Systèmes de racines sur un anneau commutatif totalement ordonné

A theory of root systems over a totally ordered commutative ring is developed. This theory includes, in particular, the usual finite root systems and the Kac-Moody real root systems. It is adapted to the construction of twisted Kac-Moody groups.

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Dédié à Jacques Tits à l'occasion de son soixantième anniversaire     

A new approach to the construction of subsystems of complex root systems

Dynkin has shown how subsystems of real root systems may be constructed. As the concept of subsystems of complex root systems is not as well developed as in the real case, in this paper we give an algorithm to classify the proper subsystems of complex proper root systems. Furthermore, as an application of this algorithm, we determine the proper subsystems of imprimitive complex proper root systems. These proper subsystems are useful in giving combinatorial constructions of irreducible representations of properly generated finite complex reflection groups.     

Lie algebras graded by finite root systems and intersection matrix algebras

This paper classifies the Lie algebras graded by doubly-laced finite root systems and applies this classification to identify the intersection matrix algebras arising from multiply affinized Cartan matrices of types B,C,F, and G. This completes the determination of the Lie algebras graded by finite root systems initiated by Berman and Moody who studied the simply-laced finite root systems of rank ≧2. Oblatum 1-XI-1994 & 22-I-1996     

Locally finite root supersystems

We 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.     

On Rank 2 Complex Reflection Groups

We describe a class of groups with the property that the finite ones among them are precisely the complex reflection groups of rank 2. This situation is reminiscent of Coxeter groups, among which the finite ones are precisely the real reflection groups. We also study braid relations between complex reflections and indicate connections to an axiomatic study of root systems and to the Shephard–Todd “collineation groups.”     

Root Systems and Length Functions

We introduce root systems for those imprimitive complex reflection groups which are generated by involutory reflections, and study the associated length functions. These have many properties in common with the usual length functions for finite Weyl groups.     

Classification of arithmetic root systems

Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. They can also be considered as generalizations of ordinary root systems with rich structure and many new examples. On the other hand, Nichols algebras are fundamental objects in the construction of quantized enveloping algebras, in the noncommutative differential geometry of quantum groups, and in the classification of pointed Hopf algebras by the lifting method of Andruskiewitsch and Schneider. In the present paper arithmetic root systems are classified in full generality. As a byproduct many new finite dimensional pointed Hopf algebras are obtained.     

Convex Geometries on Root Systems

This article investigates several convex closure operators on a finite root system. It is shown that a natural closure operator on the positive root system of a finite Weyl group satisfies the anti-exchange condition for all root systems except type F ₄.     

Torsion de groupes munis d'une donnée radicielle

We consider groups endowed with root data associated with non-necessarily finite root systems. We generalise to these groups the twisting methods of Chevalley groups initiated by Steinberg and Ree. The resulting theorem (proved in 1988) can be applied to Kac–Moody groups: see for instance two papers published by J. Ramagge in 1995 [J. Ramagge, On certain fixed point subgroups of affine Kac–Moody groups, J. Algebra 171 (2) (1995) 473–514; J. Ramagge, A realization of certain affine Kac–Moody groups of types II and III, J. Algebra 171 (3) (1995) 713–806].     

Orthogonal subsets of classical root systems and coadjoint orbits of unipotent groups

We consider a specific class of coadjoint orbits of maximal unipotent subgroups in classical groups over a finite field; namely, orbits associated with orthogonal subsets in root systems. We derive a formula for the dimension of these orbits in terms of the Weyl group and construct polarizations for canonical forms on the orbits. As a consequence, we describe all possible dimensions of irreducible representations of such unipotent groups.     

Root Systems for Asymmetric Geometric Representations of Coxeter Groups

Results are obtained concerning root systems for asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a Coxeter group in such a way as to include certain restrictions of all Kac–Moody Weyl groups. In particular, a characterization of when a nontrivial multiple of a root may also be a root is given in the general context. Characterizations of when the number of such multiples of a root is finite and when the number of positive roots sent to negative roots by a group element is finite are also given. These characterizations are stated in terms of combinatorial conditions on a graph closely related to the Coxeter graph for the group. Other finiteness results for the symmetric case which are connected to the Tits cone and to a natural partial order on positive roots are extended to this asymmetric setting.     

Cohomology of moduli spaces of Del Pezzo surfaces

We compute the rational Betti cohomology groups of the coarse moduli spaces of geometrically marked Del Pezzo surfaces of degree 3 and 4 as representations of the Weyl groups of the corresponding root systems. The proof uses a blend of methods from point counting over finite fields and techniques from arrangement complements.     

Finite Aomoto-Ito-Macdonald sums

We present finite truncations of the Aomoto-Ito-Macdonald sums associated with root systems through a two-step reduction procedure. The first reduction restricts the sum from the root lattice to a Weyl chamber; the second reduction arises after imposing a truncation condition on the parameters, and gives rise to a finite sum over a Weyl alcove.

     

Rank 2 Nichols Algebras with Finite Arithmetic Root System

The concept of arithmetic root systems is introduced. It is shown that there is a one-to-one correspondence between arithmetic root systems and Nichols algebras of diagonal type having a finite set of (restricted) Poincaré–Birkhoff–Witt generators. This has strong consequences for both objects. As an application all rank 2 Nichols algebras of diagonal type having a finite set of (restricted) Poincaré–Birkhoff–Witt generators are determined. Supported by the European Community under a Marie Curie Intra-European Fellowship.     

Weyl groupoids with at most three objects

We adapt the generalization of root systems by the second author and H. Yamane to the terminology of category theory. We introduce Cartan schemes, associated root systems and Weyl groupoids. After some preliminary general results, we completely classify all finite Weyl groupoids with at most three objects. The classification yields the result that there exist infinitely many “standard”, but only 9 “exceptional” cases.     

Leibniz algebras graded by finite root systems of type C l

We study Leibniz algebras graded by finite root systems of type C _l .     

Root Systems In Finite Symplectic Vector Spaces

We study realizations of root systems in possibly degenerate symplectic vector spaces over finite fields, up to symplectic isomorphisms. The main result of this article is the classification of such realizations for the field 𝔽₂. Thereby, each root system requires a specific degree of degeneracy of the symplectic vector space. Our main motivation for this article is that for each such realization of a root system we can construct a Nichols algebra over a nonabelian group.     

The root class residuality of Baumslag–Solitar groups

Given a homomorphically closed root class K of groups, we find a criterion for a Baumslag–Solitar group to be a residually K-group. In particular, we establish that all Baumslag–Solitar groups are residually soluble and a Baumslag–Solitar group is residually finite soluble if and only if it is residually finite.     

Isomorphisms of Kac-Moody groups which preserve bounded subgroups

A subgroup of a Kac-Moody group is called bounded if it is contained in the intersection of two finite type parabolic subgroups of opposite signs. In this paper, we study the isomorphisms between Kac-Moody groups over arbitrary fields of cardinality at least 4, which preserve the set of bounded subgroups. We show that such an isomorphism between two such Kac-Moody groups induces an isomorphism between the respective twin root data of these groups. As a consequence, we obtain the solution of the isomorphism problem for Kac-Moody groups over finite fields of cardinality at least 4.     

Morita duality and graded rings

ABSTRACT

We find representatives of all the equivalence classes of simple root systems (or r.e.s. for brevity) for the complex reflection groups G ₁₂ , G ₂₄ , G ₂₅ and G ₂₆ . Then we give representatives of all the congruence classes of (essential) presentations (or r.c.p. (r.c.e.p.) for brevity) for these groups by generators and relations. The method used in the paper is applicable to any finite (complex) reflection groups.