Triality, Exceptional Lie Algebras and Deligne Dimension Formulas

Abstract

We give a computer-free proof of the Deligne, Cohen and de Man formulas for the dimensions of the irreducible g-modules appearing in g k ; k44; where g ranges over the exceptional complex simple Lie algebras. We give additional dimension formulas for the exceptional series, as well as uniform dimension formulas for other representations distinguished by Freudenthal along the rows of his magic chart. Our proofs use the triality model of the magic square, which we review and present a simplified proof of its validity. We conclude with some general remarks about obtaining ''series'' of Lie algebras in the spirit of Deligne and Vogel. # 2002 Elsevier Science (USA)

The extended Freudenthal Magic Square and Jordan algebras

“…This construction is a superization of the algebra construction in [Eld04], which in turn is based on previous constructions (with Hurwitz algebras) in [BS,BS03,LM02,LM04]. It gives a symmetric and simple construction of Freudenthal Magic Square.…”

“…A more symmetric construction, based on two unital composition algebras, which play symmetric roles, and their triality Lie algebras, has been given recently by several authors ( [AF93], [BS], [LM02]). Among other things, this construction has the advantage of being valid too in characteristic 3.…”

On universal knot polynomials

“…Closer to our purposes in this paper, universal Vogel's [7] and Landsberg and Manivel's [8] formulas (see also [2][3][4][5][6]) deal with (some) irreducible representations in an arbitrary powers of adjoint representation of Lie algebra, while its extension to all representations, including fundamental, is problematic. In this section we collect just some of the relevant technical details.…”

Development of a Unified Tensor Calculus for the Exceptional Lie Algebras

“…We employ (3), with ℓ 2 + ℓ 3 and ℓ 2 ℓ 3 given by (18) and (46), to evaluate the right side (53). We can complete the evaluation with the aid of (1), (6) and (21), obtaining…”