Finite Root System Research Articles

Locally finite root supersystems

ABSTRACTWe introduce the notion of locally finite root supersystems as a generalization of both locally finite root systems and generalized root systems. We classify irreducible locally finite root supersystems.

Quivers with relations for symmetrizable Cartan matrices I: Foundations

We introduce and study a class of Iwanaga-Gorenstein algebras defined via quivers with relations associated with symmetrizable Cartan matrices. These algebras generalize the path algebras of quivers associated with symmetric Cartan matrices. We also define a corresponding class of generalized preprojective algebras. Without any assumption on the ground field, we obtain new representation-theoretic realizations of all finite root systems.

Representations of Frobenius-type triangular matrix algebras

The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any field. Finally, we give an analog of APR-tilting module for this class of algebras. The major conclusions contains the known results as special cases, e.g., that for path algebras over an algebraically closed field and for path algebras with relations from symmetrizable cartan matrices. Meanwhile, it means the corresponding results for some other important classes of algebras, that is, the path algebras of quivers over Frobenius algebras and the generalized path algebras endowed by Frobenius algebras at vertices.

A pattern avoidance criterion for free inversion arrangements

We show that the hyperplane arrangement of a coconvex set in a finite root system is free if and only if it is free in corank 4. As a consequence, we show that the inversion arrangement of a Weyl group element w is free if and only if w avoids a finite list of root system patterns. As a key part of the proof, we use a recent theorem of Abe and Yoshinaga to show that if the root system does not contain any factors of type C or F, then Peterson translation of coconvex sets preserves freeness. This also allows us to give a Kostant---Shapiro---Steinberg rule for the coexponents of a free inversion arrangement in any type.

Twisted strong Macdonald theorems and adjoint orbits

The strong Macdonald theorems state that, for L reductive and s an odd variable, the cohomology algebras H⁎(L[z]/zN) and H⁎(L[z,s]) are freely generated, and describe the cohomological, s-, and z-degrees of the generators. The resulting identity for the z-weighted Euler characteristic is equivalent to Macdonald's constant term identity for a finite root system. We calculate H⁎(p/zNp) and H⁎(p[s]) for p a standard parahoric in a twisted loop algebra, giving strong Macdonald theorems that take into account both a parabolic component and a possible diagram automorphism twist. In particular we show that H⁎(p/zNp) contains a parabolic subalgebra of the coinvariant algebra of the fixed-point subgroup of the Weyl group of L, and thus is no longer free. We also prove a strong Macdonald theorem for H⁎(b;S⁎n⁎) and H⁎(b/zNn) when b and n are Iwahori and nilpotent subalgebras respectively of a twisted loop algebra. For each strong Macdonald theorem proved, taking z-weighted Euler characteristics gives an identity equivalent to Macdonald's constant term identity for the corresponding affine root system. As part of the proof, we study the regular adjoint orbits for the adjoint action of the twisted arc group associated to L, proving an analogue of the Kostant slice theorem.

A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems

We obtain a presentation by generators and relations of any Nichols algebra of diagonal type with finite root system. We prove that the defining ideal is finitely generated. The proof is based on Kharchenko’s theory of PBW bases of Lyndon words. We prove that the lexicographic order on Lyndon words is convex for PBW generators and so the PBW basis is orthogonal with respect to the canonical non-degenerate form associated to the Nichols algebra.

Supersolvability and the Koszul property of root ideal arrangements

A root ideal arrangement A_I is the set of reflecting hyperplanes corresponding to the roots in an order ideal I of the root poset on the positive roots of a finite crystallographic root system. A characterisation of supersolvable root ideal arrangements is obtained. Namely, A_I is supersolvable if and only if I is chain peelable, meaning that it is possible to reach the empty poset from I by in each step removing a maximal chain which is also an order filter. In particular, supersolvability is preserved under taking subideals. We identify the maximal ideals that correspond to non-supersolvable arrangements. There are essentially two such ideals, one in type D_4 and one in type F_4. By showing that A_I is not line-closed if I contains one of these, we deduce that the Orlik-Solomon algebra OS(A_I) has the Koszul property if and only if A_I is supersolvable.

Nichols algebras over groups with finite root system of rank two III

We compute the finite-dimensional Nichols algebras over the sum of two simple Yetter–Drinfeld modules V and W over non-abelian epimorphic images of a certain central extension of the dihedral group of eight elements or SL(2,3), and such that the Weyl groupoid of the pair (V,W) is finite. These central extensions appear in the classification of non-elementary finite-dimensional Nichols algebras with finite Weyl groupoid of rank two. We deduce new information on the structure of primitive elements of finite-dimensional Nichols algebras over groups.

Dunkl’s theory and best approximation by entire functions of exponential type in L 2-metric with power weight

In this paper, we study the sharp Jackson inequality for the best approximation of f ∈ L2,κ(ℝd) by a subspace Eκ2(σ) (SEκ2(σ)), which is a subspace of entire functions of exponential type (spherical exponential type) at most σ. Here L2,κ(ℝd) denotes the space of all d-variate functions f endowed with the L2-norm with the weight \(v_\kappa (x) = \prod\nolimits_{\xi \in R_ + } {|(\xi ,x)|^{2\kappa (\xi )} } \), which is defined by a positive subsystem R+ of a finite root system R ⊂ ℝd and a function κ(ξ): R → ℝ+ invariant under the reflection group G(R) generated by R. In the case G(R) = ℤ2d, we get some exact results. Moreover, the deviation of best approximation by the subspace Eκ2(σ) (SEκ2(σ)) of some class of the smooth functions in the space L2,κ(ℝd) is obtained.

