On the commutant of the principal subalgebra in the A1 lattice vertex algebra

The coset (commutant) construction is a fundamental tool to construct vertex operator algebras from known vertex operator algebras. The aim of this paper is to provide a fundamental example of the commutants of vertex algebras outside vertex operator algebras. Namely, the commutant C of the principal subalgebra W of the A(1) lattice vertex operator algebra V-A1 is investigated. An explicit minimal set of generators of C, which consists of infinitely many elements and strongly generates C, is introduced. It implies that the algebra C is not finitely generated. Furthermore, Zhu's Poisson algebra of C is shown to be isomorphic to an infinite-dimensional algebra C[x(1),x(2),& mldr;]/(x(i)x(j)|i,j=1,2,& mldr;). In particular, the associated variety of C consists of a point. Moreover, W and C are verified to form a dual pair in V-A1.