scholarly journals Analytic Extensions of Representations of *-Subsemigroups Without Polar Decomposition

2020 ◽  
Author(s):  
Daniel Oeh
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Polar Decomposition ◽  
Complex Hilbert Space ◽  
Analytic Extension ◽  
Involutive Automorphism ◽  
Finite Dimensional ◽  
Continuous Unitary Representation ◽  
Continuous Representations ◽  
Connected Lie Group

Abstract Let $(G,\tau )$ be a finite-dimensional Lie group with an involutive automorphism $\tau $ of $G$ and let ${{\mathfrak{g}}} = {{\mathfrak{h}}} \oplus{{\mathfrak{q}}}$ be its corresponding Lie algebra decomposition. We show that every nondegenerate strongly continuous representation on a complex Hilbert space ${\mathcal{H}}$ of an open $^\ast $-subsemigroup $S \subset G$, where $s^{\ast } = \tau (s)^{-1}$, has an analytic extension to a strongly continuous unitary representation of the 1-connected Lie group $G_1^c$ with Lie algebra $[{{\mathfrak{q}}},{{\mathfrak{q}}}] \oplus i{{\mathfrak{q}}}$. We further examine the minimal conditions under which an analytic extension to the 1-connected Lie group $G^c$ with Lie algebra ${{\mathfrak{h}}} \oplus i{{\mathfrak{q}}}$ exists. This result generalizes the Lüscher–Mack theorem and the extensions of the Lüscher–Mack theorem for $^\ast $-subsemigroups satisfying $S = S(G^\tau )_0$ by Merigon, Neeb, and Ólafsson. Finally, we prove that nondegenerate strongly continuous representations of certain $^\ast $-subsemigroups $S$ can even be extended to representations of a generalized version of an Olshanski semigroup.

On Twisted Odd Graphs

2000 ◽  
Vol 9 (3) ◽  
pp. 227-240
Author(s):  
M. A. FIOL ◽  
E. GARRIGA ◽  
J. L. A. YEBRA
Keyword(s):  
Automorphism Group ◽  
Extremal Property ◽  
Involutive Automorphism ◽  
Basic Properties

The twisted odd graphs are obtained from the well-known odd graphs through an involutive automorphism. As expected, the twisted odd graphs share some of the interesting properties of the odd graphs but, in general, they seem to have a more involved structure. Here we study some of their basic properties, such as their automorphism group, diameter, and spectrum. They turn out to be examples of the so-called boundary graphs, which are graphs satisfying an extremal property that arises from a bound for the diameter of a graph in terms of its distinct eigenvalues.

INVOLUTIVE GRADINGS AND KERNELS IN JBW*-TRIPLES

2009 ◽  
Vol 02 (03) ◽  
pp. 453-463 ◽  
Author(s):  
Andreas A. Lubbe
Keyword(s):  
Involutive Automorphism ◽  
One To One

Involutive automorphism, or bijective triple homorphisms of order two, on a JBW *-triple are in a one-to-one correspondence with involutive gradings and bicontractive projections. Such mappings are always isometric and weak*-continuous. This paper analyses the algebraic kernels of the 1-eigenspace B+ and -1-eigenspace B- of involutive automorphisms. It is shown that B+ and B- give rise to several decompositions of A and, remarkably, Ker (B+) and Ker (B-) are Peirce inner ideals.

scholarly journals A Variant of D’alembert’s Matrix Functional Equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Youssef Aissi ◽  
Driss Zeglami ◽  
Mohamed Ayoubi
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Topological Groups ◽  
Involutive Automorphism ◽  
Continuous Solutions ◽  
Matrix Functional Equations

AbstractThe aim of this paper is to characterize the solutions Φ : G → M2(ℂ) of the following matrix functional equations {{\Phi \left( {xy} \right) + \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G, and {{\Phi \left( {xy} \right) - \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G, where G is a group that need not be abelian, and σ : G → G is an involutive automorphism of G. Our considerations are inspired by the papers [13, 14] in which the continuous solutions of the first equation on abelian topological groups were determined.

scholarly journals Trees and Reflection Groups

10.37236/1912 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Humberto Luiz Talpo ◽  
Marcelo Firer
Keyword(s):  
Fixed Points ◽  
Homogeneous Tree ◽  
Reflection Groups ◽  
Involutive Automorphism ◽  
Group Of Automorphisms ◽  
Topological Closure ◽  
Index 2

We define a reflection in a tree as an involutive automorphism whose set of fixed points is a geodesic and prove that, for the case of a homogeneous tree of degree $4k$, the topological closure of the group generated by reflections has index $2$ in the group of automorphisms of the tree.

On Orbit Closures of Symmetric Subgroups in Flag Varieties

2000 ◽  
Vol 52 (2) ◽  
pp. 265-292 ◽  
Author(s):  
Michel Brion ◽  
Aloysius G. Helminck
Keyword(s):  
Fixed Point ◽  
Borel Subgroup ◽  
Reductive Group ◽  
Double Coset ◽  
Restriction Map ◽  
Flag Varieties ◽  
Parabolic Subgroups ◽  
Involutive Automorphism ◽  
Orbit Closures ◽  
Geometric Analogue

AbstractWe study K-orbits in G/P where G is a complex connected reductive group, P ⊆ G is a parabolic subgroup, and K ⊆ G is the fixed point subgroup of an involutive automorphism θ. Generalizing work of Springer, we parametrize the (finite) orbit set K \ G/P and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of θ-stable (resp. θ-split) parabolic subgroups. We also describe the decomposition of any (K, P)-double coset in G into (K, B)-double cosets, where B ⊆ P is a Borel subgroup. Finally, for certain K-orbit closures X ⊆ G/B, and for any homogeneous line bundle on G/B having nonzero global sections, we show that the restriction map resX : H0(G/B, ) → H0(X, ) is surjective and that Hi(X, ) = 0 for i ≥ 1. Moreover, we describe the K-module H0(X, ). This gives information on the restriction to K of the simple G-module H0(G/B, ). Our construction is a geometric analogue of Vogan and Sepanski’s approach to extremal K-types.

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