Coset Realization of Unifying { w} Algebras

We construct several quantum coset { W} algebras, e.g. {widehat {sl(2,Bbb R)}}/{widehat {U( 1 ; )}}} and {widehat {sl(2,Bbb R)}} ⊕ {widehat {sl(2,Bbb R)}}/{widehat {sl(2,Bbb R)}} and argue that they are finitely nonfreely generated. Furthermore, we discuss in detail their role as unifying { W} algebras of Casimir { W} algebras. We show that it is possible to give coset realizations of various types of unifying { W} algebras; for example, the diagonal cosets based on the symplectic Lie algebras sp(2n) realize the unifying { W} algebras which have previously been introduced as {WD}_n. In addition, minimal models of {WD}_n are studied. The coset realizations provide a generalization of level-rank duality of dual coset pairs. As further examples of finitely nonfreely generated quantum { W} algebras, we discuss orbifolding of {W} algebras which on the quantum level has different properties than in the classical case. We demonstrate through some examples that the classical limit — according to Bowcock and Watts — of these finitely nonfreely generated quantum {W} algebras probably yields infinitely nonfreely generated classical {W} algebras.