scholarly journals Symmetries of the space of connections on a principal G-bundle and related symplectic structures

2019 ◽  
Vol 31 (10) ◽  
pp. 1950039
Author(s):  
Grzegorz Jakimowicz ◽  
Anatol Odzijewicz ◽  
Aneta Sliżewska
Keyword(s):  
Normal Subgroup ◽  
Cotangent Bundle ◽  
Total Space ◽  
Symplectic Structures ◽  
Structural Group ◽  
Reduction Procedure ◽  
One To One ◽  
Natural Way

There are two groups which act in a natural way on the bundle [Formula: see text] tangent to the total space [Formula: see text] of a principal [Formula: see text]-bundle [Formula: see text]: the group [Formula: see text] of automorphisms of [Formula: see text] covering the identity map of [Formula: see text] and the group [Formula: see text] tangent to the structural group [Formula: see text]. Let [Formula: see text] be the subgroup of those automorphisms which commute with the action of [Formula: see text]. In the paper, we investigate [Formula: see text]-invariant symplectic structures on the cotangent bundle [Formula: see text] which are in a one-to-one correspondence with elements of [Formula: see text]. Since, as it is shown here, the connections on [Formula: see text] are in a one-to-one correspondence with elements of the normal subgroup [Formula: see text] of [Formula: see text], so the symplectic structures related to them are also investigated. The Marsden–Weinstein reduction procedure for these symplectic structures is discussed.

scholarly journals Topological characterizations of amenability and congeniality of bases

2020 ◽  
Vol 21 (1) ◽  
pp. 1
Author(s):  
Sergio R. López-Permouth ◽  
Benjamin Stanley
Keyword(s):  
Direct Product ◽  
Topological Spaces ◽  
Countable Dimension ◽  
One To One ◽  
Natural Way

<div>We provide topological interpretations of the recently introduced notions of amenability and congeniality of bases of innite dimensional algebras. In order not to restrict our attention only to the countable dimension case, the uniformity of the topologies involved is analyzed and therefore the pertinent ideas about uniform topological spaces are surveyed.</div><div><p>A basis B over an innite dimensional F-algebra A is called amenable if F<sup>B</sup>, the direct product indexed by B of copies of the eld F, can be made into an A-module in a natural way. (Mutual) congeniality is a relation that serves to identify cases when different amenable bases yield isomorphic A-modules.</p><p>(Not necessarily mutual) congeniality between amenable bases yields an epimorphism of the modules they induce. We prove that this epimorphism is one-to-one only if the congeniality is mutual, thus establishing a precise distinction between the two notions.</p></div>

C-Valuations and Normal C-Orderings

1989 ◽  
Vol 41 (1) ◽  
pp. 14-67 ◽  
Author(s):  
M. Chacron
Keyword(s):  
Abelian Group ◽  
Normal Subgroup ◽  
Division Ring ◽  
Maximal Ideal ◽  
Quotient Group ◽  
Valuation Ring ◽  
Multiplicative Group ◽  
Invertible Elements ◽  
Natural Way

Let D stand for a division ring (or skewfield), let G stand for an ordered abelian group with positive infinity adjoined, and let ω: D → G. We call to a valuation of D with value group G, if ω is an onto mapping from D to G such that(i) ω(x) = ∞ if and only if x = 0,(ii) ω(x1 + x2) = min(ω (x1), ω (x2)), and(iii) ω (x1 x2) = ω (x1) + ω (x2).Associated to the valuation ω are its valuation ringR = ﹛x ∈ Dω(x) ≧ 0﹜,its maximal idealJ = ﹛x ∈ |ω(x) > 0﹜, and its residue division ring D = R/J.The invertible elements of the ring R are called valuation units. Clearly R and, hence, J are preserved under conjugation so that 1 + J is also preserved under conjugation. The latter is thus a normal subgroup of the multiplicative group Dm of D and hence, the quotient group D˙/1 + J makes sense (the residue group of ω). It enlarges in a natural way the residue division ring D (0 excluded, and addition “forgotten“).

scholarly journals QUANTUM HOMOLOGY OF FIBRATIONS OVER S2

2000 ◽  
Vol 11 (05) ◽  
pp. 665-721 ◽  
Author(s):  
DUSA J. MCDUFF
Keyword(s):  
Tensor Product ◽  
Vector Space ◽  
Main Tool ◽  
Ring Structure ◽  
Total Space ◽  
Structural Group ◽  
Rational Cohomology ◽  
Small Quantum ◽  
Space Tensor ◽  
Quantum Homology

This paper studies the (small) quantum homology and cohomology of fibrations p:P→S2 whose structural group is the group of Hamiltonian symplectomorphisms of the fiber (M, ω). It gives a proof that the rational cohomology splits additively as the vector space tensor product H*(M)⊗H*(S2), and investigates conditions under which the ring structure also splits, thus generalizing work of Lalonde–McDuff–Polterovich and Seidel. The main tool is a study of certain operations in the quantum homology of the total space P and of the fiber M, whose properties reflect the relations between the Gromov–Witten invariants of P and M. In order to establish these properties we further develop the language introduced in [22] to describe the virtual moduli cycle (defined by Liu–Tian, Fukaya–Ono, Li–Tian, Ruan and Siebert).

