scholarly journals Splitting definably compact groups in o-minimal structures

2011 ◽  
Vol 76 (3) ◽  
pp. 973-986
Author(s):  
Marcello Mamino
Keyword(s):  
Lie Group ◽  
Cartesian Product ◽  
Real Closed Field ◽  
Compact Groups ◽  
Closed Field ◽  
Parallel Result ◽  
Derived Subgroup ◽  
Connected Lie Group

AbstractAn argument of A. Borel [Bor-61, Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.

Equidimensional immersions of locally compact groups

1989 ◽  
Vol 105 (2) ◽  
pp. 253-261 ◽  
Author(s):  
K. H. Hofmann ◽  
T. S. Wu ◽  
J. S. Yang
Keyword(s):  
Exponential Function ◽  
Lie Group ◽  
Compact Groups ◽  
Locally Compact ◽  
Lie Group Theory ◽  
Group A ◽  
Image Position ◽  
Analytic Subgroup ◽  
Algebra Level ◽  
Connected Lie Group

Dense immersions occur frequently in Lie group theory. Suppose that exp: g → G denotes the exponential function of a Lie group and a is a Lie subalgebra of g. Then there is a unique Lie group ALie with exponential function exp:a → ALie and an immersion f:ALie→G whose induced morphism L(j) on the Lie algebra level is the inclusion a → g and which has as image an analytic subgroup A of G. The group Ā is a connected Lie group in which A is normal and dense and the corestrictionis a dense immersion. Unless A is closed, in which case f' is an isomorphism of Lie groups, dim a = dim ALie is strictly smaller than dim h = dim H.

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 On the Cardinality of A Semi-Algebraic Set

1994 ◽  
Vol 1 (3) ◽  
pp. 277-286
Author(s):  
G. Khimshiashvili
Keyword(s):  
Quadratic Forms ◽  
Real Closed Field ◽  
Closed Field ◽  
Algebraic Subset ◽  
Algebraic Set

Abstract It is shown that the cardinality of a finite semi-algebraic subset over a real closed field can be computed in terms of signatures of effectively constructed quadratic forms.

Equations and Complexity for the Dubois–Efroymson Dimension Theorem

2009 ◽  
Vol 52 (2) ◽  
pp. 224-236
Author(s):  
Riccardo Ghiloni
Keyword(s):  
Real Closed Field ◽  
Closed Field ◽  
Algebraic Subset ◽  
Algebraically Closed Field ◽  
Algebraic Set ◽  
Real Algebraic Set ◽  
Nonsingular Point ◽  
Minimum Integer ◽  
The Ideal

AbstractLetRbe a real closed field, letX⊂Rnbe an irreducible real algebraic set and letZbe an algebraic subset ofXof codimension ≥ 2. Dubois and Efroymson proved the existence of an irreducible algebraic subset ofXof codimension 1 containingZ. We improve this dimension theorem as follows. Indicate by μ the minimum integer such that the ideal of polynomials inR[x1, … ,xn] vanishing onZcan be generated by polynomials of degree ≤ μ. We prove the following two results: (1) There exists a polynomialP∈R[x1, … ,xn] of degree≤ μ+1 such thatX∩P–1(0) is an irreducible algebraic subset ofXof codimension 1 containingZ. (2) LetFbe a polynomial inR[x1, … ,xn] of degreedvanishing onZ. Suppose there exists a nonsingular pointxofXsuch thatF(x) = 0 and the differential atxof the restriction ofFtoXis nonzero. Then there exists a polynomialG∈R[x1, … ,xn] of degree ≤ max﹛d, μ + 1﹜ such that, for eacht∈ (–1, 1) \ ﹛0﹜, the set ﹛x∈X|F(x) +tG(x) = 0﹜ is an irreducible algebraic subset ofXof codimension 1 containingZ. Result (1) and a slightly different version of result (2) are valid over any algebraically closed field also.

