Hypergeometric functions and non-associative algebras (Q2782396)

With a homogeneous quadratic system of \(n\) ordinary differential equations \(X'_i=\displaystyle\sum_{j,k=1}^n a^i_{j,k} X_j X_k\) with \(i=1,\dots,n\) and \(a^i_{j,k}=a^i_{k,j} \in \mathbb C\) for all \(i,j,k\), \textit{L. Markus} [Ann. Math. Stud. 45, 185-213 (1960; Zbl 0119.29803)] associated a commutative non-associative algebra \(A\) over \(\mathbb C\) in \(n\) generators \(x_1,\dots,x_n\) with multiplication table \(x_j \cdot x_k=\sum_{i=1^n} a^i_{j,k} x_i\). He proved for \(n=2\) that (up to linear isomorphisms) there are only two such algebras with unit element, and determined the corresponding homogeneous quadratic systems. NEWLINENEWLINENEWLINEThe paper under review classifies the homogeneous quadratic systems of rank \(n=3\) whose corresponding algebra \(A\) has a unit element. The automorphism of \(A\) is finite if and only if the quadratic system is a Halphen-type system for the hypergeometric, Whittacker, Bessel, Hermite-Weber or Airy equations. If the automorphism group of \(A\) is infinite, the quadratic system is explicitely solved by elementary functions.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00029].