Cokernels of Homomorphisms from Burnside Rings to Inverse Limits

2017 ◽  
Vol 60 (1) ◽  
pp. 165-172 ◽  
Author(s):  
Masaharu Morimoto
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Inverse Limit ◽  
Minimal Normal Subgroup ◽  
Cartesian Product ◽  
Inverse Limits ◽  
Burnside Ring ◽  
A Finite Group ◽  
Burnside Rings ◽  
Natural Way

AbstractLet G be a finite group and let A(G) denote the Burnside ring of G. Then an inverse limit L(G) of the groups A(H) for proper subgroups H of G and a homomorphism res from A(G) to L(G) are obtained in a natural way. Let Q(G) denote the cokernel of res. For a prime p, let N(p) be the minimal normal subgroup of G such that the order of G/N(p) is a power of p, possibly 1. In this paper we prove that Q(G) is isomorphic to the cartesian product of the groups Q(G/N(p)), where p ranges over the primes dividing the order of G.

scholarly journals Reduction of Homogeneous Riemannian Structures

2014 ◽  
Vol 58 (1) ◽  
pp. 81-106 ◽  
Author(s):  
M. Castrillón López ◽  
I. Luján
Keyword(s):  
Normal Subgroup ◽  
Riemannian Structure ◽  
Reduction Procedure ◽  
Contact Manifolds ◽  
Hermitian Manifolds ◽  
Almost Hermitian Manifolds ◽  
Homogeneous Riemannian Structures ◽  
Homogeneous Riemannian Structure ◽  
Group Of Symmetries ◽  
Structure Tensors

AbstractThe goal of this paper is the study of homogeneous Riemannian structure tensors within the framework of reduction under a group H of isometries. In a first result, H is a normal subgroup of the group of symmetries associated with the reducing tensor . The situation when H is any group acting freely is analyzed in a second result. The invariant classes of homogeneous tensors are also investigated when reduction is performed. It turns out that the geometry of the fibres is involved in the preservation of some of them. Some classical examples illustrate the theory. Finally, the reduction procedure is applied to fibrings of almost contact manifolds over almost Hermitian manifolds. If the structure is, moreover, Sasakian, the obtained reduced tensor is homogeneous Kähler.

scholarly journals Relation between the left and right cosets of an α-normal subgroup

2021 ◽  
Vol 2106 (1) ◽  
pp. 012023
Author(s):  
Y Mahatma ◽  
I Hadi ◽  
Sudarwanto
Keyword(s):  
Normal Subgroup ◽  
Quotient Group ◽  
Left Coset ◽  
Left And Right ◽  
Natural Way

Abstract Let G be a group and α be an automorphism of G. In 2016, Ganjali and Erfanian introduced the notion of a normal subgroup related to α, called the α-normal subgroup. It is basically known that if N is an ordinary normal subgroup of G then every right coset Ng is actually the left coset gN. This fact allows us to define the product of two right cosets naturally, thus inducing the quotient group. This research investigates the relation between the left and right cosets of the relative normal subgroup. As we have done in the classic version, we then define the product of two right cosets in a natural way and continue with the construction of a, say, relative quotient group.

scholarly journals Group closures of one-to-one transformations

2001 ◽  
Vol 64 (2) ◽  
pp. 177-188 ◽  
Author(s):  
Inessa Levi
Keyword(s):  
Normal Subgroup ◽  
Symmetric Group ◽  
Permutation Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Alternating Group ◽  
One To One ◽  
Necessary And Sufficient ◽  
Infinite Set

For a semigroup S of transformations of an infinite set X let Gs be the group of all the permutations of X that preserve S under conjugation. Fix a permutation group H on X and a transformation f of X, and let 〈f: H〉 = 〈{hfh−1: h ∈ H}〉 be the H-closure of f. We find necessary and sufficient conditions on a one-to-one transformation f and a normal subgroup H of the symmetric group on X to satisfy G〈f:H〉 = H. We also show that if S is a semigroup of one-to-one transformations of X and GS contains the alternating group on X then Aut(S) = Inn(S) ≅ GS.

scholarly journals Symplectic structures and symmetries of solutions of the complex Monge-Ampére equation

1998 ◽  
Vol 150 ◽  
pp. 63-83
Author(s):  
Stanley M. Einstein-Matthews
Keyword(s):  
Dirichlet Problem ◽  
Cotangent Bundle ◽  
Symplectic Structure ◽  
Symplectic Structures ◽  
Absolutely Continuous ◽  
Nonlinear Dirichlet Problem ◽  
Homogeneous Complex ◽  
Regularity Results ◽  
Monge Ampere

Abstract.The graphs that arise from the gradients of solutions u of the homogeneous complex Monge-Ampère equation are characterized in terms of the natural symplectic structure on the cotangent bundle. This characterization is invariant under symplectic biholomorphisms. Using the symplectic structures we construct symmetries (to be called Lempert transformations) for real valued functions u which are absolutely continuous on lines. We then use these symmetries to generate interesting solutions to the homogeneous complex Monge-Ampère equation and to transform the Poincaré-Lelong equation and the ∂-equation. An example of Lempert transform is given and the main theorem is applied to prove regularity results for exterior nonlinear Dirichlet problem for the homogeneous complex Monge-Ampère equation.

Une caractérisation du fibré cotangent

1988 ◽  
Vol 38 (3) ◽  
pp. 465-472 ◽  
Author(s):  
Tong van Duc
Keyword(s):  
Lie Algebra ◽  
Direct Product ◽  
Cotangent Bundle ◽  
Vector Fields ◽  
Total Space ◽  
Lie Derivation ◽  
Infinitesimal Automorphisms

We prove that the Lie algebra of infinitesimal automorphisms of the cotangent structure on the total space of the cotangent bundle of a manifold is isomorphic to the semi-direct product of the Lie algebra of the vector fields on the manifold by the space of closed 1-forms, the vector fields operating on the forms by Lie derivation. The derivations of this algebra Lie are completely determined and we prove that it characterises the cotangent bundle.

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