Duffin-Kemmer algebras revisited
O. A. Sánchez-Valenzuela, R E Zuazua-Vega · Journal of Physics A Mathematical and General · 1993
Duffin-Kemmer algebras are studied from a modern perspective. Complete descriptions of these algebras and their simple modules are given in terms of tensor and exterior algebras. The approach is self-contained and no reference to general results on Jordan algebras and their representation theory is required. Absolute detail is provided for me more specific examples of the Duffin-Kemmer real algebras D(q + 2, q) (for q = 0, 1, and 2) which are relevant for applications in physics. A faithful representation of D(q + 2, q) is given in the space of real 2 q+1 * 2 q+1 matrices; it is completely reducible and yields with multiplicity one all the irreducible representations of D(q + 2, q). The representation space has a natural orthogonal structure, (.,.) q of signature (2 2q+1 ,2 2q+1 ), for 9 > 0,and (4, 0), for q = D. It corresponds to the bilinear form induced by the spin group, Spin( q + 2, q), on the tensor product space, W(q + 2, y) (X) W(q + 2, q), of two copies of the fundamental module of the Clifford algebra, C(q +2, q). Explicit computations are made simple by establishing a one-to-one correspondence with the space of 2 q * 2 q matrices with quaternion coefficients.