A commutant realization of Odake's algebra
Abstract
The bcβγ-system W of rank 3 has an action of the affine vertex algebra V0(sl2), and the commutant vertex algebra C =Com(V0(sl2), W) contains copies of V-3/2(sl2) and Odake's algebra O. Odake's algebra is an extension of the N=2 superconformal algebra with c=9, and is generated by eight fields which close nonlinearly under operator product expansions. Our main result is that V-3/2(sl2) and O form a Howe pair (i.e., a pair of mutual commutants) inside C. More generally, any finite-dimensional representation of a Lie algebra g gives rise to a similar Howe pair, and this example corresponds to the adjoint representation of sl2.
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