Abstract
We investigate a class of Kac–Moody algebras previously not considered. We refer to them as n-extended Lorentzian Kac–Moody algebras defined by their Dynkin diagrams through the connection of an A_n Dynkin diagram to the node corresponding to the affine root. The cases n=1 and n=2 correspond to the well-studied over- and very-extended Kac–Moody algebras, respectively, of which the particular examples of E_{10} and E_{11} play a prominent role in string and M-theory. We construct closed generic expressions for their associated roots, fundamental weights and Weyl vectors. We use these quantities to calculate specific constants from which the nodes can be determined that when deleted decompose the n-extended Lorentzian Kac–Moody algebras into simple Lie algebras and Lorentzian Kac–Moody algebra. The signature of these constants also serves to establish whether the algebras possess SO(1, 2) and/or SO(3)-principal subalgebras.
Highlights
The symmetry algebras relevant in the formulation of fundamental theories in particle physics have become increasingly complex over the years
Before we present our extended version of the Lorentzian Kac–Moody algebra, we recall some of the known results on the extended, over-extended and very-extended root lattices to establish our conventions and notations
The signatures of the product ρ(n) · ρ(n), that is the generalisation of the Freudenthal–de Vries strange formula, led to a necessary condition for the n-extended Lorentzian Kac–Moody algebras to possess a SO(1, 2)-principal subalgebra
Summary
The symmetry algebras relevant in the formulation of fundamental theories in particle physics have become increasingly complex over the years. Hyperbolic Kac–Moody algebras have been fully classified [17] In terms of their connected Dynkin diagrams they are defined by the property that the deletion of any one node leaves a possibly disconnected set of connected Dynkin diagrams each of which is of finite type, except for at most one affine type. 5, we compute a special set of constants obtained from the inner product of the Weyl vector and a fundamental weight, whose overall signs provide necessary and sufficient conditions for the occurrence of SO(1, 2) and SO(3) principal subalgebras and the decomposition of the n-extended Lorentzian algebras, which are studied in detail in Sect. For r < 24, such a possibility exists, but ρ2 < 0 implies it does not exist when n > 12 and n > 16, for Ar and Dr , respectively. As the criterion (4.6) is only necessary, but not sufficient, let us compute the values for Di(n) to obtain the more restrictive necessary and sufficient information
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
