Representations of E7, equivalent combinatorics and algebraic varieties

2018 ◽  
Vol 17 (03) ◽  
pp. 1850045
Author(s):  
Xiaoping Xu
Keyword(s):  
Order Differential Operator ◽  
Moody Algebra ◽  
Oscillator Representation ◽  
Bosonic Field ◽  
Space Forms ◽  
Combinatorial Properties ◽  
First Order ◽  
Vector Formula ◽  
Operator Representations ◽  
Dimensional Module

In our earlier work on a new functor from [Formula: see text]-Mod to [Formula: see text]-Mod, we found a one-parameter ([Formula: see text]) family of inhomogeneous first-order differential operator representations of the simple Lie algebra of type [Formula: see text] in [Formula: see text] variables. Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector [Formula: see text], we prove that the space forms an irreducible [Formula: see text]-module for any [Formula: see text] if [Formula: see text] is not on an explicitly given projective algebraic variety. Certain equivalent combinatorial properties of the basic oscillator representation of [Formula: see text] over its 27-dimensional module play key roles in our proof. Our result can also be used to study free bosonic field irreducible representations of the corresponding affine Kac–Moody algebra.

First Order Operators on Manifolds With a Group Action

1996 ◽  
Vol 48 (4) ◽  
pp. 758-776 ◽  
Author(s):  
H. D. Fegan ◽  
B. Steer
Keyword(s):  
Differential Operator ◽  
Group Action ◽  
Lie Group ◽  
Differential Operators ◽  
Order Differential Operator ◽  
Compact Lie Group ◽  
First Order ◽  
Rank 2 ◽  
Homogeneous Operators

AbstractWe investigate questions of spectral symmetry for certain first order differential operators acting on sections of bundles over manifolds which have a group action. We show that if the manifold is in fact a group we have simple spectral symmetry for all homogeneous operators. Furthermore if the manifold is not necessarily a group but has a compact Lie group of rank 2 or greater acting on it by isometries with discrete isotropy groups, and let D be a split invariant elliptic first order differential operator, then D has equivariant spectral symmetry.

scholarly journals Spectrum of a Differential Operator with Periodic Generalized Potential

2007 ◽  
Vol 2007 ◽  
pp. 1-8
Author(s):  
Mehmet Sahin ◽  
Manaf Dzh. Manafov
Keyword(s):  
Differential Operator ◽  
Infinite Number ◽  
Periodic Potential ◽  
Generalized Functions ◽  
Order Differential Operator ◽  
Nonzero Constant ◽  
Classic Case ◽  
First Order ◽  
Generalized Potential ◽  
The Given

We study some spectral problems for a second-order differential operator with periodic potential. Notice that the given potential is a sum of zero- and first-order generalized functions. It is shown that the spectrum of the investigated operator consists of infinite number of gaps whose length limit unlike the classic case tends to nonzero constant in some place and to infinity in other place.

Vertex operator representations of Kac-Moody algebra

1991 ◽  
Vol 260 (1-2) ◽  
pp. 70-74 ◽  
Author(s):  
M. Sakamoto ◽  
M. Tabuse
Keyword(s):  
Vertex Operator ◽  
Moody Algebra ◽  
Operator Representations

scholarly journals Solvability of a first order differential operator on the two-torus

2014 ◽  
Vol 416 (1) ◽  
pp. 166-180
Author(s):  
A.P. Bergamasco ◽  
P.L. Dattori da Silva ◽  
A. Meziani
Keyword(s):  
Differential Operator ◽  
Order Differential Operator ◽  
First Order

scholarly journals Singularities of the eta function of first order differential operators

2012 ◽  
Vol 20 (2) ◽  
pp. 59-70
Author(s):  
Paul Loya ◽  
Sergiu Moroianu
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Closed Manifold ◽  
Order Differential Operator ◽  
First Order ◽  
Positive Integers

Abstract We report on a particular case of the paper [7], joint with Raphaël Ponge, showing that generically, the eta function of a first-order differential operator over a closed manifold of dimension n has first-order poles at all positive integers of the form n - 1; n - 3; n - 5;. . . .

The New Approximate Calculation Method for the First Order Reliability

2013 ◽  
Vol 694-697 ◽  
pp. 891-895
Author(s):  
Gang Li Hao ◽  
Wei Zao Wang ◽  
Xu Li Liang ◽  
Hai Bo Wang
Keyword(s):  
Chebyshev Polynomials ◽  
Approximate Calculation ◽  
The Other ◽  
Structural Function ◽  
First Order ◽  
Structure Reliability ◽  
First Order Second Moment ◽  
Convergence Method ◽  
Transformation Relation ◽  
Vector Formula

One new method is presented for computing engineering structure reliability by accelerating convergence based on the analysis of errors in the center point method and borrowing ideas form the merits of the other First-Order Second Moment (FOSM) methods. The accelerating convergence method is based on the first class Chebyshev polynomials. Firstly, the transformation relation of the first class Chebyshev polynomials and the power polynomials is given and expanded to vector formula. Secondly, the outstanding function approximate character of the first class Chebyshev polynomials can supply the high order information for the choosing of approximately surface. An example shows that the method presented in this article has well precision. Although the accelerating convergence method needs choose the format of the structural function properly for get the high precision results, the method can serve as an effective for engineering estimate owing to non-iterative formula and convenience.

Reducibility of differential operators some: examples

1979 ◽  
Vol 86 (1) ◽  
pp. 29-33 ◽  
Author(s):  
B. Fishel ◽  
N. Denkel
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Volterra Operator ◽  
Second Order ◽  
Order Differential Operator ◽  
Order Operator ◽  
Minimal Operator ◽  
Deficiency Indices ◽  
First Order ◽  
Symmetric Operators

A symmetric operator on a Hilbert space, with deficiency indices (m; m) has self-adjoint extensions. These are ‘highly reducible’. The original operator may be irreducible, (see example (i), below). Can the mechanism whereby reducibility is achieved be understood? The concrete examples most readily studied are those associated with differential operators. It is easy to obtain operators, associated with a formal linear differential operator, having deficiency indices (m; m). What of reducibility? Nothing seems to be known. In the case of the first-order operator we were able, using the Volterra operator, to establish irreducibility of the associated minimal operator. To investigate symmetric operators associated with a second-order differential operator, different methods had to be developed. They apply also to the first-order operator, and we employ them to demonstrate the irreducibility of the associated minimal operator. In the second-order case the minimal operator proves reducible, and we also exhibit examples of reducibility of associated symmetric operators. It would clearly be of interest to elucidate the influence of the boundary conditions on reducibility.

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