The Euler-twisted dimension-vector algebra of a quiver

For a finite quiver $Q$, let $R$ and the invertible class $\mathbb{L}$ be as in Relative motivic stack-function groups for quiver representation stacks. For dimension vectors $\mathbf{d},\mathbf{e}\in {\mathbb{N}}^{Q_0}$ define ${\chi }_Q(\mathbf{d},\mathbf{e})={\sum }_{i\in Q_0}{d}_i{e}_i-{\sum }_{a\in Q_1}{d}_{s(a)}{e}_{t(a)}\mathrm{.}$This is the Euler pairing for the fixed quiver representation convention: the first argument is the quotient dimension and the second is the subrepresentation dimension. It is bilinear by its displayed finite-sum formula. Let $T_Q={⨁}_{\mathbf{d}\in {\mathbb{N}}^{Q_0}}R{x}^{\mathbf{d}}$ be the free $R$-module on formal symbols $x^{\mathbf{d}}$. Equip it with the bilinear multiplication$x^{\mathbf{d}}{x}^{\mathbf{e}}={\mathbb{L}}^{-{\chi }_Q(\mathbf{d},\mathbf{e})}{x}^{\mathbf{d}+\mathbf{e}}\mathrm{.}$ The subsequent theorem verifies associativity and the unit $x^{\mathbf{0}}$. This is the Euler-twisted dimension-vector monoid algebra used as the target of the stack integration map; the formula fixes its twist convention.