The Weyl groupoid of a Nichols algebra of diagonal type

By István Heckenberger. 2004

200 citations recorded by the source corpus.

The theory of Nichols algebras of diagonal type is known to be closely related to that of semi-simple Lie algebras. In this paper the connection between both theories is made closer. For any Nichols algebra of diagonal type invertible transformations are introduced, which remind one of the action of the Weyl group on the root system associated to a semi-simple Lie algebra. They give rise to the definition of a groupoid. As an application an alternative proof of classification results of Rosso, Andruskiewitsch, and Schneider is obtained without using any technical assumptions on the braiding.

Subfields: Algebra and Number Theory, Geometry and Topology, Spectroscopy

View Paper: https://www.semanticscholar.org/paper/e4990d0b7eb1c13be20f43eda264467a2f0e4135

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Spectroscopy

Citations in the corpus

Nichols algebras of diagonal type (2011, background): “i(1 −qniiqijqji) = 0}, and set aii = 2, si ∈ Aut(Zθ) such that si(αj) = αj −aijαi. Set qrs = χ(s i(αr),s (αs)). Let V be another vector space of dimension θ, and attach to it the matrix q = (qrs). By [H2], ∆ Vi + = s i ∆V + \{α } ∪ {α }. If we consider ∆V = ∆V + ∪ (−∆V+ ), last equation lets us to d”
Classification of arithmetic root systems (2006, background): “Let H be a Hopf algebra with coradical filtration H0 ⊂ H1 ⊂ . . . such that H0 is a Hopf subalgebra of H .”
On pointed Hopf superalgebras (2010, methodology): “setsofpositive rootsby determining aX ij asin (2.3) and applying the symmetries si. Now we recall the deﬁnition of the Weyl groupoid attached to a braided vector space (V,c) of diagonal type given in [He1], see also [AA]. Fix a basis {x1,...,xθ} and scalars qij ∈ k× such that c(xi ⊗ xj) = qijxj ⊗ xi”
On finite-dimensional Hopf algebras (2014, methodology): “The proof relies on the Weyl groupoid introduced in [43], a particular case of Theorem 286 3.”
Finite-dimensional pointed Hopf algebras with alternating groups are trivial (2008, unknown): “−1 −1 −1 1 1 −1 −1 −1 −1 1 −1 1 −1 1 −1 −1 −1 1 −1 1 −1 −1 . Hence M(O,ρ) is of Cartan type with Dynkin diagram given by Figure 2; this is not of ﬁnite type, and dimB(O,ρ) = ∞, by [H1]. ✉ ✉ ✉ ✉ ✉ ✉ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ Figure 2. 2.5. The group SL(2,3) ×Z/2. In this subsection, we present a usef”
Finite Weyl groupoids (2010, unknown): “e natural problem of classifying crystallographic arrangements by considering Cartan schemes, their root systems, and their Weyl groupoids. Weyl groupoids have been introduced by the second author in [Hec06] to obtain ﬁniteness properties of Nichols algebras of diagonal type. The Weyl groupoid provi”
On finite dimensional Nichols algebras of diagonal type (2017, background): “ not of ﬁnite type, since otherwise qii 6= 1 for all i and hence qγ,γ 6= 1 for all roots γ. If A is of aﬃne type, then GKdimB(V) = ∞ by Proposition 3.1. We assume then that A is of indeﬁnite type. By [H1], sα(∆V + − {α}) = ∆V + − {α} for all simple roots α. Since qwγ,wγ = qγ,γ = 1 for all w ∈ W, all”
On Nichols algebras associated to simple racks (2010, methodology): “roup type with many quadratic relations, preprint: arXiv:1004.3723v1. [GV] M. Gran˜a and L. Vendramin, RiG, A GAP package for racks and Nichols Algebras, available at http://mate.dm.uba.ar/~lvendram. [H1] I. Heckenberger, The Weyl groupoid of a Nichols algebra of diagonal type, Invent. Math. 164(200”
The Nichols algebra of a semisimple Yetter-Drinfeld module (2008, methodology): “In order to prove Theorem 1, we have to overcome several difficulties. The proof of [ H06 , Prop.”
Nichols algebras of group type with many quadratic relations (2010, methodology): “The proof of this remarkable result makes heavily use of the classification of Nichols algebras of diagonal type [16, 17].”
A generalization of Coxeter groups, root systems, and Matsumoto’s theorem (2006, methodology): “Example 4 In [9,12] arithmetic root systems are defined.”
Hopf Algebras and Root Systems (2020, methodology): “Several years after their discovery, general classiﬁcation results for pointed Hopf algebras were obtained ([ AS02 ]; [ AS04 ], [ AA08 ], [ AS10 ] depending on [ Ros98 ], [ Kha99 ], [ Hec06 ], [ Hec08 ]).”
Root systems and Weyl groupoids for Nichols algebras (2008, methodology): “Our root system generalizes the construction of the first author [11] in the case of diagonal braidings, where H is the group algebra of an abelian group over a field of characteristic zero.”