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A symplectic representation of \(E_7\). (English) Zbl 1340.17016
The real exceptional algebra \(\mathfrak{e}_{7,-25}\) has been described in a certain form as the symplectic algebra \(\mathfrak{sp}(6,\mathbb{O})\). The aim of this paper is to interpret symplectically the smallest fundamental representation of \(\mathfrak{e}_{7,-25} \), which has dimension \(56\).
Recall first the unified Freudenthal-Tits construction of the exceptional compact Lie algebras. If \(C_1\) and \(C_2\) are normed real division algebras (i.e., \(\mathbb{R}\), \(\mathbb{C}\), \(\mathbb{H}\) or \(\mathbb{O}\)), and \(H_3(C_2)\) denotes the set of \(3\times 3\) Hermitian matrices with coefficients in \(C_2\), this space is a Jordan algebra with the symmetrized product \(x\circ y=\frac12(xy+yx)\). Now (the subindex zero denoting the traceless elements) the vector space \[ \mathcal{T}(C_1,C_2)=\mathfrak{der}(C_1)\oplus\, (C_1)_0\otimes H_3(C_2)_0\,\oplus \mathfrak{der}(H_3(C_2)) \] is endowed with a Lie algebra structure in such a way that we obtain all the exceptional compact Lie algebras (up to \(\mathfrak{g}_{2}\)) when taking either \(C_1\) or \(C_2\) as the octonion (division) algebra \(\mathbb{O}\). We can replace \(C_1\) with a split real composition algebra (they are 3, \(\mathbb{C}'=\mathbb{R}\oplus\mathbb{R}\), \(\mathbb{H}'=\text{Mat}_{2\times 2}(\mathbb{R})\) and \(\mathbb{O}'\) the split octonions, also called Zorn’s vector-matrix algebra), then getting the half-split magic square: \[ \begin{matrix} C_1\backslash C_2 & \mathbb{R} & \mathbb{C} & \mathbb{H} & \mathbb{O}\\ \mathbb{R} & \mathfrak{so}_3 & \mathfrak{su}_3 & \mathfrak{sp}_6 & \mathfrak{f}_4\\ \mathbb{C}'& \mathfrak{sl}_3(\mathbb{R}) & \mathfrak{sl}_3(\mathbb{C}) & \mathfrak{sl}_3(\mathbb{H}) & \mathfrak{e}_{6,-26}\\ \mathbb{H}'& \mathfrak{sp}_6(\mathbb{R}) & \mathfrak{su}_{3,3} & \mathfrak{so}_6^*(\mathbb{H}) & \mathfrak{e}_{7,-25}\\ \mathbb{O}'& \mathfrak{f}_{4,4} & \mathfrak{e}_{6,2} & \mathfrak{e}_{7,-5} & \mathfrak{e}_{8,-24} \end{matrix} \] The first three rows of this square can be interpreted as \(\mathfrak{su}_3(C_2)\), \(\mathfrak{sl}_3(C_2)\) and \(\mathfrak{sp}_6(C_2)\), respectively. This is part of an effort of several authors to give a construction of the exceptional Lie groups by means of a magic square of Lie groups which makes use of a pair of composition algebras, but without constructing first the corresponding Lie algebras.
We can regard \(\mathfrak{e}_{7,-25}\) as the conformal algebra associated to \(\mathfrak{e}_{6,-26}=\mathfrak{der}(H_3( \mathbb{O}))\oplus R_{H_3( \mathbb{O})_0}\) (with \(R_x\) denoting the multiplication operator of the Jordan algebra), that is, \(\mathfrak{e}_{7,-25}=\mathcal{L}_{-1}\oplus \mathcal{L}_0\oplus\mathcal{L}_1\) (a \(\mathbb{Z}\)-grading) with \(\mathcal{L}_{0}=\mathfrak{e}_{6,-26}\oplus\, \mathbb{R}\) (rotations and a dilation), \(\mathcal{L}_{-1}=H_3( \mathbb{O})\) and \(\mathcal{L}_{1}=H_3( \mathbb{O})\) (translations). The smallest irreducible representation of \(\mathfrak{e}_{7,-25}\) was constructed by Freudenthal in the 50’s and it can be seen as \(\left( \begin{matrix} \mathbb{R} & H_3( \mathbb{O})\\ H_3( \mathbb{O}) & \mathbb{R} \end{matrix} \right).\) The authors think of any of its elements \(\left(\begin{matrix} p & X\\ Y & q \end{matrix} \right)\) in a different way: as a \(6\times6\times 6\) cube of octonions, divided into cubies, where each cubie is certain vector-valued 2-form obtained from \(p*I_3\), \(q*I_3\), \(*X\) and \(*Y\) (a kind of Hodge dual of an element in \(H_3(\mathbb{O})\)). Thus they write the elements of the representation in terms of covariant and contravariant tensors, giving explicitly the action of \(\mathfrak{e}_{7,-25}\) on them from the action of \(\mathfrak{e}_{6,-26}\) on the cubies. Also, the quartic invariant of \(\mathfrak{e}_{7,-25}\) is expressed in these terms. According to the authors, this interpretation is more geometrical than Freudenthal’s approach.

17B25 Exceptional (super)algebras
17A35 Nonassociative division algebras
20G41 Exceptional groups
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