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Multisymplectic forms of degree three in dimension seven. (English) Zbl 1045.53017
Bureš, Jarolím (ed.), The proceedings of the 22nd winter school “Geometry and physics”, Srní, Czech Republic, January 12–19, 2002. Palermo: Circolo Matemàtico di Palermo. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 71, 73-91 (2003).
A multisymplectic 3-structure on an \(n\)-dimensional manifold \(M\) is given by a closed smooth 3-form \(\omega\) of maximal rank on \(M\) which is of the same algebraic type at each point of \(M\), i.e. they belong to the same orbit under the action of the group \(GL(n,{\mathbb R})\). This means that for each point \(x\in M\) the form \(\omega_x\) is isomorphic to a chosen canonical 3-form on \({\mathbb R}^n\). R. Westwick [Linear Multilinear Algebra 10, 183–204 (1981; Zbl 0464.15001)] and D. Ž. Djoković [Linear Multilinear Algebra 13, 3–39 (1983; Zbl 0515.15011)] obtained the classification of 3-forms in dimension seven. Among these forms they revealed eight being canonical forms. By using these results the authors describe the isotropy groups of all canonical forms. To point out the nature of these eight groups we mention, for example: the exceptional Lie group \(G_2\), its noncompact dual \(\tilde G_2\), \(O_2=[{\mathbb R}^8\ltimes\text{Spin}(1,2))\ltimes SO(1,2)]\ltimes{\mathbb R}^{\ast}\), \(O_4=[((([{\mathbb R}^3\ltimes{\mathbb R}^6]\ltimes{\mathbb R})\ltimes{\mathbb R})\ltimes{\mathbb R})\ltimes SL(3,{\mathbb R})]\ltimes{\mathbb R}^{\ast}\), \(O_6=[((({\mathbb R}^6\ltimes{\mathbb R}^2)\ltimes{\mathbb R}^2)\ltimes SL(2,{\mathbb C})\ltimes SO(2)]\ltimes{\mathbb R}^{\ast}\).
For the entire collection see [Zbl 1014.00011].

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
15A75 Exterior algebra, Grassmann algebras
20H20 Other matrix groups over fields