scholarly journals Orthogonal Polynomials of Compact Simple Lie Groups

2011 ◽  
Vol 2011 ◽  
pp. 1-23 ◽  
Author(s):  
Maryna Nesterenko ◽  
Jiří Patera ◽  
Agnieszka Tereszkiewicz
Keyword(s):  
Simple Group ◽  
Lie Groups ◽  
Lie Group ◽  
Chebyshev Polynomials ◽  
Recurrence Relations ◽  
Uniform Method ◽  
Symmetric Polynomials ◽  
Semisimple Lie Group ◽  
Algebraic Construction ◽  
Special Case

Recursive algebraic construction of two infinite families of polynomials innvariables is proposed as a uniform method applicable to every semisimple Lie group of rankn. Its result recognizes Chebyshev polynomials of the first and second kind as the special case of the simple group of typeA1. The obtained not Laurent-type polynomials are equivalent to the partial cases of the Macdonald symmetric polynomials. Recurrence relations are shown for the Lie groups of typesA1,A2,A3,C2,C3,G2, andB3together with lowest polynomials.

Ergodic elements for actions of Lie groups

1996 ◽  
Vol 16 (4) ◽  
pp. 703-717
Author(s):  
K. Robert Gutschera
Keyword(s):  
Invariant Measure ◽  
Simple Group ◽  
Lie Groups ◽  
Lie Group ◽  
Isometry Group ◽  
Group Structure ◽  
Structure Theory ◽  
Ergodic Action ◽  
Finite Invariant Measure ◽  
Connected Lie Group

AbstractGiven a connected Lie group G acting ergodically on a space S with finite invariant measure, one can ask when G will contain single elements (or one-parameter subgroups) that still act ergodically. For a compact simple group or the isometry group of the plane, or any group projecting onto such groups, an ergodic action may have no ergodic elements, but for any other connected Lie group ergodic elements will exist. The proof uses the unitary representation theory of Lie groups and Lie group structure theory.

On Some q-Analogs of a Theorem of Kostant-Rallis

2000 ◽  
Vol 52 (2) ◽  
pp. 438-448 ◽  
Author(s):  
N. R. Wallach ◽  
J. Willenbring
Keyword(s):  
Conjugacy Class ◽  
Quantum Computation ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Cartan Subgroup ◽  
Semisimple Lie Groups ◽  
Semisimple Lie Group ◽  
Real Group ◽  
Qubit States

AbstractIn the first part of this paper generalizationsof Hesselink’s q-analog of Kostant’smultiplicity formula for the action of a semisimple Lie group on the polynomials on its Lie algebra are given in the context of the Kostant-Rallis theorem. They correspond to the cases of real semisimple Lie groups with one conjugacy class of Cartan subgroup. In the second part of the paper a q-analog of the Kostant-Rallis theorem is given for the real group SL(4, ) (that is SO(4) acting on symmetric 4 × 4 matrices). This example plays two roles. First it contrasts with the examples of the first part. Second it has implications to the study of entanglement of mixed 2 qubit states in quantum computation.

DYNAMICAL PROPERTIES OF LÉVY PROCESSES IN LIE GROUPS

2002 ◽  
Vol 02 (01) ◽  
pp. 1-23 ◽  
Author(s):  
MING LIAO
Keyword(s):  
Vector Field ◽  
Homogeneous Space ◽  
Lie Groups ◽  
Lyapunov Exponents ◽  
Lie Group ◽  
Stationary Points ◽  
Stochastic Flow ◽  
Semisimple Lie Group ◽  
Noncompact Type ◽  
Dynamical Properties

Let ϕt be a Lévy process in a semisimple Lie group G of noncompact type regarded as a stochastic flow on a homogeneous space of G, called a G-flow. We will determine the Lyapunov exponents and the stable manifolds of ϕt, and the stationary points of an associated vector field. As examples, SL (d,R)-flows and SO (1,d)-flows on SO (d) and Sd - 1 are discussed in details.

