[Paper Review] Weyl groupoids with at most three objects

Michael Cuntz, I. Heckenberger|ArXiv.org|May 13, 2008

Algebraic structures and combinatorial models4 citations

TL;DR

This paper classifies finite irreducible Weyl groupoids with at most three objects using a category-theoretic framework for generalized root systems, introducing Cartan schemes and Weyl groupoids. It proves that all such groupoids are either standard (with identical Cartan matrices) or fall into exactly nine exceptional cases, with the stabilizer groups being Coxeter groups of specific types like $B_2$, $G_2$, or $A_1 \times A_1$. The classification reveals that only finitely many non-standard cases exist, while standard cases are infinite in number for two objects.

ABSTRACT

We adapt the generalization of root systems of the second author and H. Yamane to the terminology of category theory. We introduce Cartan schemes, associated root systems and Weyl groupoids. After some preliminary general results, we completely classify all finite Weyl groupoids with at most three objects. The classification yields that there exist infinitely many standard, but only 9 exceptional cases. Key words: Nichols algebra, reflection, root system, Weyl groupoid

Motivation & Objective

• To generalize root systems and Weyl groups using category theory, introducing Cartan schemes and Weyl groupoids as foundational structures.
• To classify all finite irreducible Weyl groupoids with at most three objects, distinguishing between standard and exceptional cases.
• To determine which root systems arise from non-diagonal type Nichols algebras, identifying cases not realizable in the standard diagonal type framework.
• To characterize the stabilizer groups (Hom(a)) of objects in such Weyl groupoids, showing they are Coxeter groups of specific types.

Proposed method

• Formalizes generalized root systems via Cartan schemes, defined as families of generalized Cartan matrices indexed over a finite set of objects A, with reflection maps satisfying symmetry and involution axioms.
• Introduces Weyl groupoids as groupoids generated by reflections associated with each object and Cartan matrix, with morphisms defined via reflection actions on the set of objects.
• Applies Matsumoto’s theorem and Coxeter relations to analyze the structure of Weyl groupoids, particularly focusing on finiteness and reducibility conditions.
• Uses object change diagrams and Dynkin diagram techniques to classify root systems, especially in the irreducible case.
• Employs case-by-case analysis based on Cartan matrix entries and reflection relations to rule out infinite families or inconsistent Coxeter relations.
• Relies on Theorem 2.6 from [HY08], which states that Weyl groupoids are generated by reflections and satisfy Coxeter relations, as a key technical tool.

Experimental results

Research questions

• RQ1 What are all finite irreducible Weyl groupoids with at most three objects, and how can they be classified?
• RQ2 Which of these Weyl groupoids are standard (i.e., defined by a single Cartan matrix), and how many exceptional cases exist?
• RQ3 Can root systems associated with non-diagonal type Nichols algebras be realized as Weyl groupoids of non-standard Cartan schemes?
• RQ4 What are the possible stabilizer groups (Hom(a)) for objects in such Weyl groupoids, and what Coxeter types do they realize?
• RQ5 Are there only finitely many non-standard irreducible connected Weyl groupoids for a fixed number of objects?

Key findings

• There exist exactly nine exceptional finite irreducible Weyl groupoids with at most three objects, all distinct from standard cases.
• All standard Weyl groupoids with two objects are infinite in number, but all irreducible connected Weyl groupoids with three objects have rank ≤ 4.
• The stabilizer group Hom(a) of any object in a finite irreducible Weyl groupoid is a Coxeter group, specifically of type $B_2$, $G_2$, $B_3$, or $A_1 \times A_1$, depending on the case.
• For three-object Weyl groupoids, the root system is standard if and only if the Cartan matrices are of type $B_4$, $C_4$, $D_4$, or $F_4$, with specific object change diagrams.
• The classification shows that root systems from non-diagonal type Nichols algebras can yield non-standard Cartan schemes not realizable in the diagonal type setting.
• In the three-object case, all finite irreducible root systems are standard, and their object change diagrams are uniquely determined by the Dynkin type: $B_4/C_4$ has diagram 1,3 2, $D_4$ has 1,3,4 2, and $F_4$ has 1 2.