2016/05/23 by Diego Penta, Penta, Diego
Mathematics · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.1605.06901
In 1983 Feingold-Frenkel studied the structure of a rank 3 hyperbolic Kac-Moody algebra F containing the affine KM algebra A(1)1. In 2004 Feingold-Nicolai showed that F contains all rank 2 hyperbolic KM algebras with symmetric Cartan matrices, A=[2 · amp; -a
-a · amp; 2], a≥ 3. The case when a=3 is called Fib because of its connection with the Fibonacci numbers (Feingold 1980). Some important structural results about F come from the decomposition with respect to its affine subalgebra A(1)1. Here we study the decomposition of F with respect to its subalgebra Fib. We find that F has a grading by Fib-level, and prove that each graded piece, Fib(m) for m∈ℤ, is an integrable Fib-module. We show that for |m|>2, Fib(m) completely reduces as a direct sum of highest- and lowest-weight modules, and for |m|≤ 2, Fib(m) contains one irreducible non-standard quotient module VΛm=V(m)/U(m). We then prove that the quotient Fib(m)/V(m) completely reduces as a direct sum of one trivial module (on level 0), and standard modules. We give an algorithm for determining the inner multiplicities of any irreducible Fib-module, in particular the non-standard modules on levels |m|≤ 2. We show that multiplicities of non-standard modules on levels |m|=1,2 do not follow the Kac-Peterson recursion, but instead appear to follow a recursion similar to Racah-Speiser. We then use results of Borcherds and Frenkel-Lepowsky-Meurman and construct vertex algebras from the root lattices of F and Fib, and study the decomposition within this setting.