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We study properties of the $\alpha$-Green kernel $g_D^\alpha$ of order$0<\alpha\leqslant2$ for a domain $D\subset\mathbb R^n$, $n\geqslant3$. Thiskernel is associated with the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$,$x,y\in\mathbb R^n$, in a manner particularly well known in the case$\alpha=2$. Besides the usual principles of potential theory, we establish forthe $\alpha$-Green kernel the property of consistency. This allows us to provethe completeness of the cone of positive measures $\mu$ on $D$ with finiteenergy $g_D^\alpha(\mu,\mu):=\iint g_D^\alpha(x,y)\,d\mu(x)\,d\mu(y)$ in thetopology defined by the energy norm$\|\mu\|_{g_D^\alpha}=\sqrt{g_D^\alpha(\mu,\mu)}$, as well as the existence ofthe $\alpha$-Green equilibrium measure for a relatively closed set in $D$ offinite $\alpha$-Green capacity. The main tool is a generalization of Cartan'stheory of balayage (sweeping) for the Newtonian kernel to the $\alpha$-Rieszkernels with $0<\alpha<2$.

Balayage for Riesz kernels with application to potential theory for the associated Green kernels