Nichols algebras of diagonal type

By I. Angiono. Hopf Algebras and Root Systems, 2011

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We give an explicit and essentially minimal list of defining relations of a Nichols algebra of diagonal type with finite root system. This list contains the well-known quantum Serre relations but also many new variations. A conjecture by Andruskiewitsch and Schneider states that any finite-dimensional pointed Hopf algebra over an algebraically closed field of characteristic zero is generated as an algebra by its group-like and skew-primitive elements. As an application of our main result, we prove the conjecture when the group of group-like elements is abelian.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3540a802b765cd2cbfd1ea4b681b677b8860b757

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Citations in the corpus

Hopf Algebras and Root Systems (2020, methodology): “This theorem was originally proved by I. Angiono in [ Ang13 ] using his list of deﬁning relations of the ﬁnite-dimensional Nichols algebras classiﬁed in [ Hec09 ].”
Liftings of Nichols algebras of diagonal type II: all liftings are cocycle deformations (2016, methodology): “s in diﬀerent components. We denote by G a set of generators of the ideal deﬁning the Nichols algebra B(V). If every connected component of V belongs to the list of [H], then we take G as computed in [A] for each component, union G(0). Let Hbe a Hopf algebra. A principal YD-realization of V over His”
On finite dimensional Nichols algebras of diagonal type (2017, background): “11 The Lie algebra of a finite-dimensional Nichols algebra Afinite-dimensionalNichols algebraBq of diagonal type gives rise to some remarkable objects: its distinguished pre-Nichols algebra ̃ Bq [24,26], its Lusztig algebra Lq [7] and its associatedLie algebrag [8].”