraic geometry 182111 CH Projects Swiss National Science Foundation Algèbre, Analyse, Géométrie et Physique 178828 CH Projects 25 Section de mathématiques 11 false Bulletin of the London Mathematical Society 54 2 0f086457-f264-4726-898b-8a98a82107c4 Article (Published version) Public Mikhalkin_BLMS_202254335.pdf 348510 application/pdf CC BY-NC-ND-4.0 4a842337588597ab9f3cf870033a679bc1e05804a33e52be8577b0239df54332 openAccess Faculté des sciences / Section de mathématiques

Title	Published in	Year	Views	Downloads

Non-commutative amoebas	Bulletin of the London Mathematical Society	2022	64	124
Tropical curves, convex domains, sandpiles and amoebas		2017	984	1,134

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Shkolnikov, Mikhail