Triangular Hopf algebras with the Chevalley property

By N. Andruskiewitsch, Pavel Etingof, Shlomo Gelaki. 2000

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We say that a Hopf algebra has the Chevalley property if the tensor product of any two simple modules over this Hopf algebra is semisimple. In this paper we classify finite dimensional triangular Hopf algebras with the Chevalley property, over the field of complex numbers. Namely, we show that all of them are twists of triangular Hopf algebras with R-matrix having rank <=2, and explain that the latter ones are obtained from group algebras of finite supergroups by a simple modification procedure. We note that all examples of finite dimensional triangular Hopf algebras which are known to the authors, do have the Chevalley property, so one might expect that our classification potentially covers all finite dimensional triangular Hopf algebras.

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

View Paper: https://www.semanticscholar.org/paper/f869e0870b3305769e793c140ec7b2dfb5d90e44

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Citations in the corpus

A Tour of Representation Theory (2018, background): “In reference to Chevalley’s Theorem, and following [5], we will say that an arbitrary Hopf algebra H has the Chevalley property if the tensor product of any two completely reducible representations inRepfin H is again completely reducible.”
On finite-dimensional Hopf algebras (2014, background): “A finite-dimensional triangular Hopf algebra is a twisting of a bosonization Λ(V )#CG, where G is a finite group and V ∈ GYD has c = −τ [6].”
Fusion categories of rank 2 (2002, background): “But in a symmetric fusion category the dimensions of all objects are integers, see [ 1 ]Theorem 7.2.”
Quantum lines over non-cocommutative cosemisimple Hopf algebras (2004, methodology): “As is noted in [3], their method is valid to some extent, more generally, for Hopf alge with the Chevalley property [1], that is, for those Hopf algebras A whose coradical H =A0 is a (not necessarily cocommutative) Hopf subalgebra, since gr A then forms a graded Hop algebra which is of the formR #H ”
On pointed Hopf superalgebras (2010, background): “is also monoidal. A quasitriangular Hopf superalgebra is a pair (H,R) where H is a Hopf superalgebra and R ∈ H ⊗H is even, invertible, and satisﬁes the same axioms as in the non-super case, see e. g. [AEG]. In particular, it provides HSM with a braiding: if V,W ∈ HSM, v ∈ V and w ∈ W, then cV,W : V ”