Hamiltonian Reduction and Classical Extended Superconformal Algebras

We present a systematic construction of classical extended superconformal algebras from the hamiltonian reduction of a class of affine Lie superalgebras, which include an even subalgebra $sl(2)$. In particular, we obtain the doubly extended $N=4$ superconformal algebra $\tilde{A}_{\gamma}$ from the hamiltonian reduction of the exceptional Lie superalgebra $D(2|1;\gamma/(1-\gamma))$. We also find the Miura transformation for these algebras and give the free field representation. A $W$-algebraic generalization is discussed.