  We investigate whether there are unitary families of W-algebras with spin one
fields in the natural example of the Feigin-Semikhatov W^(2)_n-algebra. This
algebra is conjecturally a quantum Hamiltonian reduction corresponding to a
non-principal nilpotent element. We conjecture that this algebra admits a
unitary real form for even n. Our main result is that this conjecture is
consistent with the known part of the operator product algebra, and especially
it is true for n=2 and n=4. Moreover, we find certain ranges of allowed levels
where a positive definite inner product is possible. We also find a unitary
conformal field theory for every even n at the special level k+n=(n+1)/(n-1).
At these points, the W^(2)_n-algebra is nothing but a compactified free boson.
This family of W-algebras admits an 't Hooft limit that is similar to the
original minimal model 't Hooft limit. Further, in the case of n=4, we
reproduce the algebra from the higher spin gravity point of view. In general,
gravity computations allow us to reproduce some leading coefficients of the
operator product.
