Roger and Yang's Kauffman bracket arc algebra is finitely generated (Q2809230)

The paper concerns the Kauffman bracket arc algebra of punctured surfaces, defined in [\textit{J. Roger} and \textit{T. Yang}, J. Differ. Geom. 96, No. 1, 95--140 (2014; Zbl 1290.53080)]. This algebra is a generalization of the classical Kauffman bracket skein algebra, where in addition to the usual Kauffman skein relations, there are also relations involving framed arcs with endpoints at the punctures. (The Kauffman bracket skein algebra is known to have an interpretation as a deformation quantization of the \(\text{SL}_2({\mathbb C})\) character variety of the fundamental group of the surface, and the Kauffman bracket arc algebra has a similar interpretation as a quantization of the decorated Teichmüller space.)NEWLINENEWLINEThe authors show that the arc algebra is finitely generated. Specifically, for a surface of genus \(g\) and with \(n\) punctures, the paper gives an explicit generating set, consisting of \(n(4^g-1)\) knots and \( \frac{n(n-1)}{2}4^g\) arcs. The proof builds on a result by \textit{D. Bullock} [Math. Z. 231, No. 1, 91--101 (1999; Zbl 0932.57016)], who gave a finite generating set for the Kauffman bracket skein algebra.