Extending π-systems to bases of root systems

Root systems and inflations of non-negative quasi-Cartan matrices

A root system is any collection of vectors that has properties that satisfy the roots of a semi simple Lie algebra. If g is semi simple, then the root system A, (Q) can be described as a system of vectors in a Euclidean vector space that possesses some remarkable symmetries and completely defines the Lie algebra of g. The purpose of this paper is to show the essentiality of the root system on the Lie algebra. In addition, the paper will mention the connection between the root system and Ways chambers. In addition, we will show Dynkin diagrams, which are an integral part of the root system.

The affine Toda field theory is studied as a 2+1-dimensional system. The third dimension appears as the discrete space dimension, corresponding to the simple roots in the $A_N$ affine root system, enumerated according to the cyclic order on the $A_N$ affine Dynkin diagram. We show that there exists a natural discretization of the affine Toda theory, where the equations of motion are invariant with respect to permutations of all discrete coordinates. The discrete evolution operator is constructed explicitly. The thermodynamic Bethe ansatz of the affine Toda system is studied in the limit $L,N\to\infty$. Some conjectures about the structure of the spectrum of the corresponding discrete models are stated.

Given the toric (or toral) arrangement defined by a root system \Phi , we classify and count its components of each dimension. We show how to reduce to the case of 0-dimensional components, and in this case we give an explicit formula involving the maximal subdiagrams of the affine Dynkin diagram of \Phi . Then we compute the Euler characteristic and the Poincar\'{e} polynomial of the complement of the arrangement, that is the set of regular points of the torus.

The Nekrasov-Okounkov formula gives an expression for the Fourier coefficients of the Euler functions as a sum of hook length products. This formula can be deduced from a specialization in a renormalization of the affine type $A$ Weyl denominator formula and the use of a polynomial argument. In this paper, we rephrase the renormalized Weyl-Kac denominator formula as a sum parametrized by affine Grassmannian elements. This naturally gives rise to the (dual) atomic length of the root system considered introduced by Chapelier-Laget and Gerber. We then provide an interpretation of this atomic length as the cardinality of some subsets of $n$-core partitions by using foldings of affine Dynkin diagrams. This interpretation does not permit the direct use of a polynomial argument for all affine root systems. We show that this obstruction can be overcome by computing the atomic length of certain families of integer partitions. Then we show how hook-length statistics on these partitions are connected with the Coxeter length on affine Grassmannian elements and Nekrasov-Okounkov type formulas.

Our aim in this chapter is to study the classical Lie algebras sl(n, C), so(n, C), and sp(n, C) for n ≥ 2. We shall show that, with two exceptions, all these Lie algebras are simple. We shall also find their root systems and the associated Dynkin diagrams and describe their Killing forms. The main result we prove is the following theorem.

Splint is a decomposition of root system into union of root systems. Splint of root system for simple Lie algebra appears naturally in studies of (regular) embeddings of reductive subalgebras. Splint can be used to construct branching rules. We consider special embedding of Lie subalgebra to Lie algebra. We classify projections of algebra root systems using extended Dynkin diagrams and single out the conditions of splint appearance and coincidence of branching coefficients with weight multiplicities. While such a coincidence is not very common it is connected with Gelfand-Tsetlin basis.

We continue and complete a Coxeter spectral study (presented in our talk given in SYNASC11, Timisoara, September 2011 [6]) of the root systems in the sense of Bourbaki [4], the mesh geometries Γ(R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sub> , Φ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> ) of roots of Δ in the sense of [20], and matrix morsifications A ∈ Mor <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sub> , for simply-laced Dynkin diagrams Δ ∈ {A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</sub> , E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7</sub> , E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sub> }. One of the main aims of the talk is to present a progress obtained during the last year in solving the problems stated in our talk [6]. We show that the reduced mesh root systems and mesh geometries of roots for each of the Dynkin diagrams Δ can be classified for n ≤ 9 by applying symbolic computer algebra computations and numeric algorithmic computations. We also compute the isotropy group Gl(n, Z) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sub> of Δ defined in [22]-[23], determine its structure and compute the Gl(n, Z) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sub> -orbits of the morsifications A ∈ Mor <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sub> , for Δ ∈ {A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> }, with n ≤ 9 and 4 ≤ m ≤ 8. Results of our computing experiences are presented in three tables of Section 4.

