Combinatorics of extended affine root systems (type A1)

2019 ◽  
Vol 18 (03) ◽  
pp. 1950051
Author(s):  
Saeid Azam ◽  
Zahra Kharaghani
Keyword(s):  
Weyl Group ◽  
Lie Algebras ◽  
Root Systems ◽  
Extended Version ◽  
Type Formula ◽  
Exchange Condition ◽  
Affine Root Systems ◽  
Affine Reflection ◽  
Natural Way

We establish extensions of some important features of affine theory to affine reflection systems (extended affine root systems) of type [Formula: see text]. We present a positivity theory which decomposes in a natural way the nonisotropic roots into positive and negative roots, then using that, we give an extended version of the well-known exchange condition for the corresponding Weyl group, and finally give an extended version of the Bruhat ordering and the [Formula: see text]-Lemma. Furthermore, a new presentation of the Weyl group in terms of the parity permutations is given, this in turn leads to a parity theorem which gives a characterization of the reduced words in the Weyl group. All root systems involved in this work appear as the root systems of certain well-studied Lie algebras.

ROOT SYSTEMS ARISING FROM AUTOMORPHISMS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250057 ◽  
Author(s):  
SAEID AZAM ◽  
MALIHE YOUSOSFZADEH
Keyword(s):  
Lie Algebras ◽  
Root Systems ◽  
Affine Lie Algebras ◽  
Combinatorial Approach ◽  
Reflection Algebras ◽  
Affine Root Systems ◽  
Outstanding Work ◽  
Affine Reflection

We study a combinatorial approach of producing new root systems from the old ones in the context of affine root systems and their new generalizations. The appearance of this approach in the literature goes back to the outstanding work of Kac in the realization of affine Kac–Moody Lie algebras. In recent years, this approach has been appeared in many other works, including the study of affinization of extended affine Lie algebras and invariant affine reflection algebras.

scholarly journals QUASI ROOT SYSTEMS AND VERTEX OPERATOR REALIZATIONS OF THE VIRASORO ALGEBRA

2013 ◽  
Vol 12 (07) ◽  
pp. 1350028
Author(s):  
BORIS NOYVERT
Keyword(s):  
Lie Algebras ◽  
Virasoro Algebra ◽  
Root Systems ◽  
Vertex Operators ◽  
Boson Fields ◽  
Dimension 2 ◽  
Definition Of ◽  
Extensive List ◽  
Scaling Dimension ◽  
Natural Way

A construction of the Virasoro algebra in terms of free massless two-dimensional boson fields is studied. The ansatz for the Virasoro field contains the most general unitary scaling dimension 2 expression built from vertex operators. The ansatz leads in a natural way to a concept of a quasi root systems. This is a new notion generalizing the notion of a root system in the theory of Lie algebras. We introduce a definition of a quasi root systems and provide an extensive list of examples. Explicit solutions of the ansatz are presented for a range of quasi root systems.

scholarly journals ON CUBATURE RULES ASSOCIATED TO WEYL GROUP ORBIT FUNCTIONS

2016 ◽  
Vol 56 (3) ◽  
pp. 202 ◽  
Author(s):  
Lenka Háková ◽  
Jiří Hrivnák ◽  
Lenka Motlochová
Keyword(s):  
Weyl Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Jacobi Polynomials ◽  
Special Functions ◽  
Root Systems ◽  
Full Generality ◽  
Cubature Formulas ◽  
Special Cases ◽  
Group Orbit

The aim of this article is to describe several cubature formulas related to the Weyl group orbit functions, i.e. to the special cases of the Jacobi polynomials associated to root systems. The diagram containing the relations among the special functions associated to the Weyl group orbit functions is presented and the link between the Weyl group orbit functions and the Jacobi polynomials is explicitly derived in full generality. The four cubature rules corresponding to these polynomials are summarized for all simple Lie algebras and their properties simultaneously tested on model functions. The Clenshaw-Curtis method is used to obtain additional formulas connected with the simple Lie algebra <em>C</em><sub>2</sub>.

scholarly journals ISOMORPHIC INDUCED MODULES AND DYNKIN DIAGRAM AUTOMORPHISMS OF SEMISIMPLE LIE ALGEBRAS

2015 ◽  
Vol 58 (1) ◽  
pp. 187-203
Author(s):  
JÉRÉMIE GUILHOT ◽  
CÉDRIC LECOUVEY
Keyword(s):  
Tensor Product ◽  
Weyl Group ◽  
Lie Algebras ◽  
Dynkin Diagram ◽  
Root Systems ◽  
The Other ◽  
Converse Problem ◽  
Highest Weight ◽  
Diagram Automorphisms ◽  
Levi Subalgebra

AbstractConsider a simple Lie algebra $\mathfrak{g}$ and $\overline{\mathfrak{g}}$ ⊂ $\mathfrak{g}$ a Levi subalgebra. Two irreducible $\overline{\mathfrak{g}}$-modules yield isomorphic inductions to $\mathfrak{g}$ when their highest weights coincide up to conjugation by an element of the Weyl group W of $\mathfrak{g}$ which is also a Dynkin diagram automorphism of $\overline{\mathfrak{g}}$. In this paper, we study the converse problem: given two irreducible $\overline{\mathfrak{g}}$-modules of highest weight μ and ν whose inductions to $\mathfrak{g}$ are isomorphic, can we conclude that μ and ν are conjugate under the action of an element of W which is also a Dynkin diagram automorphism of $\overline{\mathfrak{g}}$? We conjecture this is true in general. We prove this conjecture in type A and, for the other root systems, in various situations providing μ and ν satisfy additional hypotheses. Our result can be interpreted as an analogue for branching coefficients of the main result of Rajan [6] on tensor product multiplicities.

