FUSION RULES AND COMPLETE REDUCIBILITY OF CERTAIN MODULES FOR AFFINE LIE ALGEBRAS

Abstract

We develop a new method for obtaining branching rules for affine Kac–Moody Lie algebras at negative integer levels. This method uses fusion rules for vertex operator algebras of affine type. We prove that an infinite family of ordinary modules for affine vertex algebra of type A investigated in our previous paper J. Algebra319 (2008) 2434–2450, is closed under fusion. Then, we apply these fusion rules on explicit bosonic realization of level -1 modules for the affine Lie algebra of type [Formula: see text], obtain a new proof of complete reducibility for these representations, and the corresponding decomposition for ℓ ≥ 3. We also obtain the complete reducibility of the associated level -1 modules for affine Lie algebra of type [Formula: see text]. Next, we notice that the category of [Formula: see text] modules at level -2ℓ + 3 has the isomorphic fusion algebra. This enables us to decompose certain [Formula: see text] and [Formula: see text]-modules at negative levels.