Gradings on the real form 𝔢6,−26

Abstract

We describe four fine gradings on the real form e 6,−26 . They are precisely the gradings whose complexifications are fine gradings on the complexified algebra e 6 . The universal grading groups are Zica, approvato con Decreto n.197 del 03/12/2014.and also Z-graded as

Gradings on the real form e6,−14

“…Such gradings and models could be potentially related with different physical phenomena. The gradings on e 6,−14 have nothing to do with the gradings on e 6,−26 obtained in [DrG16b]. Indeed, our main result, Theorem 1, states that there are 6 fine gradings (and essentially only 6) on e 6,−14 , whose universal grading groups are Z 3 2 × Z 2 3 , Z 6 2 , Z × Z 4 2 , Z 7 2 , Z × Z 5 2 and Z 2 × Z 3 2 .…”

“…Only two pairs of those gradings have the same origin, i.e., isomorphic complexifications (the two Z 2 × Z 3 2 -gradings have quite different properties). Regarding the tools used here for finding the gradings on e 6,−14 , they are indeed related to the tools in [DrG16b], but new approaches have been necessary too. For each grading, we have tried to do a self-contained treatment, providing a related suitable model of the algebra e 6,−14 .…”

“…On one hand, some recent papers devoted to the classical simple real Lie algebras are [EK18], [BKR18a] and [BKR18b]. On the other hand, as mentioned, this is the third of our papers about exceptional real Lie algebras: first, in [CalDrM10], we classified fine gradings on the real algebras of types G 2 and F 4 , and second, in [DrG16b], we described the fine gradings on the algebra e 6,−26 coming from fine gradings on the complex algebra e 6 . Let us recall that there are three non-split and non-compact real algebras of type E 6 , characterized by the signatures of their Killing forms, namely, -26, -14 and 2.…”

Inner ideals of real simple Lie algebras

“…Remark 5.8. Having these realizations of the real forms of E 6 helps to explain some of their fine gradings, described in [DG16] and [DG18]. For instance, an immediate consequence is that all the non-compact real forms of e 6 , namely, e 6,6 , e 6,2 , e 6,−14 and e 6,−26 , possess a Z × Z 4 2 -grading which does not admit proper refinements.…”

Gradings on classical central simple real Lie algebras

“…Fine gradings on real forms of the classical simple complex Lie algebras except D 4 were described in [15], but the equivalence problem remains open. Fine gradings on real forms of the simple complex Lie algebras of types G 2 and F 4 were classified up to equivalence in [7]; some partial results were obtained for type E 6 in [8].…”