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).

A Novel Approach to Extra Dimensions

“…The symmetry of the form L(v 27 ) = 1, that is the symmetry leaving det(X ) invariant, is a real form of E 6 as we briefly review here. We closely follow references [4,5,6,7,8] within which, in particular, the means of accommodating, and employing, the non-associative property of the octonions is described in detail. As a generalisation from the SL(2, C) ≡ Spin + (1, 3) Lorentz transformations of equation 19 a set of 2 × 2 matrix actions with M ∈ SL(2, O) ≡ Spin + (1, 9) can be embedded in the upper-left corner of 3 × 3 matrices M to obtain a conjugation action for the 3 × 3 case R : X → MX M † with:…”

“…A natural generalisation from the space h 3 C, underlying the vector v 9 ∈ h 3 C transformed in equation 19 as a symmetry time, is obtained by augmenting the complex numbers C to the largest division algebra, namely the octonions O [3]. The vector space h 3 O obtained corresponds to the set of 3 × 3 Hermitian matrices over the octonions with elements which can be written as (in this paper we closely follow [4], [5] chapters 3 and 4, together with [6,7,8], for all details of the E 6 structure, and generally adopt the notation therein):…”

“…where α ∈ R parametrises a given action. By studying the linear dependences of the resulting 135 vector fields it can be shown [5,6,7,8] how a linearly independent basis of 78 actions emerges which fully describes E 6 ≡ SL(3, O) as the determinant preserving transformations on h 3 O. The entire group is then described in terms of the actions of matrices M on the space h 3 O, with the preferred basis for the Lie algebra represented on T h 3 O reproduced below in table 1.…”

“…More generally little reference has been identified in the literature in which a 248-dimensional representation of E 8 is described in terms of an action on a quintic, or any other homogeneous polynomial, form. However in [17,18] a polynomial of degree eight which is invariant as a 248-dimensional representation of the compact real form of E 8 is described, and is closely related to an invariant polynomial for the real form E 8 (8) .…”

From Unified Field Theory to the Standard Model and Beyond

“…While the Lie algebras, including the five exceptional cases of G 2 , F 4 , E 6 , E 7 and E 8 , were classified by Killing and Cartan in the late 19 th century ([14] section 4 opening) an understanding of explicit expressions for certain representations of these algebras developed from the mid-20 th and continues into the 21 st century. For example the smallest non-trivial representation of E 6 can be expressed by the space of 3 × 3 Hermitian octonion matrices h 3 O, as employed in 1950 [15], with the corresponding determinant preserving E 6 ≡ SL(3, O) group action described in explicit detail more recently [16].…”