Related papers: Enhanced Dynkin diagrams and Weyl orbits

The present paper is a continuation of [5], where Lie bialgebra structures on g[u] were studied. These structures fall into different classes labelled by the vertices of the extended Dynkin diagram of g. In [5] the Lie bialgebras…

Let W be a Weyl group whose type is a simply laced Dynkin diagram. On several W-orbits of sets of mutually commuting reflections, a poset is described which plays a role in linear representatons of the corresponding Artin group A. The poset…

Motivated by work of Kac and Lusztig, we define a root system and a Weyl groupoid for a large class of semisimple Yetter-Drinfeld modules over an arbitrary Hopf algebra. The obtained combinatorial structure fits perfectly into an existing…

We extend Carter's notion of admissible diagrams and attach a "Dynkin-like" diagram to each reduced reflection factorization of an element in a finite Weyl group. We give a complete classification for the diagrams attached to reduced…

An equivalence between generalised restricted solid-on-solid (RSOS) models, associated with sets of graphs, and multi-colour loop models is established. As an application we consider solvable loop models and in this way obtain new solvable…

That finite-dimensional simple Lie algebras over the complex numbers can be classified by means of purely combinatorial and geometric objects such as Coxeter-Dynkin diagrams and indecomposable irreducible root systems, is arguably one of…

In this paper, we develop a diamond graph theory and apply the theory to the (co)homology of the Lie algebra generated by positive systems of the classical semi-simple Lie algebras over the field of complex numbers. As an application, we…

For the root system of type $B_l$ and $C_l$, we generalize the result of \cite{DZ1998} by showing the existence of a Frobenius manifold structure on the orbit space of the extended affine Weyl group that corresponds to any vertex of the…

The aim of the paper is to extend the class of generalized Weyl algebras to a larger class of rings (they are also called {\em generalized Weyl algebras}) that are determined by two ring endomorphisms rather than one as in the case of `old'…

A diagram obtained from the Carter diagram $\Gamma$ by adding one root together with its bonds such that the resulting subset of roots is linearly independent is said to be the {\it linkage diagram}. Given a linkage diagram, we associate…

Let $\Lambda$ be a simply laced root lattice and $w$ an elliptic automorphism of $\Lambda$ of order $d$. This paper gives a construction that begins with a central extension of the group of coinvariants $\Lambda_w$ and produces a semisimple…

The motivation comes from boson fermion correspondence. This article shows for each fermionic root there is correspondence a bosonic root, as a result we get for each Dynkin diagram of Lie Superalgebra a corresponding flip Dynkin…

We define pullback and separated presentations of modules over pullback rings, and, if the ring is a pullback of epimorphisms over a semisimple ring, an algorithm reducing such a presentation of a module to an $R$-diagram. The latter is the…

For any Carter diagram $\Gamma$ containing 4-cycle, we introduce the partial Cartan matrix $B_L$, which is similar to the Cartan matrix associated with a Dynkin diagram. A linkage diagram is obtained from $\Gamma$ by adding one root…

Let $R$ be an indecomposable root system. It is well known that any root is part of a basis $B$ of $R$. But when can you extend a set of two or more roots to a basis $B$ of $R$? A $\pi$-system is a linearly independent set of roots, $C$,…

We study subsets in possibly degenerate symplectic vector spaces over finite fields, which are stable under a given Coxeter/Weyl reflection group. These symplectic root systems provide crucial combinatorical data to classify…

The super Weyl group of a basic classical Lie superalgebra was introduced and studied in \cite{PS}, which turns out to play an important role for the study of representations of the basic classical Lie superalgebras and algebraic…

Any automorphism of the Dynkin diagram of a symmetrizable Kac-Moody algebra induces an automorphism of the algebra and a mapping between its highest weight modules. For a large class of such Dynkin diagram automorphisms, we can describe…

We consider the semisimple orbits of a Vinberg $\theta$-representation. First we take the complex numbers as base field. By a case by case analysis we show a technical result stating the equality of two sets of hyperplanes, one…

A higher rank generalization of the (rank one) Racah algebra is obtained as the symmetry algebra of the Laplace-Dunkl operator associated to the $\mathbb{Z}_2^n$ root system. This algebra is also the invariance algebra of the generic…