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

By István Heckenberger, H. Yamane. 2006

103 citations recorded by the source corpus.

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 simple reflections and Coxeter relations. In a broad sense 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/21515336b3ab825676867fd66910490fb3ae04bd

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Citations in the corpus

On finite dimensional Nichols algebras of diagonal type (2017, methodology): “6 fit into an axiomatic framework designed in [52] for this purpose.”
Grothendieck rings of basic classical Lie superalgebras (2007, background): “t the possibility of a groupoid version of the Weyl group for Lie superalgebras was contemplated by Serganova [24], but she had a diﬀerent picture in mind (see [25]). Recently Heckenberger and Yamane [11] introduced a groupoid related to basic classical Lie superalgebras motivated by Serganova’s wor”
Hopf Algebras and Root Systems (2020, methodology): “Most of the results in Part 2 have been already published in [ HY08 ], [ CH09b ], [ CH09a ], and [ CH12 ].”
Root systems and Weyl groupoids for Nichols algebras (2008, background): “Weyl groupoids and root systems In this section we give the definition of generalized root systems in [15] in terms of category theory (see [9]) and recall some results which will be needed in the following sections.”