BMW algebra in a fixed Kauffman skein convention

Let $R$ be a commutative unital ring, let $q,\rho \in R^{\times }$, put $z=q-q^{-1}$, and choose $\delta \in R$ with $z(\delta -1)=\rho -{\rho }^{-1}$. For $n\ge 2$, the BMW algebra $BM{W}_n(R;q,\rho ,\delta )$ is the unital associative $R$-algebra generated by invertible $g_i$ and elements $e_i$ for $1\le i<n$, subject to the following relations (an adjacent relation is imposed only when both indices exist):

• braid and distant braid: $g_i{g}_{i+1}{g}_i=g_{i+1}{g}_i{g}_{i+1}$ and $g_i{g}_j=g_j{g}_i$ for $\mathrm{\mid }i-j\mathrm{\mid }\ge 2$;
• distant cups and caps: $g_i{e}_j=e_j{g}_i$ and $e_i{e}_j=e_j{e}_i$ for $\mathrm{\mid }i-j\mathrm{\mid }\ge 2$;
• adjacent cup-cap: $e_i{e}_{i\pm 1}{e}_i=e_i$, $g_i{g}_{i\pm 1}{e}_i=e_{i\pm 1}{e}_i$, and $e_i{g}_{i\pm 1}{g}_i=e_i{e}_{i\pm 1}$;
• Kauffman skein and untwisting: $g_i-g_i^{-1}=z(1-e_i)$, $g_i{e}_i=e_i{g}_i={\rho }^{-1}{e}_i$, and $e_i{g}_{i\pm 1}{e}_i=\rho e_i$;
• loop: $e_i^2=\delta e_i$.

For the boundary ranks, set $BM{W}_0(R;q,\rho ,\delta )=BM{W}_1(R;q,\rho ,\delta )=R$, with the empty and single straight tangle as their respective units.

If $z$ is invertible, the parameter relation reads $\delta =1+(\rho -{\rho }^{-1})\mathrm{/}z$. If $z$ is not invertible, the presentation uses the universal coefficient relation $z(\delta -1)=\rho -{\rho }^{-1}$ and does not divide by $z$. The quotient by the two-sided ideal generated by all $e_i$ is the cup-cap-zero Hecke quotient; this definition does not assert a BMW diagram-basis theorem or a BMW quantum-centralizer theorem.