Mysterious mathematical trinities
Vladimir I. Arnold · Cambridge University Press eBooks · 2005
I shall try to tell about some phenomena in mathematics that make me surprised. In most cases, they are not formalized. They cannot even be formulated as conjectures. A conjecture differs in that it can be disproved; it is either true or false. We shall consider certain observations that lead to numerous theorems and conjectures, which can be proved or disproved. But these observations are most interesting when considered from a general point of view. I shall explain this general point of view for a simple example from linear algebra. The theory of linear operators is described in modern mathematics as the theory of Lie algebras of series A n , i.e., sl ( n + 1), and formulated in terms of root systems. A root system can be assigned to any Coxeter group, that is, a finite group generated by reflections (at least, to any crystallographic group). If we take a statement of linear algebra which refers to this special case of the group A n and remove all the content from its formulation, so as to banish all mentions of eigenvalues and eigenvectors and retain only roots, we will obtain something that can be applied to the other series, B n , C n , and D n , including the exceptional ones E 6 , E 7 , E 8 , F 4 , and G 2 (and, sometimes, even to all the Coxeter systems, including the noncrystallographic symmetry groups of polygons, of the icosahedron, and of the hypericosahedron, which lives is four-dimensional space).