On the Structure of the Fusion Ideal

Abstract

Abstract:We prove that there is a finite level-independent bound on the number of relations defining the fusion ring of positive energy representations of the loop group of a simple, simply connected Lie group. As an illustration, we compute the fusion ring of G 2 at all levels.

Fusion Rings of Loop Group Representations

“…Here S denotes the complement of the collection S in the affine Dynkin diagram D. This lemma is established as part of the proof of Theorem 4.3 of [5]: first observe that the differential d c 0ij, b…”

“…Though we will briefly review the necessary notation and concepts, we are working throughout in the context established in our preceding paper [5]; the reader may want to refer there for background material and for more extensive exposition. Fix a simple, simply connected Lie group G, with Lie algebra g; unless indicated otherwise, all of the following items refer to this Lie algebra.…”

“…More concretely, a specific affine Steinberg basis can be described as follows. Let F k T denote the affine subspace of the dual torus t * fixed by the reflection group W k T ; this plane is spanned by the face of the level k alcove A k associated to the collection T ⊂ D. If the representation module R k [Z T ] is untwisted, then the plane F k T contains some element ξ of the weight lattice of G-see [5] for a more detailed discussion of the circumstances under which representation modules are twisted or untwisted. Choose Weyl chambers…”

“…By an elegant Gröbner basis analysis, Bouwknegt and Ridout [2] showed that for any group G the fusion ideal I k can be generated by a collection of representations whose size grows exponentially with the level k-these representations are associated to a set of dominant weights that collectively encloses the level k Weyl alcove within the Weyl chamber. In the paper [5], we used a spectral sequence for the twisted equivariant K-homology of the group G, imported along the Freed-Hopkins-Teleman [7,8] identification of this K-homology with the fusion ring, to prove that there exists a finite level-independent bound on the number of representations necessary to generate the fusion ideal. In this sequel, we extend the methods developed there to permit a complete computation: [0, 0, 0, 0, 1, 0, k−2], [0, 1, 0, 0, 0, 1, k−3], [0, 1, 0, 0, 1, 1, k−5], [0, 1, 0, 0, 0, 0, k−1], [0, 0, 0, 1, 0, 1, k−4], [0, 1, 0, 0, 1, 0, k−3], [0, 0, 1, 0, 0, 1, k−3], [0, 1, 0, 1, 0, 1, k−5], [0, 0, 0, 1, 0, 0, k−2], [0, 0, 0, 0, 0, 1, k−1], [1, 1, 0, 0, 0, 1, k−3], [0, 0, 1, 1, 0, 1, k−5], [0, 1, 0, 0, 0, 1, k−2], [1, 0, 0, 1, 0, 1, k−4], [0, 0, 0, 1, 0, 1, k−3], [1, 0, 1, 0, 1, 1, k−5], [0, 0, 1, 0, 0, 0, k−1], [0, 0, 1, 0, 1, 1, k−4], [1, 0, 0, 0, 1, 0, k−2], [1, 0, 0, 0, 1, 1, k−3],…”

“…The paper is organized as follows. In Section 2, we recall the computational methods developed in the preceding paper [5], particularly those concerning the resolution of the fusion ring coming from the twisted Mayer-Vietoris spectral sequence. We then prove our main lemma, that the fusion ideal is the induction image of the kernel of an affine induction map to the twisted representation module of the centralizer of any one affine vertex of the Weyl alcove; for brevity we refer to this kernel as the "fusion kernel".…”