Root systems and Weyl groupoids for Nichols algebras

By István Heckenberger, Hans Jürgen Schneider. 2008

81 citations recorded by the source corpus.

Motivated by the work of Kac and Lusztig, we define a root system for a large class of semisimple Yetter–Drinfeld modules over an arbitrary Hopf algebra which admits the symmetry of the Weyl groupoid introduced by Andruskiewitsch and the authors. The obtained combinatorial structure fits perfectly into an existing framework of generalized root systems associated to a family of Cartan matrices and provides novel insight into Nichols algebras. We demonstrate the power of our construction with new results on Nichols algebras over finite non‐abelian simple groups and symmetric groups.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/02664cb49d469fe1a0c3783f916eb9f2e105a7a6

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Citations in the corpus

On finite-dimensional Hopf algebras (2014, methodology): “[46, 48] If dimB(M) < ∞, then M has a finite GRS.”
Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid (2009, background): “. 1.9. Conversely, suppose that id[N]si1 ···sim is a reduced expression. Then β1,...,βm are pairwise distinct elements in Nθ 0 by Prop. 1.4. Hence (i1,...,im) is N-admissible. We recall a notion from [HS08]. Deﬁnition 6.8. Let N ∈Fθ. Then the Nichols algebra B(N) of N is called decomposable if there”
Hopf Algebras and Root Systems (2020, result): “This is a new result; it was ﬁrst shown in [ HS10b ] for ﬁnite semi-Cartan graphs G ( M ).”