Triangular decomposition of ${\mathrm{SL}}_3$ skein algebras

Abstract

We give an ${\mathrm{SL}}_3$ analogue of the triangular decomposition of the Kauffman bracket stated skein algebras described by Lê. To any punctured bordered surface, we associate an ${\mathrm{SL}}_3$ stated skein algebra which contains the ${\mathrm{SL}}_3$ 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 ${\mathrm{SL}}_3$ 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 ${\mathrm{SL}}_3$ 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.