The Lattice property of biclosed sets for infinite root systems

A root system is a finite collection of vectors in Euclidean space satisfying certain symmetry conditions. Work by Killing and Cartan in the late 19th century showed that these come in seven families, and uses this fact to classify the semisimple Lie algebras. Root systems play an important role in many fields, including the representation theory of algebras, algebraic geometry, and particle physics. Relaxing the finiteness assumption in the definition of root system leads to the more general notion of a Kac–Moody root system. In the finite setting, root systems can be studied via their inversion sets, which are objects carrying geometric information. This does not carry over to Kac-Moody root systems. However, the inversion sets are completely characterised by two closure properties, making them so-called biclosed sets. Following work of Barkley and Speyer, we investigate biclosed sets as a generalization of inversion sets to infinite root systems, especially a family called the hyperbolic root systems. Their result that these sets form a lattice in affine type is found not to hold when the sets are allowed to contain imaginary roots, and the related “2-out-of-3 rule” also fails without this restriction. We explored distinctions between two different notions of closure used to define a biclosed set. Several interesting questions arose from these areas.