We construct several quantum coset [Formula: see text] algebras, e.g. [Formula: see text] and [Formula: see text] and argue that they are finitely nonfreely generated. Furthermore, we discuss in detail their role as unifying [Formula: see text] algebras of Casimir [Formula: see text] algebras. We show that it is possible to give coset realizations of various types of unifying [Formula: see text] algebras; for example, the diagonal cosets based on the symplectic Lie algebras sp (2n) realize the unifying [Formula: see text] algebras which have previously been introduced as [Formula: see text]. In addition, minimal models of [Formula: see text] are studied. The coset realizations provide a generalization of level-rank duality of dual coset pairs. As further examples of finitely nonfreely generated quantum [Formula: see text] algebras, we discuss orbifolding of [Formula: see text] 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 [Formula: see text] algebras probably yields infinitely nonfreely generated classical [Formula: see text] algebras.

No abstract

We show that that the Jacobi-identities for a W-algebra with primary fields of dimensions 3, 4 and 5 allow two different solutions. The first solution can be identified with WA 4 . The second is special in the sense that, even though associative for general value of the central charge, null-fields appear that violate some of the Jacobi-identities, a fact that is usually linked to exceptional W-algebras. In contrast we find for the algebra that has an additional spin 6 field only the solution WA 5 .

No abstract

We show that the structure constants of W-algebras can be grouped according to the lowest (bosonic) spin(s) of the algebra. The structure constants in each group are described by a unique formula, depending on a functional parameter h(c) that is characteristic for each algebra. As examples we give the structure constants C 4 33 and C 4 44 for the algebras of type W(2, 3, 4, . . .) (that include the WA n−1 -algebras) and the structure constant C 4 44 for the algebras of type W(2, 4, . . .), especially for all the algebras WD n , WB(0, n), WB n and WC n . It also includes the bosonic projection of the super-Virasoro algebra and a yet unexplained algebra of type W(2, 4, 6) found previously.