$E_8$, the most exceptional group

Abstract

Abstract. The five exceptional simple Lie algebras over the complex number are included one within the other as g 2 ⊂ f 4 ⊂ e 6 ⊂ e 7 ⊂ e 8 . The biggest one, e 8 , is in many ways the most mysterious. This article surveys what is known about it, including many recent results, and it focuses on the point of view of Lie algebras and algebraic groups over fields.

Experimental observations of the universal cascade of bound states in quantum Ising chain in a magnetic field and E8 symmetry

“…It turned out to be "affine Toda field theory" [3] associated with the E 8 exceptional Lie algebra symmetry (for introduction to the mathematical aspects of E 8 , see Refs. [4,5]). That is, even though the Ising model under the longitudinal field is not exactly solvable, its scaling limit is described by an exactly solvable field theory with a surprisingly large emergent symmetry.…”

A positive formula for the Ehrhart-like polynomials from root system chip-firing

“…But Lemma A.2 is all we will need. Using Lemma A.2 (together with knowledge of the simple root coordinates of the highest roots as recorded for instance in [14,§12,Table 2]) it is easy to determine all the Φ ∨ i -dominant representatives for W i -orbtis of Φ ∨ \ Φ ∨ i . For the classical types, this information is recorded in Table 5.…”

A class of continuous non-associative algebras arising from algebraic groups including

“…For 8 , it is known from [19] that it is the identity component of the stabilizer of an octic form on the Lie algebra 8 and that it is the automorphism group of the 8 -invariant algebra on its 3875-dimensional irreducible representation. (See also [18,Section 3] or [19,Section 16] for broader discussions of other realizations.) The latter description of 8 is known to be true even though this algebra is not well-understood; this paper gives explicit and effective formulas for calculating in the algebra.…”