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Fusion rules and complete reducibility of certain modules for affine Lie algebras

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 {; ; ; \it J. Algebra}; ; ; {; ; ; \bf 319}; ; ; (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 $A_{; ; ; \ell- 1}; ; ; ^{; ; ; (1)}; ; ; $, obtain a new proof of complete reducibility for these representations, and the corresponding decomposition for $\ell \ge 3$. We also obtain the complete reducibility of the associated level $-1$ modules for affine Lie algebra of type $C_{; ; ; \ell}; ; ; ^{; ; ; (1)}; ; ; $. Next we notice that the category of $D_{; ; ; 2 \ell -1}; ; ; ^{; ; ; (1)}; ; ; $ modules at level $- 2 \ell +3 $ has the isomorphic fusion algebra. This enables us to decompose certain $E_6 ^{; ; ; (1)}; ; ; $ and $F_4 ^{; ; ; (1)}; ; ; $--modules at negative levels.

vertex operator algebra; affine Kac-Moody algebra; fusion rules; bosonic realization

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