Higher-Dimensional Representations of $$\mathsf {SL}_2$$ SL 2 and its Real Forms Via Plücker Embedding

Abstract

The paper provides a geometric perspective on the inclusion map \(\mathfrak {sl}_2\cong \!\mathfrak {so}_3\!\subset \! \mathfrak {so}_n\) realized via the Plucker embedding. It relates higher-dimensional representations of the complex Lie group \(\mathsf {SO}_3\) and its real forms in terms of \(\mathsf {SO}(n)\) and \(\mathsf {SO}(p,q)\) transformations to different subspaces described by means of the well-known double fibrations used in twistor theory. Moreover, explicit matrix realizations and various factorization techniques, such as Euler and Wigner decompositions, are constructed in this generalized setting. Examples are provided for \(n=3,4\) and 5 in the context of special relativity, classical and quantum mechanics.