scholarly journals Invariant polynomials on Lie algebras of inhomogeneous unitary and special orthogonal groups

1972 ◽  
Vol 170 ◽  
pp. 221-221 ◽  
Author(s):  
S. J. Takiff
Keyword(s):  
Lie Algebras ◽  
Orthogonal Groups ◽  
Invariant Polynomials

LIE ALGEBRA OF THE q-POINCARÉ GROUP AND q-HEISENBERG COMMUTATION RELATIONS

1999 ◽  
Vol 13 (24n25) ◽  
pp. 2895-2902
Author(s):  
PAOLO ASCHIERI
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Heisenberg Algebra ◽  
Poincaré Group ◽  
Orthogonal Groups ◽  
Poincare Group ◽  
Commutation Relations ◽  
Real Forms

We discuss quantum orthogonal groups and their real forms. We review the construction of inhomogeneous orthogonal q-groups and their q-Lie algebras. The geometry of the q-Poincaré group naturally induces a well defined q-deformed Heisenberg algebra of hermitian q-Minkowski coordinates xaand momenta pa.

scholarly journals A family of simple groups associated with the Satake diagrams

1986 ◽  
Vol 41 (1) ◽  
pp. 13-43 ◽  
Author(s):  
Cheng Chonhu
Keyword(s):  
Lie Algebras ◽  
Number Field ◽  
Complex Number ◽  
Algebraic Groups ◽  
Simple Groups ◽  
Orthogonal Groups ◽  
Bruhat Decomposition ◽  
Chevalley Groups ◽  
Complex Number Field ◽  
Real Number Field

AbstractUsing the theory of the Satake diagrams associated with the non-compact simple Lie algebras over the real number field R, we shall construct a family of simple groups over a field K which are called the simple groups associated with the Satake diagrams. The list of these simple groups includes all Chevalley groups and twisted groups, and all simple algebraic groups of adjoint type defined over R if K is the complex number field C (except two types given by Table II′). Furthermore, the simple groups associated with the Satake diagrams of type AIII, BI, DI are identified with the simple groups obtained from the unitary or orthogonal groups of non-zero indices. The quasi-Bruhat decomposition of the “non-split” simple groups associated with the Satake diagrams which are not Chevalley groups or twisted groups will be given in this paper.

scholarly journals The characteristic identities and reduced matrix elements of the unitary and orthogonal groups

1978 ◽  
Vol 20 (4) ◽  
pp. 401-433 ◽  
Author(s):  
M. D. Gould
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Rational Functions ◽  
Irreducible Representations ◽  
Matrix Elements ◽  
Orthogonal Groups ◽  
Finite Dimensional ◽  
Explicit Realization ◽  
Trace Formulae ◽  
Image Position

AbstractPolynomial identities satisfied by the generators of the Lie groups O(n) and U(n) are rederived. Using these identities the reduced matrix elements of the Lie groups U(n) and O(n) are evaluated as rational functions of the IR labels occurring in the canonical chainsThis method does not require an explicit realization of the Lie algebras and their representations using bosons. Finally, trace formulae encountered previously by several authors for finite dimensional irreducible representations are shown to hold on arbitrary representations admitting an infinitesimal character.

scholarly journals Triality for Homogeneous Polynomials

2021 ◽  
Author(s):  
Laura P. Schaposnik ◽  
◽  
Sebastian Schulz ◽  
Keyword(s):  
Lie Algebras ◽  
Homogeneous Polynomials ◽  
Invariant Polynomials ◽  
Homogeneous Basis

Through the triality of SO(8,C), we study three interrelated homogeneous basis of the ring of invariant polynomials of Lie algebras, which give the basis of three Hitchin fibrations, and identify the explicit automorphisms that relate them.

The rigidity of some solvable Lie algebras

2018 ◽  
Vol 2018 (2) ◽  
pp. 43-49
Author(s):  
R.K. Gaybullaev ◽  
Kh.A. Khalkulova ◽  
J.Q. Adashev
Keyword(s):  
Lie Algebras ◽  
Solvable Lie Algebras
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