Abstract
Starting from the Jordan algebraic interpretation of the “Magic Star” embedding within the exceptional sequence of simple Lie algebras, we exploit the so-called spin factor embedding of rank-3 Jordan algebras and its consequences on the Jordan algebraic Lie symmetries, in order to provide another perspective on the origin of the Exceptional Periodicity (EP) and its “Magic Star” structure. We also highlight some properties of the special class of Vinberg’s rank-3 (dubbed exceptional) T-algebras, appearing on the tips of the “Magic Star” projection of EP(-generalized, finite-dimensional, exceptional) algebras.
Highlights
On its real forms related to a Jordan-algebraic interpretations
Given a finite-dimensional exceptional Lie algebra, the so-called “Magic Star” projection of the corresponding root lattice onto a plane determined by an a2 root sub-lattice has been introduced by Mukai2 [2], and later investigated in depth in [3], with a different approach exploiting Jordan Pairs [5]; in the case of e8, it has been recalled in another contribution to Group32 Proceedings [1]
Over R, there are two non-compact real forms of the above “enlarged” exceptional sequence {qconf (Jq3)}q=8,4,2,1,0,−1/3,−2/3,−1 which enjoy an immediate Jordan-algebraic interpretation; they are reported in Tables II and Table III
Summary
Given a finite-dimensional exceptional Lie algebra, the so-called “Magic Star” projection (depicted in Fig. 1) of the corresponding root lattice onto a plane determined by an a2 root sub-lattice has been introduced by Mukai2 [2], and later investigated in depth in [3] (see [4]), with a different approach exploiting Jordan Pairs [5]; in the case of e8, it has been recalled in another contribution to Group Proceedings [1]. The reduced structure Lie. Over R (as we shall consider throughout the present paper), there are two non-compact real forms of the above “enlarged” exceptional sequence {qconf (Jq3)}q=8,4,2,1,0,−1/3,−2/3,−1 which enjoy an immediate Jordan-algebraic interpretation; they are reported in Tables II and Table III. Over R (as we shall consider throughout the present paper), there are two non-compact real forms of the above “enlarged” exceptional sequence {qconf (Jq3)}q=8,4,2,1,0,−1/3,−2/3,−1 which enjoy an immediate Jordan-algebraic interpretation; they are reported in Tables II and Table III In both these cases, the real form of the a2 defining the plane onto which the Magic Star projection is defined is maximally non-compact (i.e., split), and the following branching (non-compact, real form of (I.2)) correspondingly holds: qconf (Jq3) = sl3,R ⊕ str0 (Jq3) ⊕ 3 × Jq3 ⊕ 3 × Jq3. 2 1 0 −1/3 −2/3 −1 qconf (Jq3) e8(−24) e7(−5) e6(2) f4(4) so so4,3 g2(2) sl3,R str0(Jq3) e6(−26) su∗6 (sl3,C)R sl3,R R ⊕ R R
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