We translate the axioms of a Weyl groupoid with (not necessarily finite) root system in terms of arrangements. The result is a correspondence between Weyl groupoids permitting a root system and Tits arrangements satisfying an integrality condition which we call the crystallographic property.As a result, most of the Nichols algebras define Tits cones. An approach to classify arbitrary Nichols algebras could now be to start with those algebras defining a nice cone. For example, if the Tits cone is a halfspace, we call the Nichols algebra 'affine'. A first classification result of affine Nichols algebras of diagonal type is [5].This paper is organized as follows. Section 2 recalls the relevant notions introduced in [11]. In Section 3, we discuss the crystallographic property and how to obtain a Weyl groupoid from a 1 crystallographic arrangement. The geometric realization of a connected simply connected Weyl groupoid is given in Section 4. Section 5 discusses arrangements induced by crystallographic arrangements by subspaces.Acknowledgement. Some of the results and ideas of this note were achieved during a Mini-Workshop on Nichols algebras and Weyl groupoids at the Mathematisches Forschungsinstitut Oberwolfach in October 2012, and during meetings in Gießen, Hannover, and Kaiserslautern supported by the Deutsche Forschungsgemeinschaft within the priority programme 1388.
Tits arrangements2.1. Hyperplane arrangements. In this section we recall the definitions and properties regarding hyperplane arrangements and Tits arrangements. For some basic examples and the proofs of the statements see [11]. Our notation follows this paper as well.Note that all topological notations we use refer to the standard topology in R r , in particular for X ⊂ R r we denote by X the closure of X.If T is unambiguous from the context, we also call the set A a hyperplane arrangement.Let X ⊂ T . Then the localisation at X (in A) is defined asIf X = {x} is a singleton, we write A x instead of A {x} and call (A x , T ) the parabolic subarrangement at x. A parabolic subarrangement of (A, T ) is a subarrangement (A ′ , T ), such that A ′ = A x for some x ∈ T . For the purpose of this paper we call the setthe section of X (in A). We will omit A when there is no danger of confusion.The support of X (in A) is defined to be the subspace supp A (X) = H∈A X H. The connected components of T \ H∈A H are called chambers, denoted with K(A, T ) or just K, if (A, T ) is unambiguous.Let K ∈ K(A, T ). Define the walls of K to be the elements ofThe radical of A is the subspace Rad(A) := H∈A H = supp A (0). We call the arrangement non-degenerate if Rad(A) = 0, and degenerate otherwise. A hyperplane arrangement is thin if W K ⊂ A for all K ∈ K.We recall a basic consequence of the notion of local finiteness.Lemma 2.2 ([11, Lemma 2.3]). Let (A, T ) be a hyperplane arrangement. Then for every point x ∈ T there exists a neighbourhood U x such that A x = sec(U x ). Furthermore the set sec(X) is finite for every compact set X ⊂ T . 2 2.2. Simplicial cones and Tits arrange...
