Seminar Abstract:

Exceptional Lie group G_{2} is a beautiful 14dimensional continuous group, having relations with such diverse notions as triality, 7dimensional cross product and exceptional holonomy. It was found abstractly by Killing in 1887 (complex case) and then realized as a symmetry group by Engel and Cartan in 1894 (real split case). Later in 1910 Cartan returned to the topic and realized split G_{2} as the maximal finitedimensional symmetry algebra of a rank 2 distribution in ú^{5}. In other words, Cartan classified all symmetry groups of Monge equations of the form y’=f(x,y,z,z’,z’’). I will discuss the higherdimensional generalization of this fact, based on the joint work with Ian Anderson. Compact real form of G_{2} was realized by Cartan as the automorphism group of octonions in 1914. In the talk I will also explain how to realize this G_{2} as the maximal symmetry group of a geometric object.
