Lie symmetries and differential Galois groups of linear equations

For a linear ordinary differential equation the Lie algebra of its infinitesimal Lie symmetries is compared with its differential Galois group. For this purpose an algebraic formulation of Lie symmetries is developed. It turns out that there is no direct relation between the two above objects. In connection with this a new algorithm for computing the Lie symmetries of a linear ordinary differential equation is presented.     

Appell systems on Lie groups

Using the probabilistic interpretation of Appell polynomials as systems of moments, we show how to define them in the noncommutative case. The method is based on certain infinite-dimensional representations of local Lie groups. For processes, limit theorems play an essential role in the construction. Polynomial matrix representations of convolution semigroups are a principal feature.     

Non-overlapping control systems on Lie groups

LetG be a Lie group with Lie algebraL(G) and let Ω be a non-empty subset ofL(G). If Ω is interpreted as the set of controls, then the set of elements attainable from the identity for the system Ω is a subsemigroup ofG. A system Ω is called anon-overlapping control system if any element attainable for Ω is only attainable at one time. In this paper, we show that a compact convex generating nonoverlapping control systems on a connected Lie group must be contained inX+E for someX∈L(G)\E, where E is a subspace of codimension one containing the commutator, and the homomorphism from the attainable semigroup intoR ⁺ extends continuously to the whole group in the case of solvable Lie groups. This work is done under the support of TGRC-KOSEF.     

Motives with exceptional Galois groups and the inverse Galois problem

We construct motivic ?-adic representations of $\textup {Gal}(\overline {\mathbb{Q}}/\mathbb{Q})$ into exceptional groups of type E ₇,E ₈ and G ₂ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and is inspired by the Langlands correspondence for function fields. As an application, we solve new cases of the inverse Galois problem: the finite simple groups $E_{8}(\mathbb{F}_{\ell})$ are Galois groups over $\mathbb{Q}$ for large enough primes ?.     

Special symplectic Lie groups and hypersymplectic Lie groups

A special symplectic Lie group is a triple ${(G,\omega,\nabla)}$ such that G is a finite-dimensional real Lie group and ω is a left invariant symplectic form on G which is parallel with respect to a left invariant affine structure ${\nabla}$ . In this paper starting from a special symplectic Lie group we show how to “deform” the standard Lie group structure on the (co)tangent bundle through the left invariant affine structure ${\nabla}$ such that the resulting Lie group admits families of left invariant hypersymplectic structures and thus becomes a hypersymplectic Lie group. We consider the affine cotangent extension problem and then introduce notions of post-affine structure and post-left-symmetric algebra which is the underlying algebraic structure of a special symplectic Lie algebra. Furthermore, we give a kind of double extensions of special symplectic Lie groups in terms of post-left-symmetric algebras.     

Integrability of Hamiltonian systems and differential Galois groups of higher variational equations

Given a complex analytical Hamiltonian system, we prove that a necessary condition for its meromorphic complete integrability is the commutativity of the identity component of the Galois group of each variational equation of arbitrary order along any integral curve. This was conjectured by the first author based on a suggestion by the third author. The first-order non-integrability criterion, obtained by the first and second authors using only first variational equations, is extended to higher orders by the present criterion. Using this result (at order two, three or higher) it is possible to solve important open problems of integrability which escaped the first order criterion.     

Optimal control on Lie groups and integrable Hamiltonian systems

Control theory, initially conceived in the 1950’s as an engineering subject motivated by the needs of automatic control, has undergone an important mathematical transformation since then, in which its basic question, understood in a larger geometric context, led to a theory that provides distinctive and innovative insights, not only to the original problems of engineering, but also to the problems of differential geometry and mechanics. This paper elaborates the contributions of control theory to geometry and mechanics by focusing on the class of problems which have played an important part in the evolution of integrable systems. In particular the paper identifies a large class of Hamiltonians obtained by the Maximum principle that admit isospectral representation on the Lie algebras \(\mathfrak{g} = \mathfrak{p} \oplus \mathfrak{k}\) of the form

$\frac{{dL_\lambda }}{{dt}} = \left[ {\Omega _\lambda ,L_\lambda } \right]L_\lambda = L_\mathfrak{p} - \lambda L_\mathfrak{k} - \left( {\lambda ^2 - s} \right)A, L_\mathfrak{p} \in \mathfrak{p}, L_\mathfrak{k} \in \mathfrak{k}.$

The spectral invariants associated with L _λ recover the integrability results of C.G.J. Jacobi concerning the geodesics on an ellipsoid as well as the results of C. Newmann for mechanical problem on the sphere with a quadratic potential. More significantly, this study reveals a large class of integrable systems in which these classical examples appear only as very special cases.