HIDA DISTRIBUTIONS ON COMPACT LIE GROUPS

2000 ◽  
Vol 03 (03) ◽  
pp. 337-362 ◽  
Author(s):  
THOMAS DECK
Keyword(s):  
Lie Group ◽  
Noise Analysis ◽  
Growth Conditions ◽  
Nuclear Space ◽  
Compact Lie Groups ◽  
Space Of Analytic Functions ◽  
S Transform ◽  
Tensor Norm ◽  
Taylor Map ◽  
Connected Lie Group

We show that a nuclear space of analytic functions on K is associated with each compact, connected Lie group K. Its dual space consists of distributions (generalized functions on K) which correspond to the Hida distributions in white noise analysis. We extend Hall's transform to the space of Hida distributions on K. This extension — the S-transform on K — is then used to characterize Hida distributions by holomorphic functions satisfying exponential growth conditions (U-functions). We also give a tensor description of Hida distributions which is induced by the Taylor map on U-functions. Finally we consider the Wiener path group over a complex, connected Lie group. We show that the Taylor map for square integrable holomorphic Wiener functions is not isometric w.r.t. the natural tensor norm. This indicates (besides other arguments) that there might be no generalization of Hida distribution theory for (noncommutative) path groups equipped with Wiener measure.

The model theory of ordered differential fields

1978 ◽  
Vol 43 (1) ◽  
pp. 82-91 ◽  
Author(s):  
Michael F. Singer
Keyword(s):  
Meromorphic Functions ◽  
Analytic Solutions ◽  
Positive Element ◽  
Real Closed Field ◽  
Real Field ◽  
Closed Field ◽  
Ordered Field ◽  
Positive Elements ◽  
Odd Degree ◽  
Differential Fields

In this paper, we show that the theory of ordered differential fields has a model completion. We also show that any real differential field, finitely generated over the rational numbers, is isomorphic to some field of real meromorphic functions. In the last section of this paper, we combine these two results and discuss the problem of deciding if a system of differential equations has real analytic solutions. The author wishes to thank G. Stengle for some stimulating and helpful conversations and for drawing our attention to fields of real meromorphic functions.§ 1. Real and ordered fields. A real field is a field in which −1 is not a sum of squares. An ordered field is a field F together with a binary relation < which totally orders F and satisfies the two properties: (1) If 0 < x and 0 < y then 0 < xy. (2) If x < y then, for all z in F, x + z < y + z. An element x of an ordered field is positive if x > 0. One can see that the square of any element is positive and that the sum of positive elements is positive. Since −1 is not positive, an ordered field is a real field. Conversely, given a real field F, it is known that one can define an ordering (not necessarily uniquely) on F [2, p. 274]. An ordered field F is a real closed field if: (1) every positive element is a square, and (2) every polynomial of odd degree with coefficients in F has a root in F. For example, the real numbers form a real closed field. Every ordered field can be embedded in a real closed field. It is also known that, in a real closed field K, polynomials satisfy the intermediate value property, i.e. if f(x) ∈ K[x] and a, b ∈ K, a < b, and f(a)f(b) < 0 then there is a c in K such that f(c) = 0.

Basic Forms

2020 ◽  
pp. 97-102
Author(s):  
Loring W. Tu
Keyword(s):  
Lie Group ◽  
Principal Bundle ◽  
Differential Forms ◽  
Total Space ◽  
Lie Derivative ◽  
Connected Lie Group

This chapter describes basic forms. On a principal bundle π‎: P → M, the differential forms on P that are pullbacks of forms ω‎ on the base M are called basic forms. The chapter characterizes basic forms in terms of the Lie derivative and interior multiplication. It shows that basic forms on a principal bundle are invariant and horizontal. To understand basic forms better, the chapter considers a simple example. The plane ℝ2 may be viewed as the total space of a principal ℝ-bundle. A connected Lie group is generated by any neighborhood of the identity. This example shows the necessity of the connectedness hypothesis.

The Pierce-Birkhoff Conjecture for Curves

1992 ◽  
Vol 44 (6) ◽  
pp. 1262-1271 ◽  
Author(s):  
Murray Marshall
Keyword(s):  
Real Closed Field ◽  
Noetherian Rings ◽  
Closed Field ◽  
Polynomial Functions ◽  
Piecewise Polynomial ◽  
Algebraic Set ◽  
Dedekind Domains ◽  
Piecewise Polynomial Functions ◽  
Relative Case

AbstractThe results obtained extend Madden’s result for Dedekind domains to more general types of 1-dimensional Noetherian rings. In particular, these results apply to piecewise polynomial functions t:C → R where R is a real closed field and C ⊆ Rn is a closed 1-dimensional semi-algebraic set, and also to the associated “relative” case where t, C are defined over some subfield K ⊆ R.

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