Representation type of finite quiver Hecke algebras of type $A^{(1)}_{\ell}$ for arbitrary parameters

TL;DR

This paper extends the Erdmann-Nakano theorem to finite quiver Hecke algebras of affine type A^{(1)}_{ ext{l}}, demonstrating that their representation type is parameter-independent and characterizing tame cases as biserial algebras.

Contribution

It generalizes the Erdmann-Nakano theorem to all parameters for these algebras, showing parameter independence of their representation type.

Findings

Representation type does not depend on the parameter.

Finite quiver Hecke algebras of nonzero parameter are biserial if of tame type.

Theorem applies to all parameter values, not just special cases.

Abstract

We give Erdmann-Nakano type theorem for the finite quiver Hecke algebras $R^{\Lambda_0}(\beta)$ of affine type $A^{(1)}_{\ell}$. Note that each finite quiver Hecke algebra lies in one parameter family, and the original Erdmann-Nakano theorem studied the finite quiver Hecke algebra at a special parameter value. We study the general case in our paper. Our result shows in particular that their representation type does not depend on the parameter. Moreover, when the parameter value is nonzero, we show that finite quiver Hecke algebras of tame representation type are biserial algebras.