On the Extended W-Algebra of Type sl₂ at Positive Rational Level

The extended W-algebra of type sl₂ at positive rational level, denoted byM_p+,p-, is a vertex operator algebra (VOA) that was originally proposed in [16]. This VOA is an extension of the minimal model VOA and plays the role of symmetry algebra for certain logarithmic conformal field theories. We give a construction of M_p+,p- in terms of screening operators and use this construction to prove that M_p+,p- satisfies Zhu's c₂-cofiniteness condition, calculate the structure of the zero mode algebra (also known as Zhu's algebra), and classify all simpleM_p+,p--modules.