Weyl group multiple Dirichlet series for symmetrizable Kac-Moody root systems

Weyl group multiple Dirichlet series, introduced by Brubaker, Bump, Chinta, Friedberg and Hoffstein, are expected to be Whittaker coefficients of Eisenstein series on metaplectic groups. Chinta and Gunnells constructed these multiple Dirichlet series for all the finite root systems using the method of averaging a Weyl group action on the field of rational functions. In this paper, we generalize Chinta and Gunnells’ work and construct Weyl group multiple Dirichlet series for the root systems associated with symmetrizable Kac-Moody algebras, and establish their functional equations and meromorphic continuation.

Nichols algebras over groups with finite root system of rank two II

Abstract We classify all non-abelian groups G for which there exists a pair (V,W) of absolutely simple Yetter–Drinfeld modules over G such that the Nichols algebra of the direct sum of V and W is finite-dimensional, under two assumptions: the square of the braiding between V and W is not the identity, and G is generated by the support of V and W. As a corollary, we prove that the dimensions of such V and W are at most six. As a tool we use the Weyl groupoid of (V,W).

On root posets for noncrystallographic root systems

We discuss properties of root posets for finite crystallographic root systems, and show that these properties uniquely determine root posets for the noncrystallographic dihedral types and type H 3 H_3 , while proving that there does not exist a poset satisfying all of the properties in type H 4 H_4 . We do this by exhaustive computer searches for posets having these properties. We further give a realization of the poset of type H 3 H_3 as restricted roots of type D 6 D_6 , and conjecture a Hilbert polynomial for the q , t q,t -Catalan numbers for type H 4 H_4 .

Root Polytopes and Borel Subalgebras

Let $\Phi$ be a finite crystallographic irreducible root system and $\mathcal P_{\Phi}$ be the convex hull of the roots in $\Phi$. We give a uniform explicit description of the polytope $\mathcal P_{\Phi}$, analyze the algebraic-combinatorial structure of its faces, and provide connections with the Borel subalgebra of the associated Lie algebra. We also give several enumerative results.

Central Extensions of Root Graded Lie Algebras

We study central extensions of Lie algebras graded by an irreducible locally finite root system.

Nichols Algebras of Unidentified Diagonal Type

The Nichols algebras of diagonal type with finite root system are either of standard, super or (yet) unidentified type. A concrete description of the defining relations of all those Nichols algebras was given in [8]. In the present paper we use this result to give an explicit presentation of all Nichols algebras of unidentified type.

Root polytopes and Abelian ideals

We study the root polytope $\mathcal P_\Phi$ of a finite irreducible crystallographic root system $\Phi$ using its relation with the abelian ideals of a Borel subalgebra of a simple Lie algebra with root system $\Phi$. We determine the hyperplane arrangement corresponding to the faces of codimension 2 of $\mathcal P_\Phi$ and analyze its relation with the facets of $\mathcal P_\Phi$. For $\Phi$ of type $A_n$ or $C_n$, we show that the orbits of some special subsets of abelian ideals under the action of the Weyl group parametrize a triangulation of $\mathcal P_\Phi$. We show that this triangulation restricts to a triangulation of the positive root polytope $\mathcal P_\Phi^+$.

Finite Repetitive Generalized Cluster Complexes and d-Cluster Categories

For any positive integer n, we construct an n-repetitive generalized cluster complex (a simplicial complex) associated with a given finite root system by defining a compatibility degree on the n-repetitive set of the colored root system. This simplicial complex includes Fomin-Reading's generalized cluster complex as a special case when n=1. We also introduce the intermediate coverings (called generalized d-cluster categories) of d-cluster categories of hereditary algebras, and study the d-cluster tilting objects and their endomorphism algebras in those categories. In particular, we show that the endomorphism algebras of d-cluster tilting objects in the generalized d-cluster categories provide the (finite) coverings of the corresponding (usual) d-cluster tilted algebras. Moreover, we prove that the generalized d-cluster categories of hereditary algebras of finite representation type provide a category model for the n-repetitive generalized cluster complexes.

On Nichols algebras of diagonal type

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.

Vecteurs de Perron–Frobenius et produits Gamma

This Note presents formulas to express the coordinates of Perron–Frobenius vectors of Cartan matrices (finite or affine) as products of Gamma values, for each finite irreducible root system of rank r.

Polyhedral models for generalized associahedra via Coxeter elements

Motivated by the theory of cluster algebras, F. Chapoton, S. Fomin and A. Zelevinsky associated to each finite type root system a simple convex polytope called \emph{generalized associahedron}. They provided an explicit realization of this polytope associated with a bipartite orientation of the corresponding Dynkin diagram. In the first part of this paper, using the parametrization of cluster variables by their $g$-vectors explicitly computed by S.-W. Yang and A. Zelevinsky, we generalize the original construction to any orientation. In the second part we show that our construction agrees with the one given by C. Hohlweg, C. Lange, and H. Thomas in the setup of Cambrian fans developed by N. Reading and D. Speyer.