Lattice vertex algebras and combinatorial bases: general case and W-algebras
Antun Milas, Michael Penn · MPG.PuRe (Max Planck Society) · 2012
We introduce what we call the principal subalgebra of a lattice vertex (super) algebra associated to an arbitrary Z-basis of the lattice.In the first part (to appear), the second author considered the case of positive bases and found a description of the principal subalgebra in terms of generators and relations.Here, in the most general case, we obtain a combinatorial basis of the principal subalgebra WL and of related modules.In particular, we substantially generalize several results in Georgiev, 1996, covering the case of the root lattice of type An, as well as some results from Calinescu, Lepowsky and Milas, 2010.We also discuss principal subalgebras inside certain extensions of affine W-algebras coming from multiples of the root lattice of type An.Contents 1. Introduction 621 2. The setting 623 3. Rank one subspaces 625 4. Higher rank subspaces 631 5. Graded dimensions 640 6. W-algebras and principal subspaces 642 References 648