scholarly journals A Length Function for Weyl Groups of Extended Affine Root Systems of Type A1

2015 ◽  
Vol 22 (04) ◽  
pp. 621-638 ◽  
Author(s):  
Saeid Azam ◽  
Mohammad Nikouei
Keyword(s):  
Weyl Group ◽  
Root Systems ◽  
Type A ◽  
Length Function ◽  
Weyl Groups ◽  
Combinatorial Properties ◽  
Affine Root Systems

In this work, we study the concept of the length function and some of its combinatorial properties for the class of extended affine root systems of type A1. We introduce a notion of root basis for these root systems, and using a unique expression of the elements of the Weyl group with respect to a set of generators for the Weyl group, we calculate the length function with respect to a very specific root basis.

On Closed Subsets of Root Systems

1994 ◽  
Vol 37 (3) ◽  
pp. 338-345 ◽  
Author(s):  
D. Ž. Doković ◽  
P. Check ◽  
J.-Y. Hée
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Irreducible Component ◽  
Product Space ◽  
Root Systems ◽  
Inner Product Space ◽  
Inner Product ◽  
Closed Sets ◽  
Finite Dimensional ◽  
Real Inner Product Space

AbstractLet R be a root system (in the sense of Bourbaki) in a finite dimensional real inner product space V. A subset P ⊂ R is closed if α, β ∊ P and α + β ∊ R imply that α + β ∊ P. In this paper we shall classify, up to conjugacy by the Weyl group W of R, all closed sets P ⊂ R such that R\P is also closed. We also show that if θ:R —> R′ is a bijection between two root systems such that both θ and θ-1 preserve closed sets, and if R has at most one irreducible component of type A1, then θ is an isomorphism of root systems.

Central automorphisms of free nilpotent Lie algebras

2017 ◽  
Vol 16 (11) ◽  
pp. 1750205
Author(s):  
Özge Öztekin ◽  
Naime Ekici
Keyword(s):  
Lie Algebras ◽  
Finite Rank ◽  
Characteristic Zero ◽  
Sufficient Conditions ◽  
Nilpotency Class ◽  
Nilpotent Lie Algebra ◽  
Central Automorphism ◽  
Nilpotent Lie Algebras ◽  
Central Automorphisms

Let [Formula: see text] be the free nilpotent Lie algebra of finite rank [Formula: see text] [Formula: see text] and nilpotency class [Formula: see text] over a field of characteristic zero. We give a characterization of central automorphisms of [Formula: see text] and we find sufficient conditions for an automorphism of [Formula: see text] to be a central automorphism.

Characterization of the bacterial flora associated with root systems of Pinus contorta var. latifolia

1978 ◽  
Vol 24 (12) ◽  
pp. 1520-1525 ◽  
Author(s):  
J. A. Dangerfield ◽  
D. W. S. Westlake ◽  
F. D. Cook
Keyword(s):  
Amino Acid ◽  
Pinus Contorta ◽  
Bacterial Flora ◽  
Root Systems ◽  
Growth Media ◽  
Agricultural Crops ◽  
Control Soil ◽  
Mature Tree ◽  
And Control

Root systems of young and mature lodgepole pine (Pinus contorta Dougl. var. latifolia Englem.) were removed from forest stands and the associated aerobic bacterial flora isolated. Characterization of rhizoplane and control soil isolates from these tree root systems demonstrated differences from that reported for agricultural crops. Ammonifying, proteolytic, and amylolytic organisms were proportionately reduced within the rhizoplane. The rhizoplane organisms grew more slowly than the control soil isolates, although they responded in greater numbers to the addition of an amino acid supplement to the growth media. The rhizoplane organisms also showed an increased ability to solubilize phosphate. The chitinolytic organisms were suppressed within the rhizoplane of the mature tree but were stimulated by the young trees. With this exception, the rhizoplane microflora of older and younger trees were similar.

Some studies on central derivation of nilpotent Lie superalgebra

2018 ◽  
Vol 13 (04) ◽  
pp. 2050068
Author(s):  
Rudra Narayan Padhan ◽  
K. C. Pati
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Nilpotent Lie Algebra ◽  
New Concepts

Many theorems and formulas of Lie superalgebras run quite parallel to Lie algebras, sometimes giving interesting results. So it is quite natural to extend the new concepts of Lie algebra immediately to Lie superalgebra case as the later type of algebras have wide applications in physics and related theories. Using the concept of isoclinism, Saeedi and Sheikh-Mohseni [A characterization of stem algebras in terms of central derivations, Algebr. Represent. Theory 20 (2017) 1143–1150; On [Formula: see text]-derivations of Filippov algebra, to appear in Asian-Eur. J. Math.; S. Sheikh-Mohseni, F. Saeedi and M. Badrkhani Asl, On special subalgebras of derivations of Lie algebras, Asian-Eur. J. Math. 8(2) (2015) 1550032] recently studied the central derivation of nilpotent Lie algebra with nilindex 2. The purpose of the present paper is to continue and extend the investigation to obtain some similar results for Lie superalgebras, as isoclinism in Lie superalgebra is being recently introduced.

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