Cosets of Bershadsky–Polyakov algebras and rational W -algebras of type A - Arakawa, Tomoyuki; Creutzig, Thomas; Linshaw, Andrew R. (Selecta Mathematica-New Series, 2017)
The Bershadsky–Polyakov algebra is the W-algebra associated to sl3 with its minimal nilpotent element fθ. For notational convenience we define Wℓ= Wℓ-3/2(sl3, fθ). The simple quotient of Wℓ is denoted by Wℓ, and for ℓ a positive integer, Wℓ is known to be C2-cofinite and rational. We prove that for all positive integers ℓ, Wℓ contains a rank one lattice vertex algebra VL, and that the coset Cℓ= Com (VL, Wℓ) is isomorphic to the principal, rational W(sl2ℓ) -algebra at level (2 ℓ+ 3) / (2 ℓ+ 1) - 2 ℓ. This was conjectured in the physics literature over 20 years ago. As a byproduct, we construct a new family of rational, C2-cofinite vertex superalgebras from Wℓ.
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Keywords: Coset construction; Vertex algebra; W-algebra