A kind of magic

Abstract

We introduce the extended Freudenthal–Rosenfeld–Tits magic square based on six algebras: the reals , complexes , ternions , quaternions , sextonions and octonions . The sextonionic row/column of the magic square appeared previously and was shown to yield the non-reductive Lie algebras, , , , and , for and respectively. The fractional ranks are used to denote the semi-direct extension of the simple Lie algebra in question by a unique (up to equivalence) Heisenberg algebra. The ternionic row/column yields the non-reductive Lie algebras, , , , and , for and respectively. The fractional ranks here are used to denote the semi-direct extension of the semi-simple Lie algebra in question by a unique (up to equivalence) nilpotent ‘Jordan’ algebra. We present all possible real forms of the extended magic square. It is demonstrated that the algebras of the extended magic square appear quite naturally as the symmetries of supergravity Lagrangians. The sextonionic row (for appropriate choices of real forms) gives the non-compact global symmetries of the Lagrangian for the maximal , magic and magic non-supersymmetric theories, obtained by dimensionally reducing the parent theories on a circle, with the graviphoton left undualised. In particular, the extremal intermediate non-reductive Lie algebra (which is not a subalgebra of ) is the non-compact global symmetry algebra of , supergravity as obtained by dimensionally reducing , supergravity with symmetry on a circle. On the other hand, the ternionic row (for appropriate choices of real forms) gives the non-compact global symmetries of the Lagrangian for the maximal , magic and magic non-supersymmetric theories, as obtained by dimensionally reducing the parent theories on a circle. In particular, the Kantor–Koecher–Tits intermediate non-reductive Lie algebra is the non-compact global symmetry algebra of , supergravity as obtained by dimensionally reducing , supergravity with symmetry on a circle.