Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid

By István Heckenberger, Hans Jürgen Schneider. 2009

79 citations recorded by the source corpus.

We study graded right coideal subalgebras of Nichols algebras of semisimple Yetter-Drinfeld modules. Assuming that the Yetter-Drinfeld module admits all reflections and the Nichols algebra is decomposable, we construct an injective order preserving and order reflecting map between morphisms of the Weyl groupoid and graded right coideal subalgebras of the Nichols algebra. Here morphisms are ordered with respect to right Duflo order and right coideal subalgebras are ordered with respect to inclusion. If the Weyl groupoid is finite, then we prove that the Nichols algebra is decomposable and the above map is bijective. In the special case of the Borel part of quantized enveloping algebras our result implies a conjecture of Kharchenko.

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

View Paper: https://www.semanticscholar.org/paper/01fc35416e315575455a692427d4c95e294e8d16

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Citations in the corpus

Root systems and Weyl groupoids for Nichols algebras (2008, methodology): “By different methods, we show in [14] that in Theorem IV one can omit the assumption that B(M) is decomposable.”
Hopf Algebras and Root Systems (2020, background): “Our work on right coideal subalgebras in [ HS13a ] was motivated by this conjecture, which is now proved as a special case of Corol-lary 14.6.8.”
On finite-dimensional Hopf algebras (2014, background): “Another fundamental ingredient is the following result, generalized at various levels in 287 [42, 46, 48].”