COSET REALIZATION OF UNIFYING ${\mathcal W}$ ALGEBRAS

Abstract

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.

The Structure of Parafermion Vertex Operator Algebras: General Case

“…The building block of the Kac-Moody Lie algebras is the 3-dimensional simple Lie algebra sl 2 associated to any real root. The generator results for the parafermion vertex operator algebras given in this paper and [6] show that the parafermion vertex operator algebras associated to the affine Lie algebra A (1) 1 are also the building block of general parafermion vertex operator algebras. We hope this fact will be important in the future study of the representation theory for the parafermion vertex operator algebra.…”

“…Then P α is isomorphic to the W -algebra W (2, 3, 4, 5) [1] by [6,Theorem 3.1] with k replaced by k α , i.e., P α is isomorphic to N (sl 2 , k α ).…”

The higher spin rectangle

“…1 However, the above analysis does capture the maximal set of independent bilinear generators that can be constructed out of N bosons subject to the symmetrization condition. In actual fact, a representation of the W e ∞ algebra at finite central charge will truncate to a representation of a smaller algebra because of the presence of null fields [40,41].…”

Bailey flows and Bose-Fermi identities for the conformal coset models (A1(1))N × (A1(1))N′/(A1(1))N+N′

“…This paved the way for the study of the coset models (1.3) with one fractional level [38] and two fractional levels [39]. Further aspects of coset (1.3), including its spin content, have been studied in [40] in the context of unifying W-algebras.…”