Abstract: Preparation of highly reactive isobutene homo- or copolymers having Mn=from 500 to 45 000 by polymerizing isobutene in the liquid phase in the presence of a dissolved, dispersed or supported boron-containing catalyst complex, by using, as the catalyst complex, a protic acid compound I [H+]m+1[R1R2R3B—(-Am+—BR5R6—)n—R4](m+1)?.Lx ??(I) R1 to R6 each independently aliphatic, heterocyclic or aromatic, fluorinated hydrocarbon radicals having in each case from 1 to 18 carbon atoms, or silyl groups comprising C1 to C18 hydrocarbon radicals, A nitrogen-containing bridging member which forms covalent bonds to the boron atoms via its nitrogen atoms, L neutral solvent molecules, n=0 or 1, m=0 or 1 and x is ?0.