     

Galois groups and complete domains

Consider a domain that is complete with respect to a non-zero prime ideal. This paper proves two Galois-theoretic results about such rings. Using Grothendieck’s Existence Theorem we prove that every finite group occurs as the Galois group of a Galois extension of . This generalizes results of David Harbater who proved the result in the case where the ideal is maximal and the domain is normal. As a consequence, we deduce that if is a Noetherian domain that is complete with respect to a non-zero prime ideal, then every finite group occurs as a Galois group over . This proves the Noetherian case of a conjecture posed by Moshe Jarden.     

Control of mechanical systems on Lie groups and ideal hydrodynamics

In contrast to the Euler–Poincaré reduction of geodesic flows of left- or right-invariant metrics on Lie groups to the corresponding Lie algebra (or its dual), one can consider the reduction of the geodesic flows to the group itself. The reduced vector field has a remarkable hydrodynamic interpretation: it is the velocity field for a stationary flow of an ideal fluid. Right- or left-invariant symmetry fields of the reduced field define vortex manifolds for such flows. Now we consider a mechanical system, whose configuration space is a Lie group and whose Lagrangian is invariant with respect to left translations on this group, and assume that the mass geometry f the system may change under the action of internal control forces. Such a system can also be reduced to a Lie group. Without controls, this mechanical system describes a geodesic flow of the left-invariant metric, given by the Lagrangian, and, therefore, its reduced flow is a stationary ideal fluid flow on the Lie group. The standard control problem for such system is to find the conditions under which the system can be brought from any initial position in the configuration space to another preassigned position by changing its mass geometry. We show that under these conditions, by changing the mass geometry, one can also bring one vortex manifold to any other preassigned vortex manifold. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

Quotients of absolute Galois groups which determine the entire Galois cohomology

For a prime power q = p ^d and a field F containing a root of unity of order q we show that the Galois cohomology ring H^*(G_F,\mathbbZ/q){H^*(G_F,\mathbb{Z}/q)} is determined by a quotient G_F^[3]{G_F^{[3]}} of the absolute Galois group G _F related to its descending q-central sequence. Conversely, we show that G_F^[3]{G_F^{[3]}} is determined by the lower cohomology of G _F . This is used to give new examples of pro-p groups which do not occur as absolute Galois groups of fields.     

Approximating absolute Galois groups

In this paper we identify a class of profinite groups (totally torsion free groups) that includes all separable Galois groups of fields containing an algebraically closed subfield, and demonstrate that it can be realized as an inverse limit of torsion free virtually finitely generated abelian (tfvfga) profinite groups. We show by examples that the condition is quite restrictive. In particular, semidirect products of torsion free abelian groups are rarely totally torsion free. The result is of importance for K-theoretic applications, since descent problems for tfvfga groups are relatively manageable.     

On alternating and symmetric groups as Galois groups

Fix an integern≧3. We show that the alternating groupA _n appears as Galois group over any Hilbertian field of characteristic different from 2. In characteristic 2, we prove the same whenn is odd. We show that any quadratic extension of Hilbertian fields of characteristic different from 2 can be embedded in anS _n-extension (i.e. a Galois extension with the symmetric groupS _n as Galois group). Forn≠6, it will follow thatA _n has the so-called GAR-property over any field of characteristic different from 2. Finally, we show that any polynomialf=X ⁿ+…+a₁X+a₀ with coefficients in a Hilbertian fieldK whose characteristic doesn’t dividen(n-1) can be changed into anS _n-polynomialf ^* (i.e the Galois group off ^* overK Gal(f ^*, K), isS _n) by a suitable replacement of the last two coefficienta ₀ anda ₁. These results are all shown using the Newton polygon. The author acknowledges the financial support provided through the European Community’s Human Potential Programme under contract HPRN-CT-2000-00114, GTEM.     

On Galois cohomology and realizability of 2-groups as Galois groups

In this paper we develop some new theoretical criteria for the realizability of p-groups as Galois groups over arbitrary fields. We provide necessary and sufficient conditions for the realizability of 14 of the 22 non-abelian 2-groups having a cyclic subgroup of index 4 that are not direct products of groups.     

Computing Galois groups over the rationals

Practical computational techniques are described to determine the Galois group of a polynomial over the rationals, and each transitive permutation group of degree 3 to 7 is realised as a Galois group over the rationals. The exact computations furnish a proof of the result.     

Lie groups

The survey deals with the investigations reviewed inReferativnyi Zhurnal Matematika between 1977–1981. In the survey there are reflected the investigations on the structure of Lie groups and Lie algebras, on their finite-dimensional linear representations and universal enveloping algebras, on the theory of invariants and Lie groups of transformations, and also on continuous and discrete subgroups of Lie groups.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 20, No. 153–192, 1982.