  We introduce the N\'eron-Severi Lie algebra of a Soergel module and we
determine it for a large class of Schubert varieties. This is achieved by
investigating which Soergel modules admit a tensor decomposition. We also use
the N\'eron-Severi Lie algebra to provide an easy proof of the well-known fact
that a Schubert variety is rationally smooth if and only if its Betti numbers
satisfy Poincar\'e duality.
