Advances in Theoretical and Mathematical Physics

Volume 15 (2011)

Number 6

$E_7$ groups from octonionic magic square

Pages: 1605 – 1654

DOI: http://dx.doi.org/10.4310/ATMP.2011.v15.n6.a2

Authors

Sergio L. Cacciatori (Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria)

Francesco Dalla Piazza (Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria)

Antonio Scotti (Dipartimento di Matematica, Università degli Studi di Milano)

Abstract

In this paper, we continue our program, started in\cite{Cacciatori:2005yb}, of building up explicitgeneralized Euler angle parameterizations for allexceptional compact Lie groups. Here we solve the problemfor $E_7$, by first providing explicit matrix realizationsof the Tits construction of a Magic Square product betweenthe exceptional octonionic algebra $\jj$ and thequaternionic algebra $\hh$, both in the adjoint and the$56$-dimensional representations. Then, we provide theEuler parametrization of $E_7$ starting from its maximalsubgroup $U=(E_6\times U(1))/\mathbb{Z}_3$. Next, we givethe constructions for all the other maximal compactsubgroups.

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