W_n⁽²⁾ algebras

We construct W-algebra generalizations of the sℓ̂(2) algebra - W algebras W_n⁽²⁾ 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 W₃⁽²⁾ algebra. We define these algebras as a centralizer (commutant) of the U_qsℓ(n|1) quantum supergroup and explicitly find the generators in a factored, "Miura-like" form. Another construction of the W_n⁽²⁾ algebras is in terms of the coset sℓ̂(n|1)/sℓ̂(n). The relation between the two constructions involves the "duality" (k + n -1) (k′ + n - 1) = 1 between levels k and k′ of two sℓ̂(n) algebras.