The Hall-normalized extended positive Borel of a loopless quiver

Let $Q$ be a finite loopless quiver over ${\mathbb{F}}_q$. Write $c_{ij}$ for the number of arrows $i\to j$, set $\nu =\sqrt{q}>1$, and define the symmetric Cartan matrix $a_{ij}=2{\delta }_{ij}-c_{ij}-c_{ji}\mathrm{.}$ Let $\mathbb{Q}(\nu )$ be the coefficient field. Define the extended positive Borel algebra $B_Q^{+}$ to be the $\mathbb{Q}(\nu )$-algebra generated by $E_i$ and invertible commuting $K_i^{\pm 1}$, with $K_i{K}_i^{-1}=1,K_i{E}_j{K}_i^{-1}={\nu }^{a_{ij}}{E}_j,$ and, for $i\mathrm{\ne }j$ and $m=1-a_{ij}$, ${\sum }_{r=0}^m(-1{)}^r{\left(\genfrac{}{}{0px}{}{m}{r}\right)}_{\nu }{E}_i^{m-r}{E}_j{E}_i^r=0.$ The Gaussian coefficients use the symmetric convention of Quantum integers, factorials, Gaussian binomials and divided powers at $q_i$. The formulas $\mathrm{\Delta }(E_i)=E_i\otimes K_i^{-1}+1\otimes E_i,\mathrm{\Delta }(K_i)=K_i\otimes K_i,$ are the coproduct convention intended for this presentation; their well-definedness on the defining relations is checked using The coproduct preserves the positive and negative quantum Serre ideals. The positive half is the subalgebra generated by the $E_i$ alone; adjoining the $K_i^{\pm 1}$ gives the extended positive Borel.