2012/08/31 by Quincy Loney, Loney, Quincy
Mathematics · #17B67 (Primary) #17B68 #17B69 (Secondary) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.QA #math.RT #msc:17B67 #msc:17B68 #msc:17B69
paper · pdf · doi:10.48550/arxiv.1209.0018
Dissertation, 106 pages
arxiv created 2012/08/31 · arxiv updated 2012/09/04
In [FFR] Feingold, Frenkel and Ries gave a spinor construction of the vertex operator para-algebra (abelian intertwining algebra) V = V0 ⊕ V1 ⊕ V2 ⊕ V3, whose summands are four level-1 irreducible representations of the affine Kac-Moody algebra D4^(1). The triality group S3 = < σ,τ| σ3 = 1 = τ2, τστ= σ-1 > in Aut(V) was constructed, preserving V0 and permuting the Vi, for i=1,2,3. V is (1/2)Z-graded where Vin denotes the n-graded subspace of Vi. Vertex operators Y(v,z) for v in V01 represent D4^(1) on V, while those for which σ(v) = v represent G2^(1). We investigate branching rules, how V decomposes into a direct sum of irreducible G2^(1) representations. We use a two-step process, first decomposing with respect to the intermediate subalgebra B3^(1), represented by Y(v,z) for τ(v) = v. There are three vertex operators, Y(ωD4,z), Y(ωB3,z), and Y(ωG2,z) each representing the Virasoro algebra given by the Sugawara constructions from the three algebras. The Goddard-Kent-Olive coset construction [GKO] gives two mutually commuting coset Virasoro representations, provided by the vertex operators Y(ωD4-ωB3,z) and Y(ωB3-ωG2,z), with central charges 1/2 and 7/10, respectively. The first one commutes with B3^(1), and the second one commutes with G2^(1). This gives the space of highest weight vectors for G2^(1) in V as tensor products of irreducible Virasoro modules L(1/2,h1/2) ⊗ L(7/10,h7/10). This dissertation explicitly constructs these coset Virasoro operators, and uses them to describe the decomposition of V with respect to G2^(1). This work also provides spinor constructions of the 7/10 Virasoro modules, and of the two level-1 representations of G2^(1) inside V.