Group Actions and Riemann Surfaces

In a context of special relativity under anisotropy, Burde recently presented in this journal a new group of coordinate transformations obtained using Lie group techniques from the principles of relativity and correspondence, which show conformal metric invariance. As a consequence, this author argued that conformal invariance is a necessary condition of any anisotropic transformation satisfying those principles, and rejected the adequacy of any transformation obtained from the Lorentz transformation by a coordinate change, e.g. resynchronization such as the [Formula: see text]-Lorentz transformation. This paper argues against these assertions, and presents, as a counterexample, a relativistic generalization of the Tangherlini transformation obtained by the same Lie group method, that satisfies the previous requirements and enjoys the metric invariance property.

In this paper, the conformal invariance and conserved quantities for higher-order holonomic systems are studied. Firstly, by establishing the differential equation of motion for the systems and introducing a one-parameter infinitesimal transformation group together with its infinitesimal generator vector, the determining equation of conformal invariance for the systems are provided, and the conformal factors expression are deduced. Secondly, the relation between conformal invariance and the Lie symmetry by the infinitesimal one-parameter point transformation group for the higher-order holonomic systems are deduced. Thirdly, the conserved quantities of the systems are derived using the structure equation satisfied by the gauge function. Lastly, an example of a higher-order holonomic mechanical system is discussed to illustrate these results.

Let X be a Riemann surface which admits a group H of automorphisms and assume that X is a branched cover of its orbit space under H.We derive necessary and sufficient conditions which determine when an involution of the orbit space lifts to an automorphism of X. We extend the well known result that if the orbit space is hyperelliptic and H is abelian and fixed point free, then the hyperelliptic involution lifts to an automorphism of X; we prove that if no Weierstrass points of the orbit space are ramified in X, and if the ramification indices of all non-Weierstrass points are relatively prime in pairs, then the hyperelliptic involution lifts for any abelian group H. It is possible to use our methods to obtain conditions for lifting automorphisms of arbitrary order and we give the conditions for lifting automorphisms of order three.

Chapter 1 Galois Theory: 1.1 Action of a Solvable Group and Representability by Radicals.- 1.2 Fixed Points under an Action of a Finite Group and Its Subgroups.- 1.3 Field Automorphisms and Relations between Elements in a Field.- 1.4 Action of a k-Solvable Group and Representability by k-Radicals.- 1.5 Galois Equations.- 1.6 Automorphisms Connected with a Galois Equation.- 1.7 The Fundamental Theorem of Galois Theory.- 1.8 A Criterion for Solvability of Equations by Radicals.- 1.9 A Criterion for Solvability of Equations by k-Radicals.- 1.10 Unsolvability of Complicated Equations by Solving Simpler Equations.- 1.11 Finite Fields.- Chapter 2 Coverings: 2.1 Coverings over Topological Spaces.- 2.2 Completion of Finite Coverings over Punctured Riemann Surfaces.- Chapter 3 Ramified Coverings and Galois Theory: 3.1 Finite Ramified Coverings and Algebraic Extensions of Fields of Meromorphic Functions.- 3.2 Geometry of Galois Theory for Extensions of a Field of Meromorphic Functions.- References.- Index

The articles in these two volumes arose from papers given at the 1991 International Symposium on Geometric Group Theory, and they represent some of the latest thinking in this area. This first volume contains contributions from many of the world's leading figures in this field, and their contributions demonstrate the many interesting facets of geometrical group theory. For anyone whose interest lies in the interplay between groups and geometry, these books will be an essential addition to their library

