Enveloping Algebras of Semi-Simple Lie Algebras

1950 ◽  
Vol 2 ◽  
pp. 257-266 ◽  
Author(s):  
N. Jacobson
Keyword(s):  
Quantum Mechanics ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Finite Basis ◽  
Multiplication Table ◽  
Polynomial Equations ◽  
Systems Of Equations ◽  
Enveloping Algebras ◽  
Starting Point ◽  
The Matrix

In a recent paper we studied systems of equations of the form(1) (2) where as usual [a,b] = ab — ba and ϕ(λ) is a polynomial. Equations of this type have arisen in quantum mechanics. In our paper we gave a method of determining the matrix solutions of such equations. The starting point of our discussion was the observation that if the elements xi satisfy (1) then the elements xi, [xj,xk] satisfy the multiplication table of a certain basis of the Lie algebra of skew symmetric (n + 1) ⨯ (n + 1) matrices. We proved that if (2) is imposed as an added condition, then the algebra generated by the has a finite basis, and we obtained the structure of the most general associative algebra that is generated in this way.

CLASSICAL LIE SUPERALGEBRAS OVER SIMPLE ASSOCIATIVE ALGEBRAS

2006 ◽  
Vol 92 (3) ◽  
pp. 581-600 ◽  
Author(s):  
GEORGIA BENKART ◽  
XIAOPING XU ◽  
KAIMING ZHAO
Keyword(s):  
Lie Algebras ◽  
Associative Algebra ◽  
Weyl Algebra ◽  
Identity Element ◽  
Lie Superalgebras ◽  
Associative Algebras ◽  
Matrix Algebras ◽  
Weyl Algebras ◽  
Conformal Algebras ◽  
The Matrix

Over arbitrary fields of characteristic not equal to 2, we construct three families of simple Lie algebras and six families of simple Lie superalgebras of matrices with entries chosen from different one-sided ideals of a simple associative algebra. These families correspond to the classical Lie algebras and superalgebras. Our constructions intermix the structure of the associative algebra and the structure of the matrix algebra in an essential, compatible way. Many examples of simple associative algebras without an identity element arise as a by-product. The study of conformal algebras and superalgebras often involves matrix algebras over associative algebras such as Weyl algebras, and for that reason, we illustrate our constructions by taking various one-sided ideals from a Weyl algebra or a quantum torus.

Embracing Uncertainty: Modeling Uncertainty in EPMA—Part II

2021 ◽  
Vol 27 (1) ◽  
pp. 74-89
Author(s):  
Nicholas W.M. Ritchie
Keyword(s):  
Beam Energy ◽  
Uncertainty Modeling ◽  
Matrix Correction ◽  
Correction Model ◽  
X Ray ◽  
Starting Point ◽  
Uncertainty Calculation ◽  
The Matrix ◽  
Measurement Design ◽  
New Framework

AbstractThis, the second in a series of articles present a new framework for considering the computation of uncertainty in electron excited X-ray microanalysis measurements, will discuss matrix correction. The framework presented in the first article will be applied to the matrix correction model called “Pouchou and Pichoir's Simplified Model” or simply “XPP.” This uncertainty calculation will consider the influence of beam energy, take-off angle, mass absorption coefficient, surface roughness, and other parameters. Since uncertainty calculations and measurement optimization are so intimately related, it also provides a starting point for optimizing accuracy through choice of measurement design.

scholarly journals Going Up for Enveloping Algebras of Lie Algebras

2001 ◽  
Vol 243 (2) ◽  
pp. 539-550 ◽  
Author(s):  
Boris Širola
Keyword(s):  
Lie Algebras ◽  
Enveloping Algebras

Trace Forms on Lie Algebras

1962 ◽  
Vol 14 ◽  
pp. 553-564 ◽  
Author(s):  
Richard Block
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Symmetric Bilinear Form ◽  
Trace Form ◽  
Finite Degree ◽  
Trace Forms ◽  
Matrix Products ◽  
The Matrix ◽  
Image Position

If L is a Lie algebra with a representation Δ a→aΔ (a in L) (of finite degree), then by the trace form f = fΔ of Δ is meant the symmetric bilinear form on L obtained by taking the trace of the matrix products:Then f is invariant, that is, f is symmetric and f(ab, c) — f(a, bc) for all a, b, c in L. By the Δ-radical L⊥ = L⊥ of L is meant the set of a in L such that f(a, b) = 0 for all b in L. Then L⊥ is an ideal and f induces a bilinear form , called a quotient trace form, on L/L⊥. Thus an algebra has a quotient trace form if and only if there exists a Lie algebra L with a representation Δ such that

Euclidean path integrals and quantum mechanics (QM)

2021 ◽  
pp. 18-41
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Quantum Mechanics ◽  
Path Integral ◽  
Correlation Functions ◽  
Path Integrals ◽  
Continuous Phase ◽  
Functional Integral ◽  
Smooth Transition ◽  
Path Integral Representation ◽  
The Matrix ◽  
Functional Measure

Functional integrals are basic tools to study first quantum mechanics (QM), and quantum field theory (QFT). The path integral formulation of QM is well suited to the study of systems with an arbitrary number of degrees of freedom. It makes a smooth transition between nonrelativistic QM and QFT possible. The Euclidean functional integral also emphasizes the deep connection between QFT and the statistical physics of systems with short-range interactions near a continuous phase transition. The path integral representation of the matrix elements of the quantum statistical operator e-β H for Hamiltonians of the simple separable form p2/2m +V(q) is derived. To the path integral corresponds a functional measure and expectation values called correlation functions, which are generalized moments, and related to quantum observables, after an analytic continuation in time. The path integral corresponding to the Euclidean action of a harmonic oscillator, to which is added a time-dependent external force, is calculated explicitly. The result is used to generate Gaussian correlation functions and also to reduce the evaluation of path integrals to perturbation theory. The path integral also provides a convenient tool to derive semi-classical approximations.

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