Exceptionally simple PDE

Abstract

Abstract. We give local descriptions of parabolic contact structures and show how their flat models yield explicit PDE having symmetry algebras isomorphic to all complex simple Lie algebras except sl2. This yields a remarkably uniform generalization of the Cartan-Engel models from 1893 in the G2 case. We give a formula for the harmonic curvature of a G2-contact structure and describe submaximally symmetric models for general G-contact structures.

Contact manifolds, Lagrangian Grassmannians and PDEs

“…The adjoint contact manifold of PGL n+2 (C). In this Section we show that there is essentially a unique 2 nd order PDE E ⊂ M (1) , whose group of symmetries (see Section 6) is precisely G. This example fits into a very extensive research programme, that is, finding PDEs with a prescribed Lie group of symmetries, which originated in the work of Lie, Darboux, Cartan and others (see [2,31]).…”

“…By the definition 1 of Monge-Ampère equations, these produce in turn two SL n -invariant Monge-Ampère equations. Their "speciality" is due to the fact that 1 is the minimal degree of an invariant hypersurface in PΛ n 0 (g * −1 ) (see also [31]). 11.3.…”

Lowest degree invariant second-order PDEs over rational homogeneous contact manifolds

“…Remarkably, it turns out that in this context-with the exception of groups G of type A and, for trivial reasons, C-a G-invariant second order PDE has precisely the Lie algebra g of G as its local infinitesimal symmetries at any point of M . This has been observed and used by D. The in [23] to realise the simple Lie algebras not of type C as infinitesimal symmetries of 2 nd order PDEs, a problem with a long tradition, inaugurated by the 1893 works by Cartan and Engel, 1 and recently recast in full generality by P. Nurowski in the context of the so-called parabolic contact geometries (see [8], sec. 4.2).…”

Geometry of Lagrangian Grassmannians and nonlinear PDEs

“…Another really intriguing feature of LGr (3,6), or rather of its Plücker embedding in P 13 , is that such an embedding can be regarded as an appropriate generalisation of the twisted cubic in P 3 , whereby the field of complex number has been replaced by the Jordan algebra of symmetric 3 × 3 matrix. This analogy played a fundamental role in a recent analysis of PDEs with prescribed group of symmetries [51]. A gentle introduction to it can be found in [46, Section 5].…”

“…Concerning the Lagrangian Chow form and the correspondence between substructures of the contact distribution and second order PDEs, it is worth to mention the work [51] by The and the almost simultaneous work [3] by the authors and Alekseevsky. The problem dealt with there is that of constructing a PDE admitting a prescribed simple (complex) Lie group of symmetries.…”