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 ON GENERALIZATION OF SPECIAL FUNCTIONS RELATED TO WEYL GROUPS

2016 ◽  
Vol 56 (6) ◽  
pp. 440-447
Author(s):  
Lenka Háková ◽  
Agnieszka Tereszkiewicz
Keyword(s):  
Weyl Group ◽  
Lie Algebras ◽  
Special Functions ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
One Dimensional ◽  
Complex Functions ◽  
Group Orbit ◽  
Rank 2 ◽  
Definition Of

Weyl group orbit functions are defined in the context of Weyl groups of simple Lie algebras. They are multivariable complex functions possessing remarkable properties such as (anti)invariance with respect to the corresponding Weyl group, continuous and discrete orthogonality. A crucial tool in their definition are so-called sign homomorphisms, which coincide with one-dimensional irreducible representations. In this work we generalize the definition of orbit functions using characters of irreducible representations of higher dimensions. We describe their properties and give examples for Weyl groups of rank 2 and 3.

Symmetrizable Intersection Matrix Lie Algebras

2011 ◽  
Vol 18 (04) ◽  
pp. 639-646 ◽  
Author(s):  
Mang Xu ◽  
Liangang Peng
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cartan Matrix ◽  
Intersection Matrix ◽  
Special Cases ◽  
Generalized Cartan Matrix

Associated to every symmetrizable generalized intersection matrix A, we define a Lie algebra, called an SIM-Lie algebra. We prove that SIM-Lie algebras keep unchange under braid-equivalences. Two special cases are considered. In the case when A is a symmetrizable generalized Cartan matrix, we show that the corresponding SIM-Lie algebra is just the Kac–Moody Lie algebra. In another case when A is an intersection matrix, we prove that the corresponding SIM-Lie algebra is just the intersection matrix Lie algebra in the sense of Slodowy.

scholarly journals Further results on q-Lie groups, q-Lie algebras and q-homogeneous spaces

2021 ◽  
Vol 9 (1) ◽  
pp. 119-148
Author(s):  
Thomas Ernst
Keyword(s):  
Weyl Group ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Cartan Subalgebra ◽  
Homogeneous Spaces ◽  
Iwasawa Decomposition ◽  
Base Field ◽  
Killing Form ◽  
Exact Sequences

Abstract We introduce most of the concepts for q-Lie algebras in a way independent of the base field K. Again it turns out that we can keep the same Lie algebra with a small modification. We use very similar definitions for all quantities, which means that the proofs are similar. In particular, the quantities solvable, nilpotent, semisimple q-Lie algebra, Weyl group and Weyl chamber are identical with the ordinary case q = 1. The computations of sample q-roots for certain well-known q-Lie groups contain an extra q-addition, and consequently, for most of the quantities which are q-deformed, we add a prefix q in the respective name. Important examples are the q-Cartan subalgebra and the q-Cartan Killing form. We introduce the concept q-homogeneous spaces in a formal way exemplified by the examples S U q ( 1 , 1 ) S O q ( 2 ) {{S{U_q}\left( {1,1} \right)} \over {S{O_q}\left( 2 \right)}} and S O q ( 3 ) S O q ( 2 ) {{S{O_q}\left( 3 \right)} \over {S{O_q}\left( 2 \right)}} with corresponding q-Lie groups and q-geodesics. By introducing a q-deformed semidirect product, we can define exact sequences of q-Lie groups and some other interesting q-homogeneous spaces. We give an example of the corresponding q-Iwasawa decomposition for SLq(2).

SYMMETRY INSIGHTS FOR DESIGN OF SUPERCOMPUTER NETWORK TOPOLOGIES: ROOTS AND WEIGHTS LATTICES

2012 ◽  
Vol 26 (31) ◽  
pp. 1250169 ◽  
Author(s):  
YUEFAN DENG ◽  
ALEXANDRE F. RAMOS ◽  
JOSÉ EDUARDO M. HORNOS
Keyword(s):  
Weyl Group ◽  
Lie Algebra ◽  
Root Systems ◽  
Hardware Architecture ◽  
Classification Theorem ◽  
Simple Complex ◽  
Half Integer ◽  
Network Topologies ◽  
Integer Weight

We present a family of networks whose local interconnection topologies are generated by the root vectors of a semi-simple complex Lie algebra. Cartan classification theorem of those algebras ensures those families of interconnection topologies to be exhaustive. The global arrangement of the network is defined in terms of integer or half-integer weight lattices. The mesh or torus topologies that network millions of processing cores, such as those in the IBM BlueGene series, are the simplest member of that category. The symmetries of the root systems of an algebra, manifested by their Weyl group, lends great convenience for the design and analysis of hardware architecture, algorithms and programs.

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.

Phase operators and phase states associated with the su(n + 1) Lie algebra

2020 ◽  
Vol 17 (14) ◽  
pp. 2050209
Author(s):  
S. Hajji ◽  
B. Maroufi ◽  
M. Mansour ◽  
M. Daoud
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Stable Phase ◽  
Finite Dimensional ◽  
Special Cases ◽  
Phase Operators

The main aim of this work is to build unitary phase operators and the corresponding temporally stable phase states for the [Formula: see text] Lie algebra. We first introduce an irreducible finite-dimensional Hilbertian representation of the [Formula: see text] Lie algebra which is suitable for our purpose. The phase operators obtained from the [Formula: see text] generators are defined and the phase states are derived as eigenstates associated to these unitary phase operators. The special cases of [Formula: see text] and [Formula: see text] Lie algebras are also explicitly investigated.

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 Characteristic cycles and the microlocal geometry of the Gauss map, II

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Thomas Krämer
Keyword(s):  
Weak Solution ◽  
Weyl Group ◽  
Lie Algebras ◽  
Abelian Variety ◽  
Cotangent Bundle ◽  
Abelian Varieties ◽  
Gauss Map ◽  
Schottky Problem ◽  
Group Orbit

Abstract We show that any Weyl group orbit of weights for the Tannakian group of semisimple holonomic 𝒟 {{\mathscr{D}}} -modules on an abelian variety is realized by a Lagrangian cycle on the cotangent bundle. As applications we discuss a weak solution to the Schottky problem in genus five, an obstruction for the existence of summands of subvarieties on abelian varieties, and a criterion for the simplicity of the arising Lie algebras.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

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