THE SUPER W1+∞ ALGEBRA WITH INTEGRAL CENTRAL CHARGE

The Lie superalgebra SD of regular differential operators on the super circle has a universal central extension (SD) over cap. For each c is an element of C, the vacuum module M-c((SD) over cap) of central charge c admits a vertex superalgebra structure, and M-c((SD) over cap) congruent to M-c((SD) over cap). The irreducible quotient V-c((SD) over cap) of the vacuum module is known as the super W1+infinity algebra. We show that for each integer n > 0, V-n((SD) over cap) has a minimal strong generating set consisting of 4n fields, and we identify it with aW-algebra associated to the purely odd simple root system of gl(n vertical bar n). Finally, we realize V-n((SD) over cap) as the limit of a family of commutant vertex algebras that generically have the same graded character and possess a minimal strong generating set of the same cardinality.