Characteristics of pairs of operators, Lie hybrids, Poisson brackets and nonlinear geometric algebra

The article is devoted to various algebraic, geometric and geometroalgebraic structures, which appear in the context of the problem of description of pairs of linear operators. The relations between the problem and investigations of I. Batalin, A. Weinstein, M. V. Karasev and V. P. Maslov on the analogs of Lie theory for nonlinear Poisson brackets, L. V. Sabinin's program of the nonlinear geometric algebra and the infinite-dimensional symplectic geometry are explicated.