Higher Structures in Geometry and Physics [electronic resource] : In Honor of Murray Gerstenhaber and Jim Stasheff / edited by Alberto S. Cattaneo, Anthony Giaquinto, Ping Xu.

Contributor(s): Cattaneo, Alberto S [editor.] | Giaquinto, Anthony [editor.] | Xu, Ping [editor.] | SpringerLink (Online service)
Material type: TextTextSeries: Progress in Mathematics: 287Publisher: Boston : Birkhäuser Boston, 2011Description: XV, 362 p. 92 illus. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9780817647353Subject(s): Mathematics | Algebraic geometry | Group theory | Topological groups | Lie groups | Applied mathematics | Engineering mathematics | Physics | Mathematics | Topological Groups, Lie Groups | Group Theory and Generalizations | Algebraic Geometry | Mathematical Methods in Physics | Applications of MathematicsAdditional physical formats: Printed edition:: No titleDDC classification: 512.55 | 512.482 LOC classification: QA252.3QA387Online resources: Click here to access online
Contents:
Topics in Algebraic deformation theory -- Origins and breadth of the theory of higher homotopies -- The deformation philosophy, quantization and noncommutative space-time structures -- Differential geometry of Gerbes and differential forms -- Symplectic connections of Ricci type and star products -- Effective Batalin–Vilkovisky theories, equivariant configuration spaces and cyclic chains -- Noncommutative calculus and the Gauss-Manin connection -- The Lie algebra perturbation lemma -- Twisting Elements in Homotopy G-algebras -- Homological perturbation theory and homological mirror symmetry -- Categorification of acyclic cluster algebras: an introduction -- Poisson and symplectic functions in Lie algebroid theory -- The diagonal of the Stasheff polytope -- Permutahedra, HKR isomorphism and polydifferential Gerstenhaber-Schack complex -- Applications de la bi-quantification a la théorie de Lie -- Higher homotopy Hopf algebras found: A ten year retrospective.
In: Springer eBooksSummary: This book is centered around higher algebraic structures stemming from the work of Murray Gerstenhaber and Jim Stasheff that are now ubiquitous in various areas of mathematics— such as algebra, algebraic topology, differential geometry, algebraic geometry, mathematical physics— and in theoretical physics such as quantum field theory and string theory. These higher algebraic structures provide a common language essential in the study of deformation quantization, theory of algebroids and groupoids, symplectic field theory, and much more. The ideas of higher homotopies and algebraic deformation have a growing number of theoretical applications and have played a prominent role in recent mathematical advances. For example, algebraic versions of higher homotopies have led eventually to the proof of the formality conjecture and the deformation quantization of Poisson manifolds. As observed in deformations and deformation philosophy, a basic observation is that higher homotopy structures behave much better than strict structures. Each contribution in this volume expands on the ideas of Gerstenhaber and Stasheff. Higher Structures in Geometry and Physics is intended for post-graduate students, mathematical and theoretical physicists, and mathematicians interested in higher structures. Contributors: L. Breen, A.S. Cattaneo, M. Cahen, V.A. Dolgushev, G. Felder, A. Giaquinto, S. Gutt, J. Huebschmann, T. Kadeishvili, H. Kajiura, B. Keller, Y. Kosmann-Schwarzbach, J.-L. Loday, S.A. Merkulov, D. Sternheimer, D.E. Tamarkin, C. Torossian, B.L. Tsygan, S. Waldmann, R.N. Umble.
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Topics in Algebraic deformation theory -- Origins and breadth of the theory of higher homotopies -- The deformation philosophy, quantization and noncommutative space-time structures -- Differential geometry of Gerbes and differential forms -- Symplectic connections of Ricci type and star products -- Effective Batalin–Vilkovisky theories, equivariant configuration spaces and cyclic chains -- Noncommutative calculus and the Gauss-Manin connection -- The Lie algebra perturbation lemma -- Twisting Elements in Homotopy G-algebras -- Homological perturbation theory and homological mirror symmetry -- Categorification of acyclic cluster algebras: an introduction -- Poisson and symplectic functions in Lie algebroid theory -- The diagonal of the Stasheff polytope -- Permutahedra, HKR isomorphism and polydifferential Gerstenhaber-Schack complex -- Applications de la bi-quantification a la théorie de Lie -- Higher homotopy Hopf algebras found: A ten year retrospective.

This book is centered around higher algebraic structures stemming from the work of Murray Gerstenhaber and Jim Stasheff that are now ubiquitous in various areas of mathematics— such as algebra, algebraic topology, differential geometry, algebraic geometry, mathematical physics— and in theoretical physics such as quantum field theory and string theory. These higher algebraic structures provide a common language essential in the study of deformation quantization, theory of algebroids and groupoids, symplectic field theory, and much more. The ideas of higher homotopies and algebraic deformation have a growing number of theoretical applications and have played a prominent role in recent mathematical advances. For example, algebraic versions of higher homotopies have led eventually to the proof of the formality conjecture and the deformation quantization of Poisson manifolds. As observed in deformations and deformation philosophy, a basic observation is that higher homotopy structures behave much better than strict structures. Each contribution in this volume expands on the ideas of Gerstenhaber and Stasheff. Higher Structures in Geometry and Physics is intended for post-graduate students, mathematical and theoretical physicists, and mathematicians interested in higher structures. Contributors: L. Breen, A.S. Cattaneo, M. Cahen, V.A. Dolgushev, G. Felder, A. Giaquinto, S. Gutt, J. Huebschmann, T. Kadeishvili, H. Kajiura, B. Keller, Y. Kosmann-Schwarzbach, J.-L. Loday, S.A. Merkulov, D. Sternheimer, D.E. Tamarkin, C. Torossian, B.L. Tsygan, S. Waldmann, R.N. Umble.

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