2007/08/29 by Ralph M. Kaufmann, RALPH M. KAUFMANN, David Pham +1
Mathematics · #Algebraic structures and combinatorial models #Finite Group Theory Research #Homotopy and Cohomology in Algebraic Topology #math.AG #math.KT #math.QA
paper · pdf · doi:10.1142/s0129167x09005431
published as Int.J.Math.20:623-657,2009 · 35 pages, no figures
arxiv created 2007/08/29 · openalex publication_date 2009/05/01 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
This paper exposes the fundamental role that the Drinfel'd double D(k[G]) of the group ring of a finite group G and its twists D β (k[G]), β ∈ Z 3 (G,k*) as defined by Dijkgraaf–Pasquier–Roche play in stringy orbifold theories and their twistings. The results pertain to three different aspects of the theory. First, we show that G-Frobenius algebras arising in global orbifold cohomology or K-theory are most naturally defined as elements in the braided category of D(k[G])-modules. Secondly, we obtain a geometric realization of the Drinfel'd double as the global orbifold K-theory of global quotient given by the inertia variety of a point with a G action on the one hand and more stunningly a geometric realization of its representation ring in the braided category sense as the full K-theory of the stack [pt/G]. Finally, we show how one can use the co-cycles β above to twist the global orbifold K-theory of the inertia of a global quotient and more importantly, the stacky K-theory of a global quotient [X/G]. This corresponds to twistings with a special type of two-gerbe.