Completing Verlinde Algebras

We compute the completion of the Verlinde algebra of a simply connected simple compact Lie group $G$ at the augmentation ideal of the representation ring. By results of Freed, Hopkins, Teleman and C.Dwyer and Lahtinen, this gives a computation of (non-equivariant) twisted $K$-theory of the free loop space of $BG$.

💡 Research Summary

The paper “Completing Verlinde Algebras” addresses a precise algebraic problem that has far‑reaching implications in topology and mathematical physics. For a simply‑connected, simple compact Lie group (G), the Verlinde algebra (V_k(G)) encodes the fusion rules of level‑(k) positive energy representations of the loop group (LG). It is naturally a module over the representation ring (R(G)). The authors consider the augmentation ideal (I\subset R(G)) (the kernel of the rank homomorphism) and study the (I)‑adic completion \