Geometry of Lie integrability by quadratures

Integrability of Lie Brackets

In this paper we present the solution to a longstanding problem of differential geometry: Lie’s third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigma-mode...

Lectures on Integrability of Lie Brackets

Stationary mKdV hierarchy and integrability of the Dirac equations by quadratures

where ξ is a real-valued function and η is a 2 × 2 matrix complex-valued function, a Lie symmetry of system (1) if commutation relation [L,X] = R(x)L, (4) holds with some 2× 2 matrix function R(x) (for details, see, e.g., Ref. [3]). A simple computation shows that if X is a Lie symmetry of system (1), then an operator X + r(x)L with a smooth function r(x) is its Lie symmetry as well. Hence we c...

On the Integrability of Lie Subalgebroids

Let G be a Lie groupoid with Lie algebroid g. It is known that, unlike in the case of Lie groups, not every subalgebroid of g can be integrated by a subgroupoid of G. In this paper we study conditions on the invariant foliation defined by a given subalgebroid under which such an integration is possible. We also consider the problem of integrability by closed subgroupoids, and we give conditions...

The Nonabelian Liouville-Arnold Integrability by Quadratures Problem: a Symplectic Approach

A symplectic theory approach is devised for solving the problem of algebraic-analytical construction of integral submanifold imbeddings for integrable (via the nonabelian Liouville-Arnold theorem) Hamiltonian systems on canonically symplectic phase spaces.