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

Abstract

We present the subalgebra structure of sl(3, O), a particular real form of e 6 chosen for its relevance to particle physics and its close relation to generalized Lorentz groups. We use an explicit representation of the Lie group SL(3, O) to construct the multiplication table of the corresponding Lie algebra sl(3, O). Both the multiplication table and the group are then utilized to find various nested chains of subalgebras of sl(3, O), in which the corresponding Cartan subalgebras are also nested where possible. Because our construction involves the Lie group, we simultaneously obtain an explicit representation of the corresponding nested chains of subgroups of SL(3, O).

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

“…ξm, ] ⊂ ξ 2 p = (−1)p (8) then p ⊕ ξm also has real structure constants, and is therefore also a real form of the complex Lie algebra g C . Note that φ * is a vector space isomorphism, but not a Lie algebra isomorphism -as must be the case if it takes one real form to another.…”

“…A description of the group E 6(−26) as SL(3, O) was given in [5], generalizing the interpretation of SL(2, O) as (the double cover of) SO(9, 1) discussed in [6]. An interpretation combining spinor and vector representations of the Lorentz group in 10 spacetime dimensions was described in [7], and in [8] we obtained nested chains of subgroups of SL (3, O) that respect this Lorentzian structure.…”

“…In the present work, we use various Cartan decompositions of the Lie algebra e 6 to further extend this construction in two distinct ways. First, we identify additional real subalgebras of sl (3, O), thus completing the explicit construction of nested chains of subgroups of SL(3, O) begun in [8], building on the Lorentzian structure of SL (2, O). In particular, we locate the "missing" C 4 subgroups of SL(3, O) referred to in [8], in the form SU (3, 1, H).…”

“…First, we identify additional real subalgebras of sl (3, O), thus completing the explicit construction of nested chains of subgroups of SL(3, O) begun in [8], building on the Lorentzian structure of SL (2, O). In particular, we locate the "missing" C 4 subgroups of SL(3, O) referred to in [8], in the form SU (3, 1, H). We then reinterpret the Cartan decompositions used in our construction as (vector space) isomorphisms of the complexified Lie algebra e C 6 , yielding an abelian group of such isomorphisms that acts on the 5 real forms of e 6 , allowing us to identify 8 particular copies of these real forms, based on their relationship to the original copy of sl (3, O).…”

“…We briefly review the basic structure of SL (3, O) in Section 2.1, and the basic properties of Cartan decompositions in Section 2.2. We construct three important Cartan decompositions of e 6 in Section 3, and use them to construct nested chains of subalgebras of sl (3, O) in Section 4, which are used in Section 4.2 to construct a tower of subalgebras in sl(3, O) based upon our preferred basis of the Cartan subalgebra, completing the program begun in [8]. Finally, we explore the relationships between the real forms of e 6 in Section 5.…”