CARTAN DECOMPOSITIONS AND ROOT SYSTEMS OF SEMISIMPLE LIE ALGEBRAS

Lie Algebras ◽  
1975 ◽  
pp. 48-73
Author(s):  
ZHE-XIAN WAN
Keyword(s):  
Lie Algebras ◽  
Root Systems ◽  
Semisimple Lie Algebras

Multiplicity-Free Gradings on Semisimple Lie and Jordan Algebras and Skew Root Systems

2019 ◽  
Vol 26 (01) ◽  
pp. 123-138
Author(s):  
Gang Han ◽  
Yucheng Liu ◽  
Kang Lu
Keyword(s):  
Abelian Group ◽  
Lie Algebras ◽  
Root Systems ◽  
Jordan Algebras ◽  
Homogeneous Component ◽  
New Approach ◽  
Jordan Type ◽  
Semisimple Lie Algebras ◽  
Group Gradings ◽  
Multiplicity Free

A G-grading on an algebra, where G is an abelian group, is called multiplicity-free if each homogeneous component of the grading is 1-dimensional. We introduce skew root systems of Lie type and skew root systems of Jordan type, and use them to construct multiplicity-free gradings on semisimple Lie algebras and on semisimple Jordan algebras respectively. Under certain conditions the corresponding Lie (resp., Jordan) algebras are simple. Two families of skew root systems of Lie type (resp., of Jordan type) are constructed and the corresponding Lie (resp., Jordan) algebras are identified. This is a new approach to study abelian group gradings on Lie and Jordan algebras.

Semisimple Lie Algebras of Operators

2001 ◽  
pp. 181-202
Author(s):  
Daniel Beltiţă ◽  
Mihai Şabac
Keyword(s):  
Lie Algebras ◽  
Semisimple Lie Algebras

Semisimple Lie Algebras

2020 ◽  
pp. 71-134
Author(s):  
Morikuni Goto ◽  
Frank D. Grosshans
Keyword(s):  
Lie Algebras ◽  
Semisimple Lie Algebras

Glider representations of a chain of semisimple Lie algebras

2019 ◽  
pp. 153-178
Author(s):  
Frederik Caenepeel ◽  
Fred Van Oystaeyen
Keyword(s):  
Lie Algebras ◽  
Semisimple Lie Algebras ◽  
A Chain

scholarly journals Lorentzian Toda field theories

2021 ◽  
pp. 2150017
Author(s):  
Andreas Fring ◽  
Samuel Whittington
Keyword(s):  
Lie Algebras ◽  
Mass Spectra ◽  
Energy Momentum Tensor ◽  
Root Systems ◽  
Momentum Tensor ◽  
Field Theories ◽  
Simple Lie Algebras ◽  
Different Types ◽  
Algebraic Framework ◽  
Complex Valued

We propose several different types of construction principles for new classes of Toda field theories based on root systems defined on Lorentzian lattices. In analogy to conformal and affine Toda theories based on root systems of semi-simple Lie algebras, also their Lorentzian extensions come about in conformal and massive variants. We carry out the Painlevé integrability test for the proposed theories, finding in general only one integer valued resonance corresponding to the energy-momentum tensor. Thus most of the Lorentzian Toda field theories are not integrable, as the remaining resonances, that grade the spins of the W-algebras in the semi-simple cases, are either non-integer or complex valued. We analyze in detail the classical mass spectra of several massive variants. Lorentzian Toda field theories may be viewed as perturbed versions of integrable theories equipped with an algebraic framework.

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