Spin Representations of $\mathfrak{s}{\mathfrak{d}_m}\mathbb{C}$

Abstract

In this lecture we complete the picture of the representations of the orthogonal Lie algebras by constructing the spin representations${S^ \pm }$of$\mathfrak{s}{\mathfrak{d}_m}\mathbb{C}$this also yields a description of the spin groups$Spi{n_m}\mathbb{C}$. Since the representation-theoretic analysis of the spaces${S^ \pm }$was carried out in the preceding lecture, we are concerned here primarily with the algebra involved in their construction. Thus, §20.1 and §20.2, while elementary, involve some fairly serious algebra. Section20.3, where we briefly sketch the notion of triality, may seem mysterious to the reader (this is at least in part because it is so to the authors); if so, it may be skipped. Finally, we should say that the subject of the spin representations of $\mathfrak{s}{{\mathfrak{o}}_{m}}\mathbb{C}$ is a very rich one, and one that accommodates many different points of view; the reader who is interested is encouraged to try some of the other approaches that may be found in the literature.