Representation type of finite quiver Hecke algebras of type \(D^{(2)}_{\ell +1}\) (Q2790589)

Let \(I=\left\{0,1,\ldots, l \right\}\) be an index set. Let (\texttt{A}, \texttt{P}, \(\Pi, \Pi^{\vee})\) be the affine Cartan datum of type \(D^{(2)}_{l+1}\). Fix a certain choice of polynomials \(Q_{i,j}(u,v) \in \mathbf k[u,v]\) for \(i,j \in I\). The cyclotomic quiver Hecke algebra \(R^{\Lambda}(n)\) is a \(\mathbb Z\)-graded \(\mathbf k\)-algebra with generators labeled by \(\left\{ e(\nu) : \nu \in I^n \right\}\), \(\left\{ x_k: 1 \leq k \leq n \right\}\), \(\left\{ \psi_l: 1 \leq l \leq n-1 \right\}\) subject to certain relations depending on the fundamental integral weight \(\Lambda\). This algebra is known to be a finite-dimensional self-injective algebra and is related to the categorification of integrable highest weight modules over the Kac-Moody Lie algebras.NEWLINENEWLINEWhen \(\beta\) is in the positive cone \texttt{Q}\(^+\) of the root lattice, it define a central idempotent \(e(\beta)\). Let \(R^{\Lambda}(\beta)=R^{\Lambda}(n)e(\beta)\). The paper under review studies the case when \(\Lambda=\Lambda_0\), where \(\Lambda_0\) is the fundamental weight. The algebra \(R^{\Lambda_0}(\beta)\) is called the finite quiver Hecke algebra.NEWLINENEWLINEThe goal of the paper is to study the representation type for \(R^{\Lambda_0}(\beta)\) and establish an Erdmann-Nakano type theorem analogous to the one in classical Hecke algebra of symmetric groups. Let \(\delta\) be the null root, which is the sum of all simple roots. The main result of the paper is as follows:NEWLINENEWLINETheorem. For \(\kappa \in W\Lambda_0\) and \(k \in \mathbb Z_{\geq 0}\), \(R^{\Lambda_0}(\Lambda_0-\kappa+k\delta)\) is {\parindent=0.7cm \begin{itemize}\item[1.] simple if \(k=0\), \item[2.] of finite representation type but not semisimple if \(k=1\), \item[3.] of tame representation type if \(k=2\); \item[4.] of wild representation type if \(k \geq 3\). NEWLINENEWLINE\end{itemize}}