Construct a good basis for the SL_n-skein algebra when n>3

Construct a good, computationally tractable basis for the ${\rm SL}_n$-skein algebra $_{\bar\omega}(S)$ for $n>3$ (e.g., a spanning set with manageable multiplication and compatibility with the quantum trace), in order to enable computation of its center via the quantum trace map ${\rm tr}_\lambda$ and the center of the balanced Fock–Goncharov algebra $\mathcal{Z}_{\hat\omega}^{\rm bl}(S,\lambda)$.

Background

The paper develops tools to compute the center and rank of the balanced Fock–Goncharov algebra and uses these to paper representations and connections to the ${\rm SL}_n$-skein algebra. For $n=2,3$, known bases and the quantum trace provide effective methods to analyze the center of the skein algebra.

For $n>3$, the authors emphasize that the lack of a suitable basis impedes transferring their results to the skein setting. A good basis would allow computing the center of the ${\rm SL}_n$-skein algebra via the quantum trace map and the balanced Fock–Goncharov algebra, mirroring the $n=2,3$ approach.