Gröbner-Shirshov basis of derived Hall algebra of type $\boldsymbol~B_{\boldsymbol2}$

Abstract

In the Ringel-Hall algebra of Dynkin type, the set of all skew commutator relations among the iso-classes of indecomposable modules forms aminimal Gröbner-Shirshov basis and the corresponding irreducible elements form a PBW (Poincaré-Birkhoff-Witt) type basis of the Ringel-Hall algebra. We aim to generalize this result to the derived Hall algebra $DH(B_2)$. First, we compute all the skew commutator relations among the iso-classes of indecomposable objects in the bounded derived category $D^b(B_2)$, and then we prove that all the possible compositions among these skew commutator relations are trivial. Finally, we construct a PBW type basis of $DH(B_2)$.