The Poisson Linearization Problem for 𝔰𝔩₂(ℂ)

In this paper, we prove a version of Conn’s linearization theorem for the Lie algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German s German l Subscript 2 Baseline left-parenthesis double-struck upper C right-parenthesis asymptotically-equals German s German o left-parenthesis 3 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≃ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">o</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {sl}_2(\mathbb {C})\simeq \mathfrak {so}(3,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Namely, we show that any Poisson structure whose linear approximation at a zero is isomorphic to the Poisson structure associated to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German s German l Subscript 2 Baseline left-parenthesis double-struck upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {sl}_2(\mathbb {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is linearizable. In the first part, we calculate the Poisson cohomology associated to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German s German l Subscript 2 Baseline left-parenthesis double-struck upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {sl}_2(\mathbb {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and we construct bounded homotopy operators for the Poisson complex of multivector fields that are flat at the origin. In the second part, we obtain the linearization result, which works for a more general class of Lie algebras. For the proof, we develop a Nash-Moser method for functions that are flat at a point.