A Homological Version of the Implicit Function Theorem

A general mathematical framework for the super Lie groups of supersymmetric theories is presented. The definition of super Lie group is given in terms of supermanifolds, and two theorems (analogous to theorems in classical Lie group theory) are proved. The relationship of the super Lie groups defined here to the formal groups of Berezin and Kac and the graded Lie groups of Kostant is analyzed.

In this paper, we study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal. We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of [P. Hall and C. R. Kulatilaka, A property of locally finite groups, J. Lond. Math. Soc. 39 1964, 235–239] and a characterization of a certain class of Lie groups, due to [S. K. Grosser and W. N. Herfort, Abelian subgroups of topological groups, Trans. Amer. Math. Soc. 283 1984, 1, 211–223], we prove that a c-minimal locally solvable Lie group is compact. It is shown that a topological group G is c-(totally) minimal if and only if G has a compact normal subgroup N such that G / N G/N is c-(totally) minimal. Applying this result, we prove that a locally compact group G is c-totally minimal if and only if its connected component c ⁢ ( G ) c(G) is compact and G / c ⁢ ( G ) G/c(G) is c-totally minimal. Moreover, a c-totally minimal group that is either complete solvable or strongly compactly covered must be compact. Negatively answering [D. Dikranjan and M. Megrelishvili, Minimality conditions in topological groups, Recent Progress in General Topology. III, Atlantis Press, Paris 2014, 229–327, Question 3.10 (b)], we find, in contrast, a totally minimal solvable (even metabelian) Lie group that is not compact.

The implicit function theorem and the global methods of cohomology

Since the solution of the celebrated Hilbert’s Fifth Problem, one knows that a locally euclidean topological group is a Lie group. In accordance with Hilbert’s original idea of getting rid of the differentiability hypothesis in Lie theory, one can look for a homotopy analogue of this result. Unfortunately the most straightforward idea is false: Hilton, Roitberg and Stasheff have exhibited a topological group having the homotopy type of a compact manifold and which is not homotopy equivalent to any Lie group (see [Sta69]). Nevertheless, the problem of finding a homotopy characterisation of Lie groups remains open and has been intensively investigated for the last three decades. Obviously, the first step of such a programme is a better understanding of the homotopy properties of Lie groups. One has to restrict this study to the class of compact connected Lie groups: the Iwasawa decomposition tells us that any connected Lie group has the homotopy type of a compact one.

Many areas of mathematics were deeply influenced or even founded by Hermann Weyl, including geometric foundations of manifolds and physics, topological groups, Lie groups and representation theory, harmonic analysis and analytic number theory as well as foundations of mathematics. In this volume, leading experts present his lasting influence on current mathematics, often connecting Weyl's theorems with cutting edge research in dynamical systems, invariant theory, and partial differential equations. In a broad and accessible presentation, survey chapters describe the historical development of each area alongside up-to-the-minute results, focussing on the mathematical roots evident within Weyl's work.

First integrals, solutions and conservation laws of the derivative nonlinear Schrödinger equation

Classical Lie group theory provides a universal tool for calculatingsymmetry groups for systems of differential equations. However Lie'smethod is not as much effective in the case of integral orintegro-differential equations as well as in the case of infinitesystems of differential equations.

The Connected Finite Dimensional Lie Sub-Groups of the Group Gh(n,c)

Let (X, ω) be an integral symplectic manifold and let (L, ∇) be a quantum line bundle, with connection, over X having ω as curvature. With this data one can define an induced symplectic manifold \( (\tilde X,\omega _{\tilde X} ) \), where dim \( \tilde X = 2 + \dim X \). It is then shown that prequantization on X becomes classical Poisson bracket on \( \tilde X \). We consider the possibility that if X is the coadjoint orbit of a Lie group K, then \( \tilde X \) is the coadjoint orbit of some larger Lie group G. We show that this is the case if G is a noncompact simple Lie group with a finite center and K is the maximal compact subgroup of G. The coadjoint orbit X arises (Borel-Weil) from the action of K on \( \mathfrak{p} \), where \( \mathfrak{g} = \mathfrak{k} + \mathfrak{p} \) is a Cartan decomposition. Using the Kostant-Sekiguchi correspondence and a diffeomorphism result of M. Vergne we establish a symplectic isomorphism \( (\tilde X,\omega _{\tilde X} ) \cong (Z,\omega _Z ) \), where Z is a nonzero minimal "nilpotent" coadjoint orbit of G. This is applied to show that the split forms of the five exceptional Lie groups arise symplectically from the symplectic induction of coadjoint orbits of certain classical groups.

The purpose of this study was to compare clinical and radiographic outcomes of open and mini-open arthroscopic arthrodesis. Twelve patients underwent arthroscopically assisted glenohumeral arthrodesis over a 5-year period; none were lost to follow-up. Surgery was performed with the patient in the lateral decubitus position, with the arm positioned in 30° each of flexion, internal rotation, and abduction. The articular cartilage was arthroscopically removed from the humerus and glenoid, creating flat opposing surfaces. One to 2 Kirschner wires were inserted percutaneously through the deltoid and across the glenohumeral joint in the center of the articulation; screws were then inserted arthroscopically. The glenohumeral joint was accessed through a mini-open posterior approach beneath the deltoid and was then reassessed before cannulated screws were tightened completely to compress the joint. Two dynamic compression plates were applied to the posterior glenohumeral joint to neutralize rotatory forces. Each patient was immobilized for 6weeks. Follow-up radiographic imaging was performed at 2, 6, 12, and 24months after surgery. These patients were then compared with a similar group who underwent a classic open approach. Success of arthrodesis was determined by bone growth across the glenohumeral joint as visualized on axillary radiographs. At 2-year follow-up, complete fusion was achieved in 12 (100%) arthroscopically treated patients, with 2 patients (17%) having early bone grafting (within 6weeks) through a percutaneous approach. Four patients in the classic open approach group ("open group") required additional grafting. Two patients in the group undergoing arthroscopic surgery ("arthroscopic group") had solid fusion but persisting infection from previously failed operations. Comparison of the 2 groups showed no difference in patient satisfaction or infection rates (2 in each group, all of whom had active infection at the time of the index surgery). One patient in the open group sustained a humeral shaft fracture 5years after arthrodesis, which required additional surgery. No patient who underwent an arthroscopic procedure required additional surgery other than the 2 early bone grafts. Arthroscopically assisted mini-open glenohumeral arthrodesis provides results that are at least equal to those of open arthrodesis, with a much less invasive approach. Success and complication rates are the same as for the completely open procedure. Level IV, therapeutic case series.

