Triangular decomposition of SL3 skein algebras

We give an SL3 analogue of the triangular decomposition of the Kauffman bracket stated skein algebras described by Le. To any punctured bordered surface, we associate an SL3 stated skein algebra which contains the SL3 skein algebra of closed webs. These algebras admit natural algebra morphisms associated to the splitting of surfaces along ideal arcs. We give an explicit basis for the SL3 stated skein algebra and show that the splitting morphisms are injective and describe their images. By splitting a surface along the edges of an ideal triangulation, we see that the SL3 stated skein algebra of any ideal triangulable surface embeds into a tensor product of stated skein algebras of triangles. As applications, we prove that the stated skein algebras do not have zero divisors, we construct Frobenius maps at roots of unity, and we obtain a new proof that Kuperberg's web relations generate the kernel of the Reshetikhin-Turaev functor.