An introduction to Casimir bilinear pairings and some arithmetic applications

Carlos Rivera-Guaca, Guillermo Mantilla-Soler

Published 2018-12-07Version 1

Inspired by the Casimir invariant of a semisimple Lie algebra we introduce the notion of Casimir bilinear pairing; the classical Casimir invariant is just a value taking by the pairing in a very specific case. The pairing defined here not only refines the invariant of Lie algebras but it also has arithmetic applications. By studying the properties of Casimir pairings we show that a number field that is totally real and has fundamental discriminant is completely determined by the integral quadratic form given by the trace pairing over its maximal order.