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An irreducible module for the parafermion vertex operator algebra$K(\mathfrak{sl}_2,k)$ is said to be of $\sigma$-type if an automorphism of thefusion algebra of $K(\mathfrak{sl}_2,k)$ of order $k$ is trivial on it. For anyinteger $k \ge 3$, we show that there exists an automorphism of order $2$ ofthe subalgebra of the fusion algebra of $K(\mathfrak{sl}_2,k)^{\langle \theta\rangle}$ spanned by the irreducible direct summands of $\sigma$-typeirreducible $K(\mathfrak{sl}_2,k)$-modules, where $\theta$ is an involution of$K(\mathfrak{sl}_2,k)$. We discuss some examples of such an automorphism aswell.

Sigma involutions associated with parafermion vertex operator algebra $K(\mathfrak{sl}_2,k)$