scholarly journals Free resolutions and modules with a semisimple Lie group action

2015 ◽  
Vol 7 (1) ◽  
pp. 17-29 ◽  
Author(s):  
Federico Galetto
Keyword(s):  
Group Action ◽  
Lie Group ◽  
Free Resolutions ◽  
Semisimple Lie Group ◽  
Lie Group Action

scholarly journals The stratified spaces of a symplectic lie group action

2006 ◽  
Vol 58 (1) ◽  
pp. 51-75 ◽  
Author(s):  
Juan-Pablo Ortega ◽  
Tudor S. Ratiu
Keyword(s):  
Group Action ◽  
Lie Group ◽  
Stratified Spaces ◽  
Lie Group Action

SINGULAR SWITCHED LINEAR SYSTEMS: SOME GEOMETRIC ASPECTS

2012 ◽  
Vol 26 (25) ◽  
pp. 1246006
Author(s):  
H. DIEZ-MACHÍO ◽  
J. CLOTET ◽  
M. I. GARCÍA-PLANAS ◽  
M. D. MAGRET ◽  
M. E. MONTORO
Keyword(s):  
Linear Systems ◽  
Group Action ◽  
Lie Group ◽  
Equivalence Relation ◽  
Geometric Approach ◽  
Differentiable Manifold ◽  
Equivalence Classes ◽  
Switched Linear Systems ◽  
Lie Group Action

We present a geometric approach to the study of singular switched linear systems, defining a Lie group action on the differentiable manifold consisting of the matrices defining their subsystems with orbits coinciding with equivalence classes under an equivalence relation which preserves reachability and derive miniversal (orthogonal) deformations of the system. We relate this with some new results on reachability of such systems.

scholarly journals Certain Types of Complex Lie Group Action

2018 ◽  
Vol 10 (3) ◽  
Author(s):  
Ahmed Khalaf Radhi ◽  
Taghreed Hur Majeed
Keyword(s):  
Tensor Product ◽  
Group Action ◽  
Lie Group ◽  
The Other ◽  
Smooth Representation ◽  
Equivalent Relation ◽  
Complex Group ◽  
Lie Group Action ◽  
Group 10 ◽  
Definition Of

     The main aim in this paper is to look for a novel action with new properties on       from the  , the Literature are concerned with studying the action of  of two representations , one is usual and the other is the dual, while our  interest in this work  is focused on some actions on complex Lie group[10] . Let G be a matrix complex  group , and  is representation of   In this study we will present and analytic  the  concepts of action of complex  group on    We recall the definition of  tensor  product of two representations of  group and construct  the definition of action of   group on , then by using the equivalent  relation   between  and  , we get a new action : The two actions are forming smooth  representation of    This  we have new action which called     denoted by    which acting on      This  is smooth representation of   The theoretical Justifications are developed and prove supported by some concluding  remarks and illustrations.

On Certain Polynomials of Gaussian Type

1979 ◽  
Vol 31 (2) ◽  
pp. 274-281 ◽  
Author(s):  
Daniel Reich
Keyword(s):  
Group Action ◽  
Lie Group ◽  
Orbit Space ◽  
Partition Functions ◽  
Poincaré Polynomial ◽  
Complete Answer ◽  
Lie Group Action ◽  
Image Position ◽  
Positive Coefficients ◽  
Positive Integers

Introduction. We shall consider functions of the formwhere {ri} and {si} are sets of positive integers. Such functions were studied by E. Grosswald in [2], who took {si} to be pairwise relatively prime, and asked the following two questions:(a) When is ƒ(t) a polynomial?(b) When does ƒ(t) have positive coefficients?These questions arise naturally from the work of Allday and Halperin, who show in [1] that under suitable circumstance ƒ(t) will be the Poincare polynomial of the orbit space of a certain Lie group action. Grosswald gives a complete answer to (a), but (b) is a much harder question, and a complete answer is provided only for the case m = 2. His treatment involves the representation of the coefficients of ƒ(t) by partition functions, and uses a classical description by Sylvester of the semigroup generated by {si}.

Integration of Equivariant Forms

2020 ◽  
pp. 228-232
Author(s):  
Loring W. Tu
Keyword(s):  
Group Action ◽  
Lie Group ◽  
Differential Form ◽  
Circle Action ◽  
Manifolds With Boundary ◽  
Boundary Points ◽  
Manifold With Boundary ◽  
Lie Group Action ◽  
Cartan Model ◽  
Interior Points

This chapter illustrates integration of equivariant forms. An equivariant differential form is an element of the Cartan model. For a circle action on a manifold M, it is a polynomial in u with coefficients that are invariant forms on M. Such a form can be integrated by integrating the coefficients. This can be called equivariant integration. The chapter shows that under equivariant integration, Stokes's theorem still holds. Everything done so far in this book concerning a Lie group action on a manifold can be generalized to a manifold with boundary. An important fact concerning manifolds with boundary is that a diffeomorphism of a manifold with boundary takes interior points to interior points and boundary points to boundary points.

scholarly journals MOVING FRAMES AND NOETHER’S CONSERVATION LAWS—THE GENERAL CASE

2016 ◽  
Vol 4 ◽  
Author(s):  
TÂNIA M. N. GONÇALVES ◽  
ELIZABETH L. MANSFIELD
Keyword(s):  
Conservation Laws ◽  
Group Action ◽  
Lie Group ◽  
Moving Frame ◽  
Moving Frames ◽  
Lagrange Equations ◽  
Linear Action ◽  
Independent Variables ◽  
Lie Group Action ◽  
Standard Linear

In recent works [Gonçalves and Mansfield, Stud. Appl. Math., 128 (2012), 1–29; Mansfield, A Practical Guide to the Invariant Calculus (Cambridge University Press, Cambridge, 2010)], the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler–Lagrange equations and the space of conservation laws in terms of vectors of invariants and the Adjoint representation of a moving frame. In this paper, we show how these calculations extend to the general case where the independent variables may participate in the action. We take for our main expository example the standard linear action of SL(2) on the two independent variables. This choice is motivated by applications to variational fluid problems which conserve potential vorticity. We also give the results for Lagrangians invariant under the standard linear action of SL(3) on the three independent variables.

The Cartan Model in General

2020 ◽  
pp. 167-180
Author(s):  
Loring W. Tu
Keyword(s):  
Group Action ◽  
Lie Group ◽  
Differential Form ◽  
Crucial Role ◽  
Equivariant Cohomology ◽  
Circle Action ◽  
Lie Group Action ◽  
Cartan Model ◽  
Connected Lie Group

This chapter looks at the Cartan model. Specifically, it generalizes the Cartan model from a circle action to a connected Lie group action. The chapter assumes the Lie group to be connected, because the condition that LX α‎ = 0 is sufficient for a differential form α‎ on M to be invariant holds only for a connected Lie group. It also considers the theorem that marks the transition from the Weil model to the Cartan model. It is due to Henri Cartan, who played a crucial role in the development of equivariant cohomology. The chapter then studies the Weil-Cartan isomorphism.

scholarly journals RELATIONSHIP BETWEEN EQUIVARIANT GERBES AND GERBES OVER THE QUOTIENT SPACE

2005 ◽  
Vol 07 (02) ◽  
pp. 207-226 ◽  
Author(s):  
KIYONORI GOMI
Keyword(s):  
Group Action ◽  
Lie Group ◽  
Quotient Space ◽  
Cohomology Groups ◽  
Lie Group Action

By means of cohomology groups, we study relationships between equivariant gerbes with connection over a manifold with a Lie group action and gerbes with connection over the quotient space.

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