Hasse-Schmidt Derivations on Grassmann Algebras

With Applications to Vertex Operators
 
 
Springer (Verlag)
  • erschienen am 9. August 2016
 
  • Buch
  • |
  • Hardcover
  • |
  • XXIV, 195 Seiten
978-3-319-31841-7 (ISBN)
 
This book provides a comprehensive advanced multi-linear algebra course based on the concept of Hasse-Schmidt derivations on a Grassmann algebra (an analogue of the Taylor expansion for real-valued functions), and shows how this notion provides a natural framework for many ostensibly unrelated subjects: traces of an endomorphism and the Cayley-Hamilton theorem, generic linear ODEs and their Wronskians, the exponential of a matrix with indeterminate entries (Putzer's method revisited), universal decomposition of a polynomial in the product of two monic polynomials of fixed smaller degree, Schubert calculus for Grassmannian varieties, and vertex operators obtained with the help of Schubert calculus tools (Giambelli's formula). Significant emphasis is placed on the characterization of decomposable tensors of an exterior power of a free abelian group of possibly infinite rank, which then leads to the celebrated Hirota bilinear form of the Kadomtsev-Petviashvili (KP) hierarchy describing the Plücker embedding of an infinite-dimensional Grassmannian. By gathering ostensibly disparate issues together under a unified perspective, the book reveals how even the most advanced topics can be discovered at the elementary level.
Book
1st ed. 2016
  • Englisch
  • Cham
  • |
  • Schweiz
Springer International Publishing
  • Für Beruf und Forschung
  • 16 s/w Abbildungen
  • |
  • 6 schwarz-weiße Abbildungen, Bibliographie
  • Höhe: 241 mm
  • |
  • Breite: 159 mm
  • |
  • Dicke: 20 mm
  • 483 gr
978-3-319-31841-7 (9783319318417)
10.1007/978-3-319-31842-4
weitere Ausgaben werden ermittelt
Letterio Gatto received his PhD in mathematics from the University of Torino in 1993, and since then has held permanent positions at the Department of Mathematical Sciences of the Politecnico di Torino. He is currently associate professor at the Politecnico, where he offers courses on linear algebra and geometry for students in Engineering. His research interests range from Schubert calculus (classical, equivariant and quantum), algebraic curves and moduli (families of Weierstrauss points and jets of line bundles on Gorenstein curves) and integrable systems from the algebro-geometrical point of view. Parham Salehyan graduated from Sharif University of Technology, Iran, and received his PhD in mathematics from the IMPA, Brazil, in 2003. He holds a permanent position at the Department of Mathematics of São Paulo State University - Unesp, São José do Rio Preto Campus. His research interests lie in algebraic geometry, mainly in the theory of Weierstrauss points on families of curves and, more recently, in the algebro-combinatorial aspects related to Schubert calculus.
Prologue.- Generic Linear Recurrence Sequences.- Algebras and Derivations.- Hasse-Schmidt Derivations on Exterior Algebras.- Schubert Derivations.- Decomposable Tensors in Exterior Powers.- Vertex Operators via Generic LRS.- Index.
"It provides a comprehensive advanced multi-linear algebra course based on the concept of Hasse-Schmidt derivations on a Grassmann algebra ... . It covers a wealth of important material in a concise, nevertheless instructive manner, and as such it may serve as an excellent guide to further, more advanced and detailed reading in this fundamental area of contemporary mathematics." (Ahmed Lesfari, Mathematical Reviews, June, 2017)

"It is entirely self-contained, and at the same time advanced in that it touches on many different areas in the fields of differential equations and mathematical physics and has further notes and references at the end of every chapter, as well as exercises highlighting further connections. ... This book will be welcomed not only by scholars interested in generalized global coordinate-free settings, but also by students wishing to become acquainted with advanced areas of multilinear algebra and their applications." (Rabe von Randow, zbMATH 1350.15001, 2017)
This book provides a comprehensive advanced linear algebra
course based on the concept of Hasse-Schmidt derivations on a Grassmann algebra
(an analogue of the Taylor expansion for real-valued functions), and shows how
this notion provides a natural framework for many ostensibly unrelated
subjects: traces of an endomorphism and the Cayley-Hamilton theorem, generic
linear ODEs and their Wronskians, the exponential of a matrix with
indeterminate entries (Putzer's method revisited), universal decomposition of a
polynomial in the product of two monic polynomials of fixed smaller degree,
Schubert calculus for Grassmannian varieties, and vertex operators obtained
with the help of Schubert calculus tools (Giambelli's formula). Significant
emphasis is placed on the characterization of decomposable tensors of an
exterior power of a free abelian group of possibly infinite rank, which then
leads to the celebrated Hirota bilinear form of the Kadomtsev-Petviashvili (KP)
hierarchy describing the Plücker embedding of an infinite-dimensional
Grassmannian. By gathering ostensibly disparate issues together under a unified
perspective, the book reveals how even the most advanced topics can be
discovered at the elementary level.

Versand in 10-15 Tagen

48,14 €
inkl. 7% MwSt.
Sonderpreis bis 30.06.2020
Aktion Yellow Sale | statt 96,29 €
in den Warenkorb