Stretching and welding indecomposable quadratic forms (Q2639083)

A \({\mathbb Z}\)-lattice \(L\) on a rational quadratic space \((V,{\mathbb Q})\) is said to be indecomposable (with respect to \({\mathbb Q})\) if there exists a non-trivial orthogonal splitting of \(L\) into sublattices. In this paper, methods for the construction of new indecomposable positive definite integral quadratic lattices from known ones are presented. The main results give sufficient conditions under which lattices arising from either of two constructions are guaranteed to be indecomposable. In the first of these, it is assumed that two positive definite quadratic lattices \(L_ 1\) and \(L_ 2\), which are indecomposable with respect to quadratic forms \(Q_ 1\) and \(Q_ 2\), respectively, are given. A form \(Q\) defined on \(L_ 1\oplus L_ 2\) which is a nonorthogonal sum of \(Q_ 1\) and \(Q_ 2\) is then investigated. In the second, a positive definite form \(Q_ 0\) and a positive semi-definite form \(Q_ 1\) both acting on the same lattice \(L\) are given, and the form \(Q=Q_ 0+Q_ 1\) on \(L\) is investigated. Several interesting examples are given to illustrate the use of these constructions. For instance, an infinite family of indecomposable lattices of rank 8 is produced which, in particular, contains \(\Gamma_ 8\), the unique positive definite even unimodular lattice of rank 8.