The polymer quantization of cosmological backgrounds provides an alternative path to the original Wheeler-de Witt (WdW) quantum cosmology, based on a different representation the commutation relations of the canonical variables. This polymer representation allows to capture the lattice like structure of the quantum geometry and leads to a radically different quantum cosmology compared to the WdW construction. This new quantization scheme has attracted considerable attention due to the singularity resolution it allows in a wide class of symmetry reduced gravitational systems, in particular where the WdW scheme fails. However, as any canonical quantization scheme, ambiguities in the construction of the quantum theory, being regularization or factor-ordering ones, can drastically modify the resulting quantum dynamics. In this work, we propose a new criteria to restrict the quantization ambiguities in the simplest model of polymer quantum cosmology, for homogeneous and isotropic General Relativity minimally coupled to a massless scalar field. This new criteria is based on an underlying SL(2,ℝ) structure present in the phase space of this simple cosmological model. By preserving the symmetry of this cosmological system under this 1d conformal group, we derive a new regularization of the phase space. We perform both its polymer quantization and a quantization scheme directly providing a representation of the SL(2,ℝ) group action. The resulting quantum cosmology can be viewed as a lattice-like quantum mechanics with an SL(2,ℝ) invariance. This provides a new version of Loop Quantum Cosmology consistent with the conformal symmetry. This alternative construction opens new directions, among which a possible mapping with the conformal quantum mechanics as well as with recent matrix or tensor models constructions for quantum cosmological space-time.

In this paper we introduce the new concept of the conformal invariance and the conserved quantities for Appell systems under second-class Mei symmetry. The one-parameter infinitesimal transformation group and infinitesimal transformation vector of generator are described in detail. The conformal factor in the determining equations under second-class Mei symmetry is found. The relationship between Appell system's conformal invariance and Mei symmetry are discussed. And Appell system's conformal invariance under second-class Mei symmetry may lead to corresponding Hojman conserved quantities when the conformal invariance satisfies some conditions. Lastly, an example is provided to illustrate the application of the result.

Let $(F,J,\omega)$ be an almost K\"ahler manifold, $\alpha$ a $J$-holomorphic action of a compact Lie group $\hat K$ on $F$, and $K$ a closed normal subgroup of $\hat K$ which leaves $\omega$ invariant. We introduce gauge theoretical invariants for such triples $(F,\alpha,K)$. The invariants are associated with moduli spaces of solutions of a certain vortex type equation on a Riemann surface. We give explicite descriptions of the moduli spaces associated with the triple $(\Hom(\C^r,\C^{r_0}), \alpha_{\rm can},U(r))$, where $\alpha_{\rm can}$ denotes the canonical action of $\hat K=U(r)\times U(r_0)$ on $\Hom(\C^r,\C^{r_0})$. In the abelian case $r=1$, the new invariants can be computed expliciteley and identified with the full Seiberg-Witten invariants of ruled surfaces.

We compute the integer cohomology rings of the “polygon spaces”introduced in [F. Kirwan, Cohomology rings of moduli spaces of vector bundles over Riemann surfaces, J. Amer. Math. Soc., 5 (1992), 853-906] and [M. Kapovich & J. Millson, the symplectic geometry of polygons in Euclidean space, J. of Diff. Geometry, 44 (1996), 479-513]. This is done by embedding them in certain toric varieties; the restriction map on cohomology is surjective and we calculate its kernel using ideas from the theory of Gröbner bases. Since we do not invert the prime 2, we can tensor with Z 2 ; halving all degrees we show this produces the Z 2 cohomology rings of the planar polygon spaces. In the equilateral case, where there is an action of the symmetric group permuting the edges, we show that the induced action on the integer cohomology is not the standard one, despite it being so on the rational cohomology, cf.F. Kirwan, op. loc. Finally, our formulae for the Poincaré polynomials are more computationally effective than those known, cf.F. Kirwan op. loc.

