Duality for Differential-Difference Systems over Lie Groups

In modern mathematical systems theory, there exist two consistent ways of defining and describing a linear system: (i) in the behavioral approach, a linear system is the kernel $\mathfrak{B}$ of a matrix-valued operator $R$ in a power of a signal space $W$; (ii) in the module-theoretic setting, a linear system is the cokernel $M$ of the above matrix $R$. These two formulations have connections. The minimal conditions under which they are equivalent are investigated in this paper. The general theory is applied to differential-difference systems over Lie groups.