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For the cyclic group $\mathbb{Z}_3$ and positive integer $k$, we study therepresentations of the orbifold vertex operator algebra$L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$. All the irreduciblemodules for $L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$ are classifiedand constructed explicitly. Quantum dimensions and fusion rules for theorbifold vertex operator algebra$L_{\widehat{\mathfrak{sl}_2}}(k,0)^{\mathbb{Z}_3}$ are completely determined.

Representations and fusion rules for the orbifold vertex operator algebras $L_{\widehat{\frak{sl}_2}}(k,0)^{\mathbb{Z}_3}$