  These notes are an extended version of the talks given by the authors at the
XIV International Workshop on Lie Theory and Its Applications in Physics,
Sofia, Bulgaria, 20-26 June 2021. The concise version published in the
proceedings of the workshop contains additional discussions for the q-deformed
scenario: https://link.springer.com/chapter/10.1007/978-981-19-4751-3_23
  In these notes we identify KLT kernel with the Shapovalov form on Verma
module with its highest/lowest weight given by the reference momentum and rest
of the momenta as roots. We then take a step forward and show how the Feynman
diagrams emerge naturally as the Shapovalov duals of the Verma module basis
vectors. We show such algebraic construct offers a compact expression for the
BCJ numerators. Explicit examples are shown for nonlinear sigma model and HEFT
pre-numerators.
