$W_n^{(2)}$ algebras

We construct W-algebra generalizations of the ^sl(2) algebra – W-algebras W^{(2)}_n generated by two currents E and F with the highest pole of order n in their OPE. The n=3 term in this series is the Bershadsky–Polyakov algebra. We define these algebras as a centralizer (commutant) of the $U_{q}sl(n\mathrm{\mid }1)$ super quantum group and explicitly find the generators in a factored, Miura-like'' form. Another construction of W^{(2)}_n is in terms of the coset ^sl(n|1)/^sl(n). The relation between the two constructions involves the duality’’ (k+n-1)(k’+n-1)=1 between levels k and k’ of two ^sl(n) algebras.