Weyl Groupoids Research Articles

Hamiltonian Cycles for Finite Weyl Groupoids

Hamiltonian Cycles for Finite Weyl Groupoids

The Weyl groupoids of [formula omitted] and [formula omitted

We provide a convenient formulation of the definition of Cartan graphs and Weyl groupoids introduced by Heckenberger and Schneider, and construct Cartan graphs for regular symmetrizable contragredient Lie superalgebras. For sl(m|n), osp(2m+1|2n) and osp(2m|2n) we explicitly describe the Cartan graph in terms of partitions and determine the relations between the generators of their Weyl groupoids.

Affine super-Yangian and a quantum Weyl groupoid

Affine super-Yangian and a quantum Weyl groupoid

Higher Braidings of Diagonal Type

Heckenberger introduced the Weyl groupoid of a finite-dimensional Nichols algebra of diagonal type. We replace the matrix of its braiding by a higher tensor and present a construction which yields further Weyl groupoids. Abelian cohomology theory gives evidence for the existence of a higher braiding associated to such a tensor.

The Haagerup property for twisted groupoid dynamical systems

We introduce the Haagerup property for twisted groupoid C⁎-dynamical systems in terms of naturally defined positive-definite operator-valued multipliers. By developing a version of ‘the Haagerup trick’ we prove that this property is equivalent to the Haagerup property of the reduced crossed product C⁎-algebra with respect to the canonical conditional expectation E. This extends a theorem of Dong and Ruan for discrete group actions, and implies that a given Cartan inclusion of separable C⁎-algebras has the Haagerup property if and only if the associated Weyl groupoid has the Haagerup property in the sense of Tu. We use the latter statement to prove that every separable C⁎-algebra which has the Haagerup property with respect to some Cartan subalgebra satisfies the Universal Coefficient Theorem. This generalises a recent result of Barlak and Li on the UCT for nuclear Cartan pairs.

Congruence Normality of Simplicial Hyperplane Arrangements via Oriented Matroids

A catalogue of simplicial hyperplane arrangements was first given by Grünbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and also the weak order through the poset of regions. The weak order is known to be a congruence normal lattice, and congruence normality of lattices of regions of simplicial arrangements can be determined using polyhedral cones called shards. In this article, we update Grünbaum’s catalogue by providing normals realizing all known simplicial arrangements with up to 37 lines and key invariants. Then we add structure to this catalogue by determining which arrangements always/sometimes/never lead to congruence normal lattices of regions. To this end, we use oriented matroids to recast shards as covectors to determine congruence normality of large hyperplane arrangements. We also show that lattices of regions coming from finite Weyl groupoids of any rank are always congruence normal.

Reconstruction of groupoids and C⁎-rigidity of dynamical systems

We show how to construct a graded locally compact Hausdorff étale groupoid from a C⁎-algebra carrying a coaction of a discrete group, together with a suitable abelian subalgebra. We call this groupoid the extended Weyl groupoid. When the coaction is trivial and the subalgebra is Cartan, our groupoid agrees with Renault's Weyl groupoid. We prove that if G is a second-countable locally compact étale groupoid carrying a grading of a discrete group, and if the interior of the trivially graded isotropy is abelian and torsion free, then the extended Weyl groupoid of its reduced C⁎-algebra is isomorphic as a graded groupoid to G. In particular, two such groupoids are isomorphic as graded groupoids if and only if there is an equivariant diagonal-preserving isomorphism of their reduced C⁎-algebras. We introduce graded equivalence of groupoids, and establish that two graded groupoids in which the trivially graded isotropy has torsion-free abelian interior are graded equivalent if and only if there is an equivariant diagonal-preserving Morita equivalence between their reduced C⁎-algebras. We use these results to establish rigidity results for a number of classes of dynamical systems, including all actions of the natural numbers by local homeomorphisms of locally compact Hausdorff spaces.