INVERSION OF HOROSPHERICAL INTEGRAL TRANSFORM ON REAL SEMISIMPLE LIE GROUPS

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 1047-1071 ◽  
Author(s):  
Anton ZORICH
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Inversion Formula ◽  
Integral Transform ◽  
Regular Representation ◽  
Integral Transforms ◽  
Semisimple Lie Groups ◽  
Semisimple Lie Group ◽  
Complex Semisimple ◽  
Local Inversion

There exists the wonderful integral transform on complex semisimple Lie groups, which assigns to a function on the group the set of its integrals over "generalized horospheres" — some specific submanifolds of the Lie group. The local inversion formula for this integral transform, discovered in 50's for [Formula: see text] by Gel'fand and Graev, made it possible to decompose the regular representation on [Formula: see text] into irreducible ones. In case of real semisimple Lie group the situation becomes more complicated, and usually there is no reasonable analogous integral transform at all. Nevertheless, in the present paper we succeed to define the integral transforms on the Lorentz group and some other real semisimple Lie groups, which are in a sense analogous to the integration over "horospheres". We obtain the inversion formulas for these integral transforms.

scholarly journals Differential equations and an analog of the Paley-Wiener theorem for linear semisimple Lie groups

1976 ◽  
Vol 64 ◽  
pp. 17-29 ◽  
Author(s):  
Kenneth D. Johnson
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Group ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Nilpotent Subgroup ◽  
Iwasawa Decomposition ◽  
Semisimple Lie Groups ◽  
Semisimple Lie Group ◽  
Simply Connected

Let G be a noncompact linear semisimple Lie group. Fix G = KAN an Iwasawa decomposition of G. That is, K is a maximal compact subgroup of G, A is a vector subgroup with AdA consisting of semisimple transformations and A normalizes N, a simply connected nilpotent subgroup of G.

scholarly journals Central lacunary sets for Lie groups

1988 ◽  
Vol 45 (1) ◽  
pp. 30-45 ◽  
Author(s):  
A. H. Dooley
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Maximal Torus ◽  
Infinite Subset ◽  
Semisimple Lie Group ◽  
Infinite Sets ◽  
Connected Lie Group

AbstractIf G is a compact connected Lie group every infinite subset of Ĝ contains an infinite central Λ(p) set, for p < 2 + 2 rank G/(dim G - rank G). A subset R of Ĝ is of type central Λ(2) if and only if the associated set of characters on the maximal torus is of type Λ(2). The dual of a compact connected semisimple Lie group contains infinite sets which are central p-Sidon for all p > 1. Every infinite subset of the dual of Su(2) contains such a set.

scholarly journals Products of protopological groups

2001 ◽  
Vol 28 (7) ◽  
pp. 433-435
Author(s):  
Julie C. Jones
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Characterization Theorem ◽  
Invariant Subgroup ◽  
Topological Groups ◽  
Special Case

Montgomery and Zippin saied that a group is approximated by Lie groups if every neighborhood of the identity contains an invariant subgroupHsuch thatG/His topologically isomorphic to a Lie group. Bagley, Wu, and Yang gave a similar definition, which they called a pro-Lie group. Covington extended this concept to a protopological group. Covington showed that protopological groups possess many of the characteristics of topological groups. In particular, Covington showed that in a special case, the product of protopological groups is a protopological group. In this note, we give a characterization theorem for protopological groups and use it to generalize her result about products to the category of all protopological groups.

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

scholarly journals The Higher Order Riesz Transform andBMOType Space Associated with Schrödinger Operators on Stratified Lie Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Yu Liu ◽  
Jianfeng Dong
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Riesz Transform ◽  
Integral Operators ◽  
Higher Order ◽  
Reverse Hölder Class ◽  
Stratified Lie Group ◽  
Homogeneous Dimension ◽  
Stratified Lie Groups ◽  
Reverse Hölder

Assume thatGis a stratified Lie group andQis the homogeneous dimension ofG. Let-Δbe the sub-Laplacian onGandW≢0a nonnegative potential belonging to certain reverse Hölder classBsfors≥Q/2. LetL=-Δ+Wbe a Schrödinger operator on the stratified Lie groupG. In this paper, we prove the boundedness of some integral operators related toL, such asL-1∇2,L-1W, andL-1(-Δ) on the spaceBMOL(G).

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