Related papers: Reflectable bases for affine reflection systems
Extended affine root systems appear as the root systems of extended affine Lie algebras. A subclass of extended affine root systems, whose elements are called ``minimal" turns out to be of special interest mostly because of the geometric…
Following the definition of a root basis of an affine root system, we define a base of the root system of an affine Lie superalgebra to be a linearly independent subset $B$ of its root system such that each root can be written as a linear…
The interaction of a Lie algebra $\LL,$ having a weight space decomposition with respect to a nonzero toral subalgebra, with its corresponding root system forms a powerful tool in the study of the structure of $\LL.$ This, in particular,…
We tackle several problems related to a finite irreducible crystallographic root system $\Phi$ in the real vector space $\mathbb E$. In particular, we study the combinatorial structure of the subsets of $\Phi$ cut by affine subspaces of…
We discuss the classification of reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case free proof is given of the well known classification of the isomorphism classes of reflection…
We study a combinatorial object, which we call a GRRS (generalized reflection root system); the classical root systems and GRSs introduced by V. Serganova are examples of finite GRRSs. A GRRS is finite if it contains a finite number of…
This is an exposition in order to give an explicit way to understand (1) a non-topological proof for an existence of a base of an affine root system, (2) a Serre-type definition of an elliptic Lie algebra with rank =>2, and (3) the…
We develop a Borel-de Siebenthal theory for affine reflection systems by classifying their maximal closed subroot systems. Affine reflection systems (introduced by Loos and Neher) provide a unifying framework for root systems of…
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 introduce the notion of extended affine Lie superalgebras and investigate the properties of their root systems. Extended affine Lie algebras, invariant affine reflection algebras, finite dimensional basic classical simple Lie…
A root systems in Carroll spaces with degenerate metric are defined. It is shown that their Cartan matrices and reflection groups are affine. With the help of the geometric consideration the root system structure of affine algebras is…
We generalize the definition and properties of root systems to complex reflection groups - roots become rank one projective modules over the ring of integers of a number field k. In the irreducible case, we provide a classification of root…
We revisit the concept of a minimal basis through the lens of the theory of modules over a commutative ring $R$. We first review the conditions for the existence of a basis for submodules of $R^n$ where $R$ is a B\'{e}zout domain. Then, we…
We introduce and study a combinatorially defined notion of root basis of a (real) root system of a possibly infinite Coxeter group. Known results on conjugacy up to sign of root bases of certain irreducible finite rank real root systems are…
We investigate a class of Lie algebras which we call {\it generalized reductive Lie algebras}. These are generalizations of semi-simple, reductive, and affine Kac-Moody Lie algebras. A generalized reductive Lie algebra which has an…
It is shown that graphs that generalize the ADE Dynkin diagrams and have appeared in various contexts of two-dimensional field theory may be regarded in a natural way as encoding the geometry of a root system. After recalling what are the…
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…
We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system.
The reflection principle is the statement that if a sentence is provable then it is true. Reflection principles have been studied for first-order theories, but they also play an important role in propositional proof complexity. In this…
The principle of boundary bootstrap plays a significant role in the algebraic study of the purely elastic boundary reflection matrix $K_a(\theta)$ for integrable quantum field theory defined on a space-time with a boundary. However, general…