Discovering real lie subalgebras of $\mathfrak {e}_6$e6 using Cartan decompositions

Abstract

The process of complexification is used to classify a Lie algebra and identify its Cartan subalgebra. However, this method does not distinguish between real forms of a complex Lie algebra, which can differ in signature. In this paper, we show how Cartan decompositions of a complexified Lie algebra can be combined with information from the Killing form to identify real forms of a given Lie algebra. We apply this technique to sl(3, O), a real form of e 6 with signature (52,26), thereby identifying chains of real subalgebras and their corresponding Cartan subalgebras within e 6 . Motivated by an explicit construction of sl(3, O), we then construct an abelian group of order 8 which acts on the real forms of e 6 , leading to the identification of 8 particular copies of the 5 real forms of e 6 , which can be distinguished by their relationship to the original copy of sl(3, O).

E6, the group: The structure of SL(3, 𝕆)

“…In particular, we have not yet identified any of the C 4 subgroups of E 6 . In other work [14], we extend, and in a sense complete, the present investigation by constructing and discussing chains of subgroups adapted to these other subgroups. We hope that the resulting maps of E 6(−26) will prove useful in further attempts to apply the exceptional groups to nature.…”

The Magic Square of Lie Groups: The 2 × 2 Case

“…It appears to be straightforward to reinterpret our previous description (12) of E 6(−26) [11,12,13,14] http://arxiv.org/ps/2009.00390v1…”

“…Because all other division algebras are subalgebras of the octonions, these two constructions fully capture the first two rows of the 2 × 2 magic square of Lie groups shown in Table 5. More recently, Dray and Manogue [11,12] have extended these results to the exceptional Lie group E 6 , using the framework described in more detail by Wangberg and Dray [13,14] and in Wangberg's thesis [15]. All of these results rely on the description of certain groups using matrices over division algebras.…”

“…More recently, Dray and Manogue [11,12] have extended these results to the exceptional Lie group E 6 , using the framework described in more detail by Wangberg and Dray [13,14] and in Wangberg's thesis [15]. All of these results rely on the description of certain groups using matrices over division algebras.…”

E6, the Group: The structure of SL(3,O)

“…Our preferred quaternionic subalgebra of O is H = 1, k, kℓ, ℓ , so we discard transformations involving i, j, iℓ, or jℓ. We therefore discard 3 × 4 = 12 boosts, 3 × 4 = 12 simple rotations involving z, and 4 simple rotations involving x -but we must add back in 3 rotations involving x of type 2, since we can no longer use the middle relation in (12) to eliminate them. Turning to the transverse rotations, we need only consider transformations sl(2, O)…”