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Minuscule embeddings. (English) Zbl 07394423

Summary: We study embeddings \(J \to G\) of simple linear algebraic groups with the following property: the simple components of the \(J\) module \(\operatorname{Lie}(G)/\operatorname{Lie}(J)\) are all minuscule representations of \(J\). One family of examples occurs when the group \(G\) has roots of two different lengths and \(J\) is the subgroup generated by the long roots. We classify all such embeddings when \(J = \mathrm{SL}_2\) and \(J = \mathrm{SL}_3\), show how each embedding implies the existence of exceptional algebraic structures on the graded components of \(\operatorname{Lie}(G)\), and relate properties of those structures to the existence of various twisted forms of \(G\) with certain relative root systems.

MSC:

20G05 Representation theory for linear algebraic groups
20G15 Linear algebraic groups over arbitrary fields
17B45 Lie algebras of linear algebraic groups
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