A generalization of Coxeter groups, root systems, and Matsumoto's theorem

2006/10/27 by Heckenberger, I., Yamane, H. · 5 citations
#17B20 #20F55 #20M05 #FOS: Mathematics #Group Theory (math.GR) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.math/0610823

Abstract

The root systems appearing in the theory of Lie superalgebras and Nichols algebras admit a large symmetry extending properly the one coming from the Weyl group. Based on this observation we set up a general framework in which the symmetry object is a groupoid. We prove that in our context the groupoid is generated by reflections and Coxeter relations. This answers a question of Serganova. Our weak version of the exchange condition allows us to prove Matsumoto's theorem. Therefore the word problem is solved for the groupoid.

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• Categorification of generic Su-Zhang character formula
• \mathfrak b₁-Verma \mathfrak b₂-dual Verma supermodules
• Pointed Hopf algebras revisited, with a view from tensor categories
• Nichols algebras, tensor categories and Kazhdan-Lusztig correspondences
• Computing finite Weyl groupoids

Related

• Root groupoid and related Lie superalgebras
• Generalized root systems
• Matsumoto theorem for skeleta
• Weyl groupoids with at most three objects
• Root systems, symmetries and linear representations of Artin groups