Deformation Theory of Algebras and Structures and Applications

ISBN-13: 9789401078757 / Angielski / Miękka / 2011 / 1030 str.

Michiel Hazewinkel;Murray Gerstenhaber

Opis

This volume is a result of a meeting which took place in June 1986 at 'll Ciocco" in Italy entitled 'Deformation theory of algebras and structures and applications'. It appears somewhat later than is perhaps desirable for a volume resulting from a summer school. In return it contains a good many results which were not yet available at the time of the meeting. In particular it is now abundantly clear that the Deformation theory of algebras is indeed central to the whole philosophy of deformations/perturbations/stability. This is one of the main results of the 254 page paper below (practically a book in itself) by Gerstenhaber and Shack entitled "Algebraic cohomology and defor mation theory." Two of the main philosphical-methodological pillars on which deformation theory rests are the fol lowing (Pure) To study a highly complicated object, it is fruitful to study the ways in which it can arise as a limit of a family of simpler objects: "the unraveling of complicated structures" . (Applied) If a mathematical model is to be applied to the real world there will usually be such things as coefficients which are imperfectly known. Thus it is important to know how the behaviour of a model changes as it is perturbed (deformed)."

Spis treści

The philosophy of deformations: introductory remarks and a guide to this volume.- A. Deformations of algebras.- Algebraic cohomology and deformation theory.- Perturbations of Lie algebra structures.- Cohomology of current Lie algebras.- An example of formal deformations of Lie algebras.- On the rigidity of solvable Lie algebras.- Triangular algebras.- B. Perturbations of algebras in functional analysis and operator theory.- Deformation theory for algebras of analytic functions.- Close operator algebras.- Perturbations of function algebras.- Perturbations of multiplication and homomorphisms.- C. Deformations and moduli in geometry and differential equations, and algebras.- Local isoformal deformation theory for meromorphic differential equations near an irregular singularity.- Geometric and Lie-theoretic principles in pure and applied deformation theory.- Complexes of differential operators and symmetric spaces.- Deformation theory of geometric and algebraic structures.- Some rigidity results in the deformation theory of symmetric spaces.- D. Deformations of algebras and mathematical and quantum physics.- Applications of the deformations of the algebraic structures to geometry and mathematical physics.- Formal deformations of the Poisson Lie algebra of a symplectic manifold and star-products. Existence, equivalence, derivations.- Invariant deformations of the Poisson Lie algebra of a symplectic manifold and star-products.- E. Deformations elsewhere.- A remarkable matrix.- Deformation stability of periodic and quasi periodic motion in dissipative systems.- List of participants.