We develop a method for describing invariant PDEs of Monge–Ampère type in the sense of Lychagin and Morimoto (MAE) on a homogeneous contact manifold N of a semisimple Lie group G, which is the contactification of the homogeneous symplectic manifold <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9999_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="176" /> </InlineMediaObject> <EquationSource Format="TEX">$M = G/H = \textrm{Ad}_G Z \subset \mathfrak {g}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo>=</mo> <mi>G</mi> <mo stretchy="false">/</mo> <mi>H</mi> <mo>=</mo> <msub> <mtext>Ad</mtext> <mi>G</mi> </msub> <mi>Z</mi> <mo>⊂</mo> <mi mathvariant="fraktur">g</mi> </mrow> </math></EquationSource> </InlineEquation>, where M is the adjoint orbit of a splittable closed element Z of the Lie algebra <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9999_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="85" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {g}= {{\,\textrm{Lie}\,}}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">g</mi> <mo>=</mo> <mrow> <mspace width="0.166667em" /> <mtext>Lie</mtext> <mspace width="0.166667em" /> </mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The method is then applied to a ten-dimensional semisimple orbit M of the exceptional Lie group <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10455_2025_9999_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\textsf{G}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="sans-serif">G</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and a complete list of mutually non-equivalent MAEs on N is obtained.

Invariant Monge–Ampère equations on contactified para–Kähler manifolds

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2025年5月29日

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Dmitri Alekseevsky,
Gianni Manno,
Giovanni Moreno

摘要

We develop a method for describing invariant PDEs of Monge–Ampère type in the sense of Lychagin and Morimoto (MAE) on a homogeneous contact manifold N of a semisimple Lie group G, which is the contactification of the homogeneous symplectic manifold $M = G/H = \textrm{Ad}_G Z \subset \mathfrak {g}$ $M = G / H = {Ad}_{G} Z \subset g$ , where M is the adjoint orbit of a splittable closed element Z of the Lie algebra $\mathfrak {g}= {{\,\textrm{Lie}\,}}(G)$ $g = Lie (G)$ . The method is then applied to a ten-dimensional semisimple orbit M of the exceptional Lie group $\textsf{G}_2$ $G_{2}$ and a complete list of mutually non-equivalent MAEs on N is obtained.