Analyzing the Weyl Construction for Dynamical Cartan Subalgebras

Abstract When the reduced twisted $C^*$-algebra $C^*_r({\mathcal{G}}, c)$ of a non-principal groupoid ${\mathcal{G}}$ admits a Cartan subalgebra, Renault’s work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r({\mathcal{G}}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid ${\mathcal{S}}$ of ${\mathcal{G}}$. In this paper, we study the relationship between the original groupoids ${\mathcal{S}}, {\mathcal{G}}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum ${\mathfrak{B}}$ of the Cartan subalgebra $C^*_r({\mathcal{S}}, c)$. We then show that the quotient groupoid ${\mathcal{G}}/{\mathcal{S}}$ acts on ${\mathfrak{B}}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly, we show that if the quotient map ${\mathcal{G}}\to{\mathcal{G}}/{\mathcal{S}}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on ${\mathcal{G}}/{\mathcal{S}} \ltimes{\mathfrak{B}}$.

An algebraic approach to the Weyl groupoid

We unify the Kumjian-Renault Weyl groupoid construction with the Lawson-Lenz version of Exel's tight groupoid construction. We do this by utilising only a weak algebraic fragment of the C*-algebra structure, namely its *-semigroup reduct. Fundamental properties like local compactness are also shown to remain valid in general classes of *-rings.

Cartan subalgebras for non-principal twisted groupoid C⁎-algebras

Renault proved in 2008 [22, Theorem 5.2] that if G is a topologically principal groupoid, then C0(G(0)) is a Cartan subalgebra in Cr⁎(G,Σ) for any twist Σ over G. However, there are many groupoids which are not topologically principal, yet their (twisted) C⁎-algebras admit Cartan subalgebras. This paper gives a dynamical description of a class of such Cartan subalgebras, by identifying conditions on a 2-cocycle c on G and a subgroupoid S⊆G under which Cr⁎(S,c) is Cartan in Cr⁎(G,c). When G is a discrete group, we also describe the Weyl groupoid and twist associated to these Cartan pairs, under mild additional hypotheses.

On the Tits cone of a Weyl groupoid

We translate the axioms of a Weyl groupoid with (not necessarily finite) root system in terms of arrangements. The result is a correspondence between Weyl groupoids permitting a root system and Tits arrangements satisfying an integrality condition which we call the crystallographic property.

Bruhat order and nil-Hecke algebras for Weyl groupoids

We introduce nil-Hecke algebras for Weyl groupoids. We describe a basis and some properties of these algebras which lead to a notion of Bruhat order for Weyl groupoids.

Quantales and Fell bundles

We study Fell bundles on groupoids from the viewpoint of quantale theory. Given any saturated upper semicontinuous Fell bundle π:E→G on an étale groupoid G with G0 locally compact Hausdorff, equipped with a suitable completion C*-algebra A of its convolution algebra, we obtain a map of involutive quantales p:MaxA→Ω(G), where MaxA consists of the closed linear subspaces of A and Ω(G) is the topology of G. We study various properties of p which mimick, to various degrees, those of open maps of topological spaces. These are closely related to properties of G, π, and A, such as G being Hausdorff, principal, or topological principal, or π being a line bundle. Under suitable conditions, which include G being Hausdorff, but without requiring saturation of the Fell bundle, A is an algebra of sections of the bundle if and only if it is the reduced C*-algebra Cr⁎(G,E). We also prove that MaxA is stably Gelfand. This implies the existence of a pseudogroup IB and of an étale groupoid B associated canonically to any sub-C*-algebra B⊂A. We study a correspondence between Fell bundles and sub-C*-algebras based on these constructions, and compare it to the construction of Weyl groupoids from Cartan subalgebras.

