On realizations and representations of simple Lie algebra s l(2, C)

We consider realizations and matrix representations of the simple threedimensional Lie algebra s l(2, C) over the field of complex numbers. Using a direct method based on solving systems of first-order partial differential equations derived from the algebra’s commutation relations, we find all faithful realizations up to equivalence. The obtained exhaustive list of realizations is compared with the one over the real field. The explicit correspondence between two realizations involving nonlinear coefficients in two variables and two matrix representations is established. We also present an example of a realization that cannot be linearized. Some possible applications of realizations and further investigations are pointed out.