Pure spinors and a construction of the $E_*$-Lie algebras

TL;DR

This paper refines the understanding of pure spinors and operators in Clifford algebras to construct exceptional Lie algebras $\,\mathfrak e_6, \mathfrak e_7, \mathfrak e_8$ within the spinor framework.

Contribution

It provides explicit formulas for the operator $L_2$ in terms of spinor inner products and uses this to construct exceptional Lie algebras within Clifford algebra theory.

Findings

Explicit formulae for the operator $L_2$ in terms of spinor inner products.

A complete Clifford algebra-based construction of $\,\mathfrak e_6, \mathfrak e_7, \mathfrak e_8$.

Enhanced understanding of pure spinors and their role in Lie algebra construction.

Abstract

Let $(V,g)$ be a $2n$-dimensional hyperbolic space and $C(V,g)$ its Clifford algebra. $C(V,g)$ has a $\mathbb Z$-grading, $C^k $, and an algebra isomorphism $C(V,g)\cong End(S)$, $S$ the space of spinors. \'E. Cartan defined operators $L_k: End(S) \to C^k$ which are involved in the definition of pure spinors. We shall give a more refined study of the operator $L_2$, in fact, obtain explicit formulae for it in terms of spinor inner products and combinatorics, as well as the matrix of it in a basis of pure spinors. Using this information we give a construction of the exceptional Lie algebras $\mathfrak e_6, \mathfrak e_7, \mathfrak e_8$ completely within the theory of Clifford algebras and spinors.