On finite dimensional Nichols algebras of diagonal type

This is a survey on Nichols algebras of diagonal type with finite dimension, or more generally with arithmetic root system. The knowledge of these algebras is the cornerstone of the classification program of pointed Hopf algebras with finite dimension, or finite Gelfand–Kirillov dimension; and their structure should be indispensable for the understanding of the representation theory, the computation of the various cohomologies, and many other aspects of finite dimensional pointed Hopf algebras. These Nichols algebras were classified in Heckenberger (Adv Math 220:59–124, 2009) as a notable application of the notions of Weyl groupoid and generalized root system (Heckenberger in Invent Math 164:175–188, 2006; Heckenberger and Yamane in Math Z 259:255–276, 2008). In the first part of this monograph, we give an overview of the theory of Nichols algebras of diagonal type. This includes a discussion of the notion of generalized root system and its appearance in the contexts of Nichols algebras of diagonal type and (modular) Lie superalgebras. In the second and third part, we describe for each Nichols algebra in the list of Heckenberger (2009) the following basic information: the generalized root system; its label in terms of Lie theory; the defining relations found in Angiono (J Eur Math Soc 17:2643–2671, 2015; J Reine Angew Math 683:189–251, 2013); the PBW-basis; the dimension or the Gelfand–Kirillov dimension; the associated Lie algebra as in Andruskiewitsch et al. (Bull Belg Math Soc Simon Stevin 24(1):15–34, 2017). Indeed the second part deals with Nichols algebras related to Lie algebras and superalgebras in arbitrary characteristic, while the third contains the information on Nichols algebras related to Lie algebras and superalgebras only in small characteristic, and the few examples yet unidentified in terms of Lie theory.

The classification of Nichols algebras over groups with finite root system of rank two

We classify all groups G and all pairs (V,W) of absolutely simple Yetter–Drinfeld modules over G such that the support of V\oplus W generates G , c_{W,V}c_{V,W}\ne\mathrm {id} , and the Nichols algebra of the direct sum of V and W admits a finite root system. As a byproduct, we determine the dimensions of such Nichols algebras, and several new families of finite-dimensional Nichols algebras are obtained. Our main tool is the Weyl groupoid of pairs of absolutely simple Yetter–Drinfeld modules over groups.

Rank three Nichols algebras of diagonal type over arbitrary fields

We classify all rank three Nichols algebras of diagonal type with a finite root system over fields of arbitrary characteristic. Our proof uses the classification of the finite Weyl groupoids of rank three.

A classification of Nichols algebras of semisimple Yetter–Drinfeld modules over non-abelian groups

Over fields of arbitrary characteristic we classify all braid-indecomposable tuples of at least two absolutely simple Yetter-Drinfeld modules over non-abelian groups such that the group is generated by the support of the tuple and the Nichols algebra of the tuple is finite-dimensional. Such tuples are classified in terms of analogs of Dynkin diagrams which encode much information about the Yetter-Drinfeld modules. We also compute the dimensions of these finite-dimensional Nichols algebras. Our proof uses the Weyl groupoid of a tuple of simple Yetter-Drinfeld modules.

On Fomin–Kirillov algebras for complex reflection groups

ABSTRACTIn this note, we apply classification results for finite-dimensional Nichols algebras to generalizations of Fomin–Kirillov algebras to complex reflection groups. First, we focus on the case of cyclic groups where the corresponding Nichols algebras are only finite-dimensional up to order four, and we include results about the existence of Weyl groupoids and finite-dimensional Nichols subalgebras for this class. Second, recent results by Heckenberger–Vendramin [ArXiv e-prints, 1412.0857 (December 2014)] on the classification of Nichols algebras of semisimple group type can be used to find that these algebras are infinite-dimensional for many non-exceptional complex reflection groups in the Shephard–Todd classification.

Orbits and Invariants of Super Weyl Groupoid

We study the orbits and polynomial invariants of certain affine action of the super Weyl groupoid of Lie superalgebra $\mathfrak {gl}(n,m)$, depending on a parameter. We show that for generic values of the parameter all the orbits are finite and separated by certain explicitly given invariants. We also describe explicitly the special set of parameters, for which the algebra of invariants is not finitely generated and does not separate the orbits, some of which are infinite.

A simplicial complex of Nichols algebras

We translate the concept of restriction of an arrangement in terms of Hopf algebras. In consequence, every Nichols algebra gives rise to a simplicial complex decorated by Nichols algebras with restricted root systems. As applications, some of these Nichols algebras provide Weyl groupoids which do not arise for diagonal Nichols algebras and in fact we realize all root systems of finite Weyl groupoids of rank greater than three. Further, our result explains the root systems of the folded Nichols algebras over nonabelian groups and of generalized Satake diagrams.