Sigma involutions associated with parafermion vertex operator algebra K(sl2, k)

An irreducible module for the parafermion vertex operator algebra K(sl(2),k) is said to be of sigma-type if an automorphism of the fusion algebra of K(sl(2),k) of order k is trivial on it. For any integer k >= 3, we show that there exists an automorphism of order 2 of the subalgebra of the fusion algebra of K(sl(2),k)(<theta >) spanned by the irreducible direct summands of sigma-type irreducible K(sl(2),k)-modules, where theta is an involution of K(sl(2), k). We discuss some examples of such an automorphism as well.