scholarly journals Continuous Homomorphisms Defined on (Dense) Submonoids of Products of Topological Monoids

Axioms ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 23
Author(s):  
Mikhail Tkachenko
Keyword(s):  
Topological Group ◽  
Lie Groups ◽  
Finite Type ◽  
Continuous Homomorphism ◽  
Open Problems ◽  
Group A ◽  
Topological Monoid

We study the factorization properties of continuous homomorphisms defined on a (dense) submonoid S of a Tychonoff product D = ∏ i ∈ I D i of topological or even topologized monoids. In a number of different situations, we establish that every continuous homomorphism f : S → K to a topological monoid (or group) K depends on at most finitely many coordinates. For example, this is the case if S is a subgroup of D and K is a first countable left topological group without small subgroups (i.e., K is an NSS group). A stronger conclusion is valid if S is a finitely retractable submonoid of D and K is a regular quasitopological NSS group of a countable pseudocharacter. In this case, every continuous homomorphism f of S to K has a finite type, which means that f admits a continuous factorization through a finite subproduct of D. A similar conclusion is obtained for continuous homomorphisms of submonoids (or subgroups) of products of topological monoids to Lie groups. Furthermore, we formulate a number of open problems intended to delimit the validity of our results.

scholarly journals DECONSTRUCTING MONOPOLES AND INSTANTONS

2000 ◽  
Vol 12 (10) ◽  
pp. 1367-1390 ◽  
Author(s):  
GIOVANNI LANDI
Keyword(s):  
Vector Bundle ◽  
Lie Groups ◽  
Finite Type ◽  
Topological Charge ◽  
Projective Modules ◽  
Dirac Monopole ◽  
Canonical Connection ◽  
Equivariant Maps ◽  
K Theory

We give a unifying description of the Dirac monopole on the 2-sphere S2, of a graded monopole on a (2, 2)-supersphere S2, 2 and of the BPST instanton on the 4-sphere S4, by constructing a suitable global projector p via equivariant maps. This projector determines the projective modules of finite type of sections of the corresponding vector bundle. The canonical connection ∇ = p ◦ d is used to compute the topological charge which is found to be equal to -1 for the three cases. The transposed projector q = pt gives the value +1 for the charges; this showing that transposition of projectors, although an isomorphism in K-theory, is not the identity map. We also study the invariance under the action of suitable Lie groups.

scholarly journals Invariant Poisson–Nijenhuis structures on Lie groups and classification

2018 ◽  
Vol 15 (04) ◽  
pp. 1850059 ◽  
Author(s):  
Zohreh Ravanpak ◽  
Adel Rezaei-Aghdam ◽  
Ghorbanali Haghighatdoost
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Symmetry Group ◽  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Text Structure ◽  
Baxter Equation ◽  
Text Structures ◽  
Group A

We study right-invariant (respectively, left-invariant) Poisson–Nijenhuis structures ([Formula: see text]-[Formula: see text]) on a Lie group [Formula: see text] and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra [Formula: see text]. We show that [Formula: see text]-[Formula: see text] structures can be used to find compatible solutions of the classical Yang–Baxter equation (CYBE). Conversely, two compatible [Formula: see text]-matrices from which one is invertible determine an [Formula: see text]-[Formula: see text] structure. We classify, up to a natural equivalence, all [Formula: see text]-matrices and all [Formula: see text]-[Formula: see text] structures with invertible [Formula: see text] on four-dimensional symplectic real Lie algebras. The result is applied to show that a number of dynamical systems which can be constructed by [Formula: see text]-matrices on a phase space whose symmetry group is Lie group a [Formula: see text], can be specifically determined.

scholarly journals Rigid hypersurfaces and infinitesimal CR automorphisms

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 77-84
Author(s):  
Atsushi Hayashimoto
Keyword(s):  
Finite Type ◽  
Open Problems ◽  
Infinite Type ◽  
Infinitesimal Cr Automorphisms

We study a survey on the relations between rigid hypersurfaces and infinitesimal CR automorphisms. After reviewing the case of hypersurfaces of finite type, we study the case of hypersurfaces of infinite type. Some open problems are posed in the last section.

Compactness of a Locally Compact Group G and Geometric Properties of Ap(G)

1996 ◽  
Vol 48 (6) ◽  
pp. 1273-1285 ◽  
Author(s):  
Tianxuan Miao
Keyword(s):  
Topological Group ◽  
Compact Group ◽  
Multiplication Operator ◽  
Locally Compact Group ◽  
Compact Groups ◽  
Locally Compact ◽  
Geometric Properties ◽  
Nikodym Property ◽  
Group A ◽  
Locally Compact Topological Group

AbstractLet G be a locally compact topological group. A number of characterizations are given of the class of compact groups in terms of the geometric properties such as Radon-Nikodym property, Dunford-Pettis property and Schur property of Ap(G), and the properties of the multiplication operator on PFp(G). We extend and improve several results of Lau and Ülger [17] to Ap(G) and Bp(G) for arbitrary p.

