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Separating invariants for the basic 𝔾ₐ-actions

2010/11/09 by Jonathan Elmer, Martin Kohls · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Geometric and Algebraic Topology #math.AC #msc:13A50 #msc:13N15

paper · pdf · doi:10.1090/s0002-9939-2011-11273-5

published as Proc. Amer. Math. Soc. 140 (2012), 135-146 · 10 pages

arxiv created 2010/11/09 · openalex publication_date 2011/07/13 · arxiv updated 2013/01/23 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We explicitly construct a finite set of separating invariants for the basic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G Subscript a"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb Ga</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -actions. These are the finite dimensional indecomposable rational linear representations of the additive group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G Subscript a"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb Ga</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a field of characteristic zero, and their invariants are the kernel of the Weitzenböck derivation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript n Baseline equals x 0 StartFraction partial-differential Over partial-differential x 1 EndFraction plus ellipsis plus x Subscript n minus 1 Baseline StartFraction partial-differential Over partial-differential x Subscript n Baseline EndFraction"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> <mml:mfrac> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mrow> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:mrow> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mo> … </mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mfrac> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mrow> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">Dn=x0\frac ∂ ∂ x1+… + xn-1\frac ∂ ∂ xn</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

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