On generalizations of root systems

We define a generalization of a root system as a set of vectors in a vector space with some symmetry property. The main difference with the usual root systems is the existence of isotropic roots. We classify irreducible generalized root systems. As follows from our classification all such root systems are root systems of contragredient Lie superalgebras which were classified by V.Kac in 1977.     

Embeddings of root systems I: Root systems over commutative rings

This paper gives examples of embeddings of root systems of Coxeter groups, including sporadic embeddings of standard real root systems in other standard real root systems, and, for a general Coxeter group, an embedding of its universal symmetric root system over commutative rings into the standard real root system of a simply laced Coxeter group.     

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.     

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.     

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     

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.     

Root subsystems of loop extensions

We completely classify the real root subsystems of root systems of loop algebras of Kac–Moody Lie algebras. This classification involves new notions of “admissible subgroups” of the coweight lattice of a root system Ψ, and “scaling functions” on Ψ. Our results generalise and simplify earlier work on subsystems of real affine root systems.     

A Euclidean interpretation of Dynkin diagrams and its relation to root systems

A simple metric property satisfied by bases of (finite, not necessarily reduced) root systems is used to define sets in Euclidean space that provide models for Dynkin diagrams and their positive semidefinite one-vertex extensions. The theory of root systems can be founded on the study of these Dynkin sets, and conversely the Dynkin sets representing connected diagrams can be characterized as the bases and extended bases of root systems. (By an extended base, we mean a base together with the lowest root of a given length.) In this correspondence the role of nonreduced root systems is natural and important.     

基于二次算子族的无界Hamilton算子根向量组的基性质

利用无界Hamilton算子导出的二次算子族,本文研究了一类无界Hamilton算子根向量组的Schauder基性质.首先,建立了无界Hamilton算子的根向量与相应的二次算子族的根向量之间的关系.其次,借助二次算子族谱的相关性质,刻画了无界Hamilton算子的本征值分布以及本征值的代数指标,并得到了无界Hamilton算子的根向量组是某个Hilbert空间的一个块状Schauder基的充要条件.最后,将所得结果应用于矩形薄板弯曲问题.     

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.     

Simply-laced isomonodromy systems

A new class of isomonodromy equations will be introduced and shown to admit Kac?CMoody Weyl group symmetries. This puts into a general context some results of Okamoto on the 4th, 5th and 6th Painlevé equations, and shows where such Kac?CMoody Weyl groups and root systems occur ??in nature??. A key point is that one may go beyond the class of affine Kac?CMoody root systems. As examples, by considering certain hyperbolic Kac?CMoody Dynkin diagrams, we find there is a sequence of higher order Painlevé systems lying over each of the classical Painlevé equations. This leads to a conjecture about the Hilbert scheme of points on some Hitchin systems.     

On constructing periodic solutions of quasi-linear non-self-contained systems with one degree of freedom, when amplitude equations possess multiple roots

Construction of periodic solutions of quasilinear non-self-contained systems with one degree of freedom, was investigated in [1 and 2]. In [1] the case of simple roots of amplitude equations was considered together with the case of a double root when the solution could be expanded into a series in integral powers of μ. In [2] the case of a double root is investigated in more detail Including expansions of solutions into series in μ^1/2. In the present paper, the case of arbitrary multiple roots for non-self-contained systems is reduced to the corresponding case for self-contained systems, which simplifies computations.     

Parabolic subgroup orbits on finite root systems

Oshima's Lemma describes the orbits of parabolic subgroups of irreducible finite Weyl groups on crystallographic root systems. This note generalises that result to all root systems of finite Coxeter groups, and provides a self contained proof, independent of the representation theory of semisimple complex Lie algebras.     

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.     

Multiplicative Invariants of Root Lattices

We describe the multiplicative invariant algebras of the root lattices of all irreducible root systems under the action of the Weyl group. In each case, a finite system of fundamental invariants is determined and the class group of the invariant algebra is calculated. In some cases, a presentation and a Hironaka decomposition of the invariant algebra is given.     

Logarithmic Frobenius structures and Coxeter discriminants

We consider a class of solutions of the WDVV equation related to the special systems of covectors (called ∨-systems) and show that the corresponding logarithmic Frobenius structures can be naturally restricted to any intersection of the corresponding hyperplanes. For the Coxeter arrangements the corresponding structures are shown to be almost dual in Dubrovin's sense to the Frobenius structures on the strata in the discriminants discussed by Strachan. For the classical Coxeter root systems this leads to the families of ∨-systems from the earlier work by Chalykh and Veselov. For the exceptional Coxeter root systems we give the complete list of the corresponding ∨-systems. We present also some new families of ∨-systems, which cannot be obtained in such a way from the Coxeter root systems.     

The algorithmic beauty of plant roots – an L-System model for dynamic root growth simulation

Understanding the impact of root architecture on plant resource efficiency is important, in particular, in the light of upcoming shortages of mineral fertilizers and changed environmental conditions. In the 1950s, a great number of root systems of European cultivated plants were excavated and studied by L. Kutschera (1960). Her work gave enormous insight into the variety of root system architectures and helped to realize the importance of belowground processes to plant productivity. We analysed the resulting hand drawings by using mathematical modelling and found root system parameters for a newly developed parametric L-System model. In this way we were able to first reproduce the illustrations, second computationally analyse root system traits and finally access the dynamic root architecture development.