arXiv subjects
Publications and source records attributed to Yvette Kosmann-Schwarzbach.
Siméon-Denis Poisson was 25 years old when he was appointed Professor of Mathematics at the École Polytechnique in 1806. Elected to the Paris Académie des Sciences six years later, he soon became one of its most influential members. The origin and later developments of the many concepts in mathematics and physics that bear his name make interesting stories, a few of which we shall attempt to sketch in this paper.
In this paper, dedicated to the memory of Kirill Mackenzie, I relate the origins and early development of the theory of graded Lie brackets, first in the publications on differential geometry of Schouten, Nijenhuis, and Frölicher--Nijenhuis, then in the work of Gerstenhaber and Nijenhuis--Richardson in cohomology theory.
I shall sketch the contents of Noether's 1918 article, "Invariante Variationsprobleme", in the context of the debate on the conservation of energy that had arisen from Einstein's general theory of relativity. How original was Noether's "Invariante Variationsprobleme"? how modern were her use of Lie groups and her introduction of generalized vector fields? and how influential was her article? I shall sketch the curious transmission of her results from 1918 to 1970, the history of the later developments of her theory in mathematics and in physics, and the ultimate recognition of the wide applicability of "the Noether theorems".
We briefly recall the history of the Nijenhuis torsion of (1,1)-tensors on manifolds and of the lesser-known Haantjes torsion. We then show how the Haantjes manifolds of Magri and the symplectic-Haantjes structures of Tempesta and Tondo generalize the classical approach to integrable systems in the bi-hamiltonian and symplectic-Nijenhuis formalisms, the sequence of powers of the recursion operator being replaced by a family of commuting Haantjes operators.
We survey the concept of multiplicativity from its initial appearance in the theory of Poisson-Lie groups to the far-reaching generalizations, for multivectors and differential forms in the geometry and the generalized geometry of Lie groupoids, as well as their infinitesimal counterparts in the theory of Lie algebroids.
An account of the careers of the five women who completed a doctorate in mathematics in France before 1960 and became internationally known scientists, followed by a more general description of the place of women on the mathematical scene in France between 1930 and 1960.
The search for a geometric interpretation of the constrained brackets of Dirac led to the definition of the Courant bracket. The search for the right notion of a "double" for Lie bialgebroids led to the definition of Courant algebroids. We recount the emergence of these concepts.
We study Nijenhuis structures on Courant algebroids in terms of the canonical Poisson bracket on their symplectic realizations. We prove that the Nijenhuis torsion of a skew-symmetric endomorphism N of a Courant algebroid is skew-symmetric if the square of N is proportional to the identity, and only in this case when the Courant algebroid is irreducible. We derive a necessary and sufficient condition for a skew-symmetric endomorphism to give rise to a deformed Courant structure. In the case of the double of a Lie bialgebroid (A,A*), given an endomorphism n of A that defines a skew-symmetric endomorphism N of the double of A, we prove that the torsion of N is the sum of the torsion of n and that of the transpose of n.
We extend the definition of the Nijenhuis torsion of an endomorphism of a Lie algebroid to that of a relation, and we prove that the torsion of the relation defined by a bi-Hamiltonian structure vanishes. Following Gelfand and Dorfman, we then define Dirac pairs, and we analyze the relationship of this general notion with the various kinds of compatible structures on manifolds, more generally on Lie algebroids.
We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a closed 2-form, a Poisson bivector or a Nijenhuis tensor, with suitable compatibility assumptions. We establish the relationships between such composite structures. We then show that the non-degenerate Monge-Ampère structures on 2-dimensional manifolds satisfying an integrability condition provide numerous examples of such structures, while in the case of 3-dimensional manifolds, such Monge-Ampère operators give rise to generalized complex structures or generalized product structures on the cotangent bundle of the manifold.
Emphasizing the role of Gerstenhaber algebras and of higher derived brackets in the theory of Lie algebroids, we show that the several Lie algebroid brackets which have been introduced in the recent literature can all be defined in terms of Poisson and pre-symplectic functions in the sense of Roytenberg and Terashima. We prove that in this very general framework there exists a one-to-one correspondence between non-degenerate Poisson functions and symplectic functions. We determine the differential associated to a Lie algebroid structure obtained by twisting a structure with background by both a Lie bialgebra action and a Poisson bivector.
We study the behavior of the modular class of a Lie algebroid under general Lie algebroid morphisms by introducing the relative modular class. We investigate the modular classes of pull-back morphisms and of base-preserving morphisms associated to Lie algebroid extensions. We also define generalized morphisms, including Morita equivalences, that act on the 1-cohomology, and observe that the relative modular class is a coboundary on the category of Lie algebroids and generalized morphisms with values in the 1-cohomology.
After a brief summary of the main properties of Poisson manifolds and Lie algebroids in general, we survey recent work on the modular classes of Poisson and twisted Poisson manifolds, of Lie algebroids with a Poisson or twisted Poisson structure, and of Poisson-Nijenhuis manifolds. A review of the spinor approach to the modular class concludes the paper.
We derive a formula for the the modular class of a Lie algebroid with a regular twisted Poisson structure in terms of a canonical Lie algebroid representation of the image of the Poisson map. We use this formula to compute the modular classes of Lie algebras with a twisted triangular r-matrix. The special case of r-matrices associated to Frobenius Lie algebras is also studied.
We prove a property of the Poisson-Nijenhuis manifolds which yields new proofs of the bihamiltonian properties of the hierarchy of modular vector fields defined by Damianou and Fernandes.
We study the geometric and algebraic properties of the twisted Poisson structures on Lie algebroids, leading to a definition of their modular class and to an explicit determination of a representative of the modular class, in particular in the case of a twisted Poisson manifold.
We study the relative modular classes of Lie algebroids, and we determine their relationship with the modular classes of Lie algebroids with a twisted Poisson structure.
We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and Vinogradov, and we prove that the Vinogradov bracket is the skew-symmetrization of a derived bracket. Odd (resp., even) Poisson brackets on supermanifolds are derived brackets of canonical even (resp., odd) Poisson brackets on their cotangent bundle (resp., parity-reversed cotangent bundle). Lie algebras have analogous properties, and the theory of Lie algebroids unifies the results valid for manifolds on the one hand, and for Lie algebras on the other. We outline the role of derived brackets in the theory of "Poisson structures with background".