Let (R, G) be a pair consisting of an elliptic root system R and a marking G of R . Assume that the attached elliptic Dynkin diagram Γ(R, G) is simply-laced. To the simply-laced elliptic root system, we associate three Lie algebras, explained in 1), 2) and 3) below. The main result of the present paper is to show that all three algebras are isomorphic.

The simple root system of each exceptional, simple Lie algebra is explicitly constructed in a variety of forms. Each construction is based on the natural embedding in the exceptional algebra of a classical, semi-simple algebra of the same rank. The procedure adopted leads to the discovery of a number of new chains of subalgebras illustrated by means of supplemented Dynkin diagrams. The various sets of simple roots are then used to determine natural labels for the irreducible representations of each of the exceptional algebras. For each such labelling scheme the modification rules for dealing with non-standard representation labels are tabulated. The connection between the natural labels and Dynkin labels is given in detail and a comparison is made with the labels of B.G. Wybourne and M.J. Bowick (1977).

Throughout this paper G(k) denotes a Chevalley group of rank n defined over the field k, where n⩾3. Let Φ be the root system associated with G(k) and let Π={α1, α2, …, αn} be a set of fundamental roots of Φ, with Φ+ being the set of positive roots of Φ with respect to Π. For α∈Π and γ∈Φ+, let nα(γ) be the coefficient of α in the expression of γ as a sum of fundamental roots; so γ=∑α∈Πnα(γ)α. Also we recall that ht(γ), the height of γ, is given by ht(γ)=∑α∈Πnα(γ). The highest root in Φ+ will be denoted by ρ. We additionally assume that the Dynkin diagram of G(k) is connected.

We show that the reduced mesh root systems and mesh geometries of roots for each of the simply-laced Dynkin diagrams can be classified by applying symbolic computer algebra computations and numeric algorithmic computations in Maple, C++ and Linux. Results of our computing experiences are presented in a table of Section 2 and in Section 6.

Extended affine Weyl groups of BCD-type: Their Frobenius manifolds and Landau–Ginzburg superpotentials

In 1982 Palev showed that the algebraic structure generated by the creation and annihilation operators of a system of m parafermions and n parabosons, satisfying the mutual parafermion relations, is the Lie superalgebra \(\mathfrak {osp}(2m+1|2n)\). The “parastatistics Fock spaces” of order p of such systems are then certain lowest weight representations of \(\mathfrak {osp}(2m+1|2n)\). We investigate now the situation when the number of parafermions and parabosons becomes infinite, which is of interest not only in a physics context but also from the mathematical point of view. In this contribution, we will discuss the various steps that are needed to understand the infinite-rank case. First, we will introduce appropriate bases and Dynkin diagrams for \(\mathfrak {B}(n,n)=\mathfrak {osp}(2n+1|2n)\) that allow us to extend \(n\rightarrow \infty \). Then we will develop a new matrix form for \(\mathfrak {B}(n,n)=\mathfrak {osp}(2n+1|2n)\), because the standard one is not appropriate for taking this limit. Following this, we construct a new Gelfand-Zetlin basis of the parastatistics Fock spaces in the finite rank case (in correspondence with this new matrix form). The new structures, related to a non-distinguished simple root system, allow the extension to \(n\rightarrow \infty \). This leads to the definition of the algebra \(\mathfrak {B}(\infty ,\infty )\) as a Lie superalgebra generated by an infinite number of creation and annihilation operators (subject to certain relations), or as an algebra of certain infinite-dimensional matrices. We study the parastatistics Fock spaces, as certain lowest weight representations of \(\mathfrak {B}(\infty ,\infty )\). In particular, we construct a basis consisting of well-described row-stable Gelfand-Zetlin patterns.