<p>This work presents five explicit constructions of the exceptional Lie algebra <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1768_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {e}_8$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="fraktur">e</mi> <mn>8</mn> </msub> </math></EquationSource> </InlineEquation>, each associated with a semisimple subalgebra of maximal rank. The provided models are based on gradings by finite abelian groups, namely, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1768_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1768_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}_5$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>5</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1768_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}_6$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mn>6</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1768_Article_IEq8.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}_3^2$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">Z</mi> <mn>3</mn> <mn>2</mn> </msubsup> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1768_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}_2\times \mathbb {Z}_4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">Z</mi> <mn>2</mn> </msub> <mo>×</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mn>4</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. In all cases, the neutral component is a direct sum of special linear algebras, while the remaining homogeneous components are irreducible modules over it.</p>

Linear models of the exceptional Lie algebra $\mathfrak {e}_8$

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2025年8月6日

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Yolanda Cabrera,
Cristina Draper,
Antonio Garvín

摘要

This work presents five explicit constructions of the exceptional Lie algebra $\mathfrak {e}_8$ $e_{8}$ , each associated with a semisimple subalgebra of maximal rank. The provided models are based on gradings by finite abelian groups, namely, $\mathbb {Z}_4$ $Z_{4}$ , $\mathbb {Z}_5$ $Z_{5}$ , $\mathbb {Z}_6$ $Z_{6}$ , $\mathbb {Z}_3^2$ $Z_{3}^{2}$ , and $\mathbb {Z}_2\times \mathbb {Z}_4$ $Z_{2} \times Z_{4}$ . In all cases, the neutral component is a direct sum of special linear algebras, while the remaining homogeneous components are irreducible modules over it.

Linear models of the exceptional Lie algebra \(\mathfrak {e}_8\)

摘要