In the moduli space {\mathcal M}_{g} of Riemann surfaces of genus g \geq 2 there is important, so-called, real locus {\mathcal R}_{g} , consisting of points representing Riemann surfaces having symmetries, by which we understand antiholomorphic involutions. {\mathcal R}_g {itself} is covered by the strata {\mathcal R}_g^{k} , each being formed by the points representing surfaces having a symmetry of given topological type k . These strata are known to be real analytic varieties of dimension 3(g-1) . Also, their topological structure is pretty well known. Natanzon–Seppälä have realized that they are connected orbifolds homeomorphic to the factors of \mathbb{R}^{3g-3} with respect to actions of some discrete groups and Goulden–Jackson–Harer and Harer–Zagier have found their Euler characteristics, expressing them through the Riemann zeta function. However, topological properties of the whole real locus {\mathcal R}_{g} were less studied. The most known fact is its connectivity, proved independently by Buser–Seppälä–Silhol, Frediani and Costa–Izquierdo. This paper can be seen as a further contribution to the study of topology of {\mathcal R}_g , {which was possible through the notion of the nerve \mathcal{N}_g , associated to {\mathcal R}_{g} and called the real nerve . We find upper bounds for its geometrical and homological dimensions and we show their sharpness for infinitely many values of {odd g . Precise values of these dimensions for even g have been found by the authors in an earlier paper.

Conformal invariance and conserved quantities for a higher-order Lagrange system by Lie point transformation of groups are studied. The differential equation of motion for the higher-order Lagrange system is introduced. The definition of conformal invariance for the system together with its determining equations and conformal factor are provided. The necessary and sufficient condition that the system's conformal invariance would be Lie symmetry by the infinitesimal one-parameter point transformation group is deduced. The conserved quantity of the system is derived using the structural equation satisfied by the gauge function. An example of a higher-order mechanical system is offered to illustrate the application of the result.

The conformal invariance and conserved quantity of Mei symmetry for a higher-order nonholonomic mechanical system are presented. Introducing an infinitesimal transformation group and infinitesimal generator vector, the definition of conformal invariance of Mei symmetry and the determining equation for the holonomic system which corresponds to a higher-order nonholonomic system are provided, and the relationship between Mei symmetry and conformal invariance of the system is discussed. The basis of restriction equations and additional restriction equations, the conformal invariance of weak and strong Mei symmetry for the higher-order nonholonomic mechanical system is constructed. With the aid of a structure equation that the gauge function satisfies, the system’s corresponding conserved quantity is derived. Finally, an example is given to illustrate the application of the method and its result.

The convex-transitivity property can be seen as a convex generalization of the almost transitive (or quasi-isotropic) group action of the isometry group of a Banach space on its unit sphere. We will show that certain Banach algebras, including conformal invariant Douglas algebras, are weak-star convex-transitive. Geometrically speaking, this means that the investigated spaces are highly symmetric. Moreover, it turns out that the symmetry property is satisfied by using only 'inner' isometries, i.e. a subgroup consisting of isometries which are homomorphisms on the algebra. In fact, weighted composition operators arising from function theory on the unit disk will do. Some interesting examples are provided at the end.

Riemann surface transformation groups

In this paper, using the method of blow-up analysis, the authors obtain an improved Trudinger–Moser inequality involving [Formula: see text]-norm ([Formula: see text]) and prove the existence of its extremal function on a closed Riemann surface [Formula: see text] with the action of a finite isometric group [Formula: see text]. To be exact, let [Formula: see text] be the usual Sobolev space, a function space [Formula: see text] and [Formula: see text] [Formula: see text] and [Formula: see text] where [Formula: see text] stands for the number of all distinct points in the set [Formula: see text]. Define [Formula: see text], where [Formula: see text] is the standard [Formula: see text]-norm on [Formula: see text]. Using blow-up analysis, we prove that if [Formula: see text], the supremum [Formula: see text] and this supremum can be attained; if [Formula: see text], the above supremum is infinite. This kind of inequality will play an important role in the study of prescribing Gaussian curvature problem and mean field equations. In particular, their result generalizes those of Chen [A Trudinger inequality on surfaces with conical singularities, Proc. Amer. Math. Soc. 108 (1990) 821–832], Yang [Extremal functions for Trudinger–Moser inequalities of Adimurthi–Druet type in dimension two, J. Differential Equations 258 (2015) 3161–3193] and Fang–Yang [Trudinger–Moser inequalities on a closed Riemannian surface with the action of a finite isometric group, Ann. Sc. Norm. Super. Pisa Cl. Sci. 20 (2020) 1295–1324].