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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_{33}^4 and C_{44}^4 for the algebras of type W(2,3,4,...) (that include the WA_{n-1}-algebras) and the structure constant C_{44}^4 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.
In a recent paper, the authors have shown that the secondary reduction of W-algebras provides a natural framework for the linearization of W-algebras. In particular, it allows in a very simple way the calculation of the linear algebra $W(G,H)_{geq0}$
We show that quantum Casimir W-algebras truncate at degenerate values of the central charge c to a smaller algebra if the rank is high enough: Choosing a suitable parametrization of the central charge in terms of the rank of the underlying simple Lie
Recently it has been discovered that the W-algebras (orbifold of) WD_n can be defined even for negative integers n by an analytic continuation of their coupling constants. In this letter we shall argue that also the algebras WA_{-n-1} can be defined
In this note, using Nekrasovs gauge origami framework, we study two differe
We construct several quantum coset W-algebras, e.g. sl(2,R)/U(1) and sl(2,R)+sl(2,R) / sl(2,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 t