By a classical group we mean one of the groups G L n ( R ) G{L_n}(R) , G L n ( C ) G{L_n}(C) , G L n ( H ) G{L_n}(H) , U ( p , q ) U(p,\,q) , O n ( C ) {O_n}(C) , O ( p , q ) O(p,\,q) , S O ∗ ( 2 n ) S{O^{\ast }}(2n) , S p 2 n ( C ) S{p_{2n}}(C) , S p 2 n ( R ) S{p_{2n}}(R) , or S p ( p , q ) Sp(p,\,q) . Let G G be a classical group and L L its Lie algebra. For each x ∈ L x \in L we determine the closure of the orbit G ⋅ x G \cdot x (for the adjoint action of G G on L L ). The problem is first reduced to the case when x x is nilpotent. By using the exponential map we also determine the closures of conjugacy classes of G G .

We study graded group-valued continuously differentiable mappings defined on stratified groups, where differentiability is understood with respect to the group structure. We characterize these mappings by a system of nonlinear first-order PDEs, establishing a quantitative estimate for their difference quotient. This provides us with a mean value estimate that allows us to prove both the inverse mapping theorem and the implicit function theorem. The latter theorem also relies on the fact that the differential admits a proper factorization of the domain into a suitable inner semidirect product. When this splitting property of the differential holds in the target group, then the inverse mapping theorem leads us to the rank theorem. Both implicit function theorem and rank theorem naturally introduce the classes of image sets and level sets. For commutative groups, these two classes of sets coincide and correspond to the usual submanifolds. In noncommutative groups, we have two distinct classes of intrinsic submanifolds. They constitute the so-called intrinsic graphs, that are defined with respect to the algebraic splitting and everywhere possess a unique metric tangent cone equipped with a natural group structure.

(2001). Quantum Error Correction: Classic Group Theory Meets a Quantum Challenge. The American Mathematical Monthly: Vol. 108, No. 10, pp. 932-962.

This note presents a canonical construction of the structures of modal Θ -valent field ( mΘf) and modal Θ -valent pseudo field (mΘpf) as defined by F. Ayissi Eteme in (9). Their respective sets of the invertibles and modal Θ -invertibles are modal Θ -valent groups ( mΘg). Many intrinsic properties of these mΘg reveal the limit of the same notions in the classical group theory. The set of modal Θ -valent automorphisms of a mΘf of characteristic p and cardinal p2r({℧k : k = 0, 1, … 2r – 2}) is a mΘg of cardinal 2r - 1. Any Galois automorphism φq of 𝔽pr: q = 0, 1, …, r - 1 is an 𝔽pr: k ∊ {0, 1, … 2r - 2}.

The material presented below amounts roughly to the so-called fundamental theorems of Lie and their implications concerning Lie algebras, Lie subgroups and subalgebras. By straightforward and fairly elementary steps, we shall extend the concept of Lie group to include groups admitting coordinate systems in which the functions defining ab, the group product of elements a, b, possess continuous first order derivatives in the coordinates of a and satisfy a Lipschitz condition in those of b. That such groups are equivalent to classical (i.e. analytical) Lie groups was announced in 1936 by van Kampen, although apparently van Kampen published nothing by way of proof beyond a certain decisive uniqueness theorem concerning systems of ordinary differential equations. (This is contained in our theorem (12.1); the proof given below is essentially that of van Kampen [3]). It is conceivable that the condition relative to coordinates of b could be weakened or dispensed with entirely. For, the only part the condition in question plays in our development is to make it certain that the inverse of a certain mapping-the mapping into canonical coordinates-is single-valued. In this connection mention should be made of the paper [1J of Garrett Birkhoff in which it is shown that the existence of continuous first order derivatives of ab with respect to the coordinates of both the a's and the b's is sufficient for defining a Lie group. We shall consider only groups with a finite number of real parameters (whereas the groups considered by Birkhoff are not necessarily finite dimensional). This makes it possible to use the standard existence theorems for systems of ordinary real differential equations. We make no use whatever of the theory of partial differential equations. We maintain throughout a purely local point of view in the sense that we consider only what happens in the neighborhood of the identity. Historical notes. The proof of the uniqueness of square roots (8.5) is due to Claude Chevalley (unpublished). The proof given below of Lie's theorem that a linear system of vector fields which is closed under commutation defines a Lie group is, we believe, due to van der Waerden [4]. We have supplied a number of preliminary lemmas which make the proof rigorously applicable to the case in which the vector fields are only assumed to possess continuous first order derivatives. The theorem in ?17 that every local subgroup of a Lie group is a Lie group is due essentially to E. Cartan [2]. 481