Locally compact groups: maximal compact subgroups and N-groups

1988 ◽  
Vol 104 (1) ◽  
pp. 47-64
Author(s):  
R. W. Bagley ◽  
T. S. Wu ◽  
J. S. Yang
Keyword(s):  
Topological Group ◽  
Normal Subgroup ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Compact Set ◽  
Locally Compact ◽  
Group A ◽  
Totally Disconnected ◽  
Compactly Generated

AbstractIf G is a locally compact group such thatG/G0contains a uniform compactly generated nilpotent subgroup, thenGhas a maximal compact normal subgroupKsuch thatG/Gis a Lie group. A topological groupGis an N-group if, for each neighbourhoodUof the identity and each compact setC⊂G, there is a neighbourhoodVof the identity such thatfor eachg∈G. Several results on N-groups are obtained and it is shown that a related weaker condition is equivalent to local finiteness for certain totally disconnected groups.

scholarly journals Moments of probability measures on a group

1981 ◽  
Vol 4 (1) ◽  
pp. 1-37 ◽  
Author(s):  
Herbert Heyer
Keyword(s):  
Topological Group ◽  
Banach Spaces ◽  
Lie Groups ◽  
Finite Groups ◽  
Lie Group ◽  
Probability Measures ◽  
Moment Conditions ◽  
New Developments ◽  
Large Numbers ◽  
Special Cases

New developments and results in the theory of expectatiors and variances for random variables with range in a topological group are presented in the following order (i) Introduction (2) Basic notions (3) The three series theorem in Banach spaces (4) Moment Conditions (5) Expectations and variances (6) A general three series theorem (7) The special cases of finite groups and Lie groups (8)The strong laws of large numbers on a Lie group (9) Further studies on moments of probability measures.

scholarly journals INVARIANT PSEUDO-SASAKIAN AND K-CONTACT STRUCTURES ON SEVEN-DIMENSIONAL NILPOTENT LIE GROUPS

2017 ◽  
pp. 91-99
Author(s):  
Nikolai Smolentsev ◽  
Nikolai Smolentsev
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Ricci Tensor ◽  
Positive Definite ◽  
Contact Structures ◽  
Nilpotent Lie Groups ◽  
Central Extensions ◽  
Metric Tensor ◽  
Group A ◽  
Contact Distribution

This paper studies the existence of left-invariant Sasaki contact structures on the seven-dimensional nilpotent Lie groups. It is shown that the only Lie group allowing Sasaki structure with a positive definite metric tensor is the Heisenberg group A complete list of 22 classes of seven-dimensional nilpotent Lie groups which admit pseudo-Riemannian Sasaki structures is found. A list of 25 classes of seven-dimensional nilpotent Lie groups admitting K-contact structures, but not pseudo-Riemannian Sasaki structures, is also presented. All the contact structures considered are central extensions of six-dimensional nilpotent symplectic Lie groups. Formulas that connect the geometric characteristics of six-dimensional nilpotent almost pseudo-Kähler Lie groups and seven-dimensional nilpotent contact Lie groups are established. As is known, for six-dimensional nilpotent pseudo-Kähler Lie groups the Ricci tensor is always zero. In contrast to the pseudo-Kӓhler case, it is shown that on contact seven-dimensional Lie algebras the Ricci tensor is nonzero even in directions of the contact distribution

Universal smooth k-tame central extension

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Finite Type ◽  
Central Extension ◽  
Group Scheme ◽  
Reductive Groups ◽  
Central Extensions ◽  
Weil Restriction ◽  
Group A ◽  
Subgroup Scheme ◽  
Minimal Type ◽  
Simply Connected

This chapter describes the construction of canonical central extensions that are analogues for perfect smooth connected affine k-groups of the simply connected central cover of a connected semisimple k-group. A commutative affine k-group scheme of finite type is k-tame if it does not contain a nontrivial unipotent k-subgroup scheme. The chapter establishes good properties of the universal smooth k-tame central extension, noting that the property “locally of minimal type” is inherited by pseudo-reductive central quotients of pseudo-reductive groups. Although inseparable Weil restriction does not generally preserve perfectness, the chapter shows that the formation of the universal smooth k-tame central extension interacts with derived groups of Weil restrictions.

On nth roots and infinitely divisible elements in a connected Lie group

1981 ◽  
Vol 89 (2) ◽  
pp. 293-299 ◽  
Author(s):  
M. McCrudden
Keyword(s):  
Topological Group ◽  
Finite Number ◽  
Lie Groups ◽  
Lie Group ◽  
Connected Components ◽  
Compact Lie Group ◽  
Infinitely Divisible ◽  
Closed Set ◽  
Image Position ◽  
Connected Lie Group

For any group G, x ∈ G and n ∈ ℕ (the natural numbers), leti.e. the set of all nth roots of x in G. If G is a Hausdorff topological group, then Rn(x, G) is a closed set in G, but may otherwise be quite complicated. However, as we have observed in (4), if G is a compact Lie group, then Rn(x, G) always has a finite number of connected components, and this result has led us to wonder about the connectedness properties of Rn(x, G) for other Lie groups G. Here is the result.

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