Lie algebra automorphisms, the Weyl group, and tables of shift vectors

1989 ◽  
Vol 30 (3) ◽  
pp. 571-584 ◽  
Author(s):  
Peter Bouwknegt
Keyword(s):  
Weyl Group ◽  
Lie Algebra

scholarly journals Combinatorial descriptions of the crystal structure on certain PBW bases

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Ben Salisbury ◽  
Adam Schultze ◽  
Peter Tingley
Keyword(s):  
Crystal Structure ◽  
Weyl Group ◽  
Lie Algebra ◽  
Simple Complex ◽  
International Audience ◽  
The Relationship

International audience Lusztig's theory of PBW bases gives a way to realize the crystal B(∞) for any simple complex Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different such realizations, one for each reduced expression for the longest element of the Weyl group. There is an algorithm to calculate the actions of the crystal operators, but it can be quite complicated. For ADE types, we give conditions on the reduced expression which ensure that the corresponding crystal operators are given by simple combinatorial bracketing rules. We then give at least one reduced expression satisfying our conditions in every type except E8, and discuss the resulting combinatorics. Finally, we describe the relationship with more standard tableaux combinatorics in types A and D.

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).

scholarly journals A simple construction of basic polynomials invariant under the Weyl group of the simple finite-dimensional complex Lie algebra

2020 ◽  
Vol 28 (3) ◽  
pp. 301-305
Author(s):  
Askold M. Perelomov
Keyword(s):  
Weyl Group ◽  
Lie Algebra ◽  
Algebra I ◽  
Simple Construction ◽  
Finite Dimensional

AbstractFor every simple finite-dimensional complex Lie algebra, I give a simple construction of all (except for the Pfaffian) basic polynomials invariant under the Weyl group. The answer is given in terms of the two basic polynomials of smallest degree.

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.

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>.

THE -SCHUR ALGEBRAS AND -SCHUR DUALITIES OF FINITE TYPE

2020 ◽  
pp. 1-32 ◽  
Author(s):  
Li Luo ◽  
Weiqiang Wang
Keyword(s):  
Weyl Group ◽  
Lie Algebra ◽  
Finite Type ◽  
Canonical Basis ◽  
Hecke Algebra ◽  
Schur Algebras ◽  
Schur Algebra ◽  
Simple Lie Algebra ◽  
Finite Set

We formulate a $q$ -Schur algebra associated with an arbitrary $W$ -invariant finite set $X_{\text{f}}$ of integral weights for a complex simple Lie algebra with Weyl group $W$ . We establish a $q$ -Schur duality between the $q$ -Schur algebra and Hecke algebra associated with $W$ . We then realize geometrically the $q$ -Schur algebra and duality and construct a canonical basis for the $q$ -Schur algebra with positivity. With suitable choices of $X_{\text{f}}$ in classical types, we recover the $q$ -Schur algebras in the literature. Our $q$ -Schur algebras are closely related to the category ${\mathcal{O}}$ , where the type $G_{2}$ is studied in detail.

Demazure Crystals and Extended Young Diagrams

2012 ◽  
Vol 19 (02) ◽  
pp. 293-304
Author(s):  
Julie C. Beier ◽  
Kailash C. Misra
Keyword(s):  
Weyl Group ◽  
Lie Algebra ◽  
Young Diagrams ◽  
Affine Lie Algebra ◽  
Demazure Modules ◽  
Suitable Sequence ◽  
Demazure Crystals

For a suitable sequence of Weyl group elements {w(l)}l≥0 we give explicit combinatorial descriptions of the crystals Bw(l)(kΛ0) for the Demazure modules Vw(l)(kΛ0) of the quantum affine Lie algebra [Formula: see text] in terms of extended Young diagrams.

scholarly journals Icosahedral Supersymmetry

2014 ◽  
Vol 2 ◽  
pp. 52-54
Author(s):  
J.A. de Wet
Keyword(s):  
Standard Model ◽  
Weyl Group ◽  
Lie Algebra ◽  
Top Quark ◽  
Elementary Particles ◽  
Mckay Correspondence ◽  
Exceptional Lie Algebra ◽  
Icosahedral Group ◽  
Supersymmetric Particles ◽  
The Masses

The Exceptional Lie Algebra E6 used by the Author as a basis forthe Standard Model of the Elementary Particles is a subalgebra of the Lie algebra E8 which in turn is the Lie algebra of the icosahedral group by the McKay correspondence. It is possible to introduce a mass proportional toan entropy given by the the number of permutations of the elements of E6, E8 labeled by the Weyl group W. In this way the masses of the top-quark pair uu and electron are derived without any appeal to QCD and a mass of approximately 19 TeV is predicted for supersymmetric particles.

scholarly journals On the adjoint quotient of Chevalley groups over arbitrary base schemes

2010 ◽  
Vol 9 (4) ◽  
pp. 673-704 ◽  
Author(s):  
Pierre-Emmanuel Chaput ◽  
Matthieu Romagny
Keyword(s):  
Weyl Group ◽  
Lie Algebra ◽  
Chevalley Group ◽  
Maximal Torus ◽  
Adjoint Action ◽  
Group Scheme ◽  
Base Change ◽  
Chevalley Groups ◽  
Base Scheme ◽  
Arbitrary Base

AbstractFor a split semisimple Chevalley group scheme G with Lie algebra $\mathfrak{g}$ over an arbitrary base scheme S, we consider the quotient of $\mathfrak{g} by the adjoint action of G. We study in detail the structure of $\mathfrak{g} over S. Given a maximal torus T with Lie algebra $\mathfrak{t}$ and associated Weyl group W, we show that the Chevalley morphism π : $\mathfrak{t}$/W → $\mathfrak{g}/G is an isomorphism except for the group Sp2n over a base with 2-torsion. In this case this morphism is only dominant and we compute it explicitly. We compute the adjoint quotient in some other classical cases, yielding examples where the formation of the quotient $\mathfrak{g} → \mathfrak{g}//G commutes, or does not commute, with base change on S.

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