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Magic coset decompositions. (English) Zbl 1295.81135
Summary: By exploiting a “mixed” non-symmetric Freudenthal-Rozenfeld-Tits magic square, two types of coset decompositions are analyzed for the non-compact special Kähler symmetric rank-3 coset $$E_{7(-25)}/[(E_{6(-78)}\times U(1))/\mathbb Z_3]$$, occurring in supergravity as the vector multiplets’ scalar manifold in $$\mathcal N=2$$, $$\mathcal D=4$$ exceptional Maxwell-Einstein theory. The first decomposition exhibits maximal manifest covariance, whereas the second (triality-symmetric) one is of Iwasawa type, with maximal $$\mathrm{SO}(8)$$ covariance. Generalizations to conformal non-compact, real forms of nondegenerate, simple groups “of type $$E_7$$” are presented for both classes of coset parametrizations, and relations to rank-3 simple Euclidean Jordan algebras and normed trialities over division algebras are also discussed.

MSC:
 81R05 Finite-dimensional groups and algebras motivated by physics and their representations 17B25 Exceptional (super)algebras 22E70 Applications of Lie groups to the sciences; explicit representations 83E50 Supergravity
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