The Lie algebra s o (N) and the Duffin-Kemmer-Petiau ring

An explicit expression is given for the unit element E of the ring generated by the Duffin-Kemmer-Petiau (DKP) operators βμ. The relation of E to the unit operator I (unit matrix in a matrix representation) is also derived. It is pointed out that one must be careful to distinguish E from I. Bhabha's observation that one may use the irreducible representations (irreps) of the Lie algebra s o (5) to obtain the irreps of the Dirac, DKP, and other algebras is given a concise and general setting in terms of a relation between the Lie algebra s o (n + 1) and a family of semisimple operator rings. We emphasize that for the case n + 1 = 5 this means that there is an underlying relationship between the physical DKP and Dirac algebras and wave equations.