Formal Verlinde Module

2014/04/18 by Yanli Song, Song, Yanli
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #K-Theory and Homology (math.KT) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1404.4850

openalex publication_date 2014/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a compact, simple and simply connected Lie group and \A be an equivariant Dixmier-Douady bundle over G. For any fixed level k, we can define a G-C*-algebra C_\A^k+h(G) as all the continuous sections of the tensor power \A^k+h vanishing at infinity. A deep theorem by Freed-Hopkins-Teleman showed that the twisted K-homology KK^G(C_\A^k+h(G), \C) is isomorphic to the level k Verlinde ring R_k(G). By the construction of crossed product, we define a C*-algebra C^*(G,C_\A^k+h(G)). We show that the K-homology KK(C^*(G,C_\A^k+h(G)),\C) is isomorphic to the formal Verlinde module R^-∞(G) ⊗_R(G) R_k(G), where R^-∞(G) = Hom_\Z(R(G),\Z) is the completion of the representation ring.

Related

• Completing Verlinde Algebras
• Verlinde modules and quantization
• Geometric K-homology and the Freed-Hopkins-Teleman theorem
• The Verlinde algebra is twisted equivariant K-theory
• Lie algebra cohomology of the positive part of twisted affine Lie algebras