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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.

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