The KLR algebras in root degrees ${\alpha }_i$ and $2{\alpha }_i$

(a) In weight ${\alpha }_i$ the only sequence is $(i)$, and there are no crossings, so the presentation reduces to the single generator $x_1$ with relation (2) $x_{1}e(i)=e(i)x_1$; hence $R({\alpha }_i)=k[x_1]e(i)$ with basis $\{x_1^a\}$, $\mathrm{deg}{x}_1=2d_i$ by The KLR grading and the diagrammatic multiplication, and $\mathrm{g}\mathrm{d}\mathrm{i}\mathrm{m}R({\alpha }_i)={\sum }_{a\ge 0}{q}^{2d_{i}a}=(1-q^{2d_i}{)}^{-1}$. A graded simple is cyclic and bounded below; the positive-degree central $x_1$ cannot act bijectively on it, so simplicity forces $x_{1}S=0$. Thus its unique graded simple is $k=R({\alpha }_i)\mathrm{/}(x_1)$ up to shift; $R({\alpha }_i)$ is free of rank one over itself, hence projective, indecomposable (its endomorphism ring is $k[x_1]$, whose only idempotents are $0,1$), and its quotient onto $k$ is a graded projective cover: if a graded submodule $U$ satisfies $U+(x_1)=R$, comparison in degree zero gives $1\in U$, hence $U=R$; the pairing value is the normalization of The KLR Cartan (graded Hom) pairing on the projective Grothendieck group: $P_i^{\psi }{\otimes }_{R({\alpha }_i)}{P}_i\cong k[x_1]$ as a graded space.

(b) In weight $2{\alpha }_i$ there is one sequence $(ii)$ and one crossing ${\tau }_1$, with relations (4) and (6) specialized to $Q_{ii}=0$: ${\tau }_1^2=0$ and, on the equal-colour component, ${\tau }_1{x}_1=x_2{\tau }_1+e(ii)$, ${\tau }_1{x}_2=x_1{\tau }_1-e(ii)$. The PBW basis [F1] is $\{x_1^a{x}_2^b,x_1^a{x}_2^b{\tau }_1{\}}_{a,b\ge 0}$, which is free of rank $2$ over $k[x_1,x_2]$; the graded dimension is the sum of the two geometric series $\frac{1}{(1-q^{2d_i}{)}^2}+\frac{q^{-2d_i}}{(1-q^{2d_i}{)}^2}=\frac{1+q^{-2d_i}}{(1-q^{2d_i}{)}^2}$ because ${\tau }_1$ has degree $-b_{ii}=-2d_i$. For the idempotent: $e_{i,2}^2=x_1{\tau }_1{x}_1{\tau }_1=x_1(x_2{\tau }_1+e(ii)){\tau }_1=x_1{x}_2{\tau }_1^2+x_1{\tau }_1=x_1{\tau }_1=e_{i,2}$ using ${\tau }_1^2=0$, and $e_{i,2}\mathrm{\ne }0$ by the PBW basis; its degree is $2d_i+(-2d_i)=0$.

(b, continued) By [F2] with $n=2$ the element $e_{i,2}=x_1{\tau }_1$ is the divided-power idempotent, $P_{i,2}=R(2{\alpha }_i)\psi (e_{i,2})\{-d_i\}$ is the unique indecomposable projective summand up to shift, and $[P_{i,2}]=\gamma ({\theta }_i^{(2)})$; the shift here is $-\left(\genfrac{}{}{0px}{}{2}{2}\right)d_i=-d_i$.