2008/12/31 by Igor B. Frenkel, Anton M. Zeitlin
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Cohomology #Connection (principal bundle) #Fock space #Homotopy and Cohomology in Algebraic Topology #Operator algebra #Quantum algebra #Quantum group #Realization (probability) #Tensor product #Vertex operator algebra #Virasoro algebra #hep-th #math-ph #math.MP #math.RT
paper · pdf · doi:10.1007/s00220-010-1055-2
published as Commun. Math. Phys.297:687-732, 2010 · 50 pages, 7 figures, minor revisions, typos corrected, Communications in Mathematical Physics, in press
arxiv created 2010/04/15 · openalex publication_date 2010/05/06 · arxiv updated 2014/11/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We obtain the quantum group SLq(2) as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges c+c=26. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the generators of SLq(2) and verify their relations. We also discuss the possible extensions of our construction and its connection to the Liouville model and minimal string theory.