Minuscule embeddings.

We study embeddings J → G of simple linear algebraic groups with the following property: the simple components of the J module Lie (G) / 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 = SL 2 and J = SL 3 , show how each embedding implies the existence of exceptional algebraic structures on the graded components of Lie (G) , and relate properties of those structures to the existence of various twisted forms of G with certain relative root systems. [ABSTRACT FROM AUTHOR]