Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Algebras and Coalgebras Read chapter Read first chapter

2015 | OriginalPaper | Chapter

4. Applications of Hopf Algebras

Author : Robert G. Underwood

Published in: Fundamentals of Hopf Algebras

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this chapter we present three diverse applications of Hopf algebras. Our first application involves almost cocommutative bialgebras and quasitriangular bialgebras. We show that a quastitriangular bialgebra determines a solution to the Quantum Yang–Baxter Equation, and we give details on how to compute quastitriangular structures for certain two-dimensional bialgebras and Hopf algebras. We show that almost cocommutative Hopf algebras generalize Hopf algebras in which the coinverse has order 2. We then define the braid group on three strands (or more simply, the braid group) and show that a quasitriangular structure determines a representation of the braid group.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Hopf Algebras
Literature
[Ab77]
E. Abe, Hopf Algebras (Cambridge University of Press, Cambridge, 1977)
[Ca98]
S. Caenepeel, Brauer Groups, Hopf Algebras and Galois Theory. K-Monographs in Mathematics (Kluwer, Dordrecht, 1998)
[CS69]
S.U. Chase, M. Sweedler, Hopf Algebras and Galois Theory. Lecture Notes in Mathematics, vol. 97 (Springer, Berlin, 1969)
[Ch87]
[Ch00]
L.N. Childs, Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory. Mathematical Surveys and Monographs, vol. 80 (American Mathematical Society, Providence, RI, 2000)
[CRV14a]
T. Crespo, A. Rio, M. Vela, From Galois to Hopf Galois: theory and practice. Contemp. Math. (to appear). arXiv:1403.6300, (2014)
[CRV14b]
T. Crespo, A. Rio, M. Vela, On the Galois correspondence theorem in separable Hopf Galois theory. arXiv:1405.0881, (2014)
[CRV14c]
T. Crespo, A. Rio, M. Vela, The Hopf Galois property in subfield lattices. arXiv:1309.5754, (2014)
[Da80]
[Dr86]
V.G. Drinfeld, Quantum groups, in Proceedings of International Congress of Mathematics, Berkeley, CA, vol. 1, 1986, pp. 789–820
[Dr90]
V.G. Drinfeld, On almost commutative Hopf algebras. Leningrad Math. J. 1, 321–342 (1990)MathSciNet
[He66]
C.S. Herz, Construction of class fields, in Seminar on Complex Multiplication. Lecture Notes in Mathematics, vol. 21, VII-1-VII-21, (Springer, Berlin, 1966)
[IR90]
K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, 2nd edn. (Springer, New York, 1990)MATHCrossRef
[Mo93]
S. Montgomery, Hopf Algebras and Their Actions on Rings. CBMS Regional Conference Series in Mathematics, vol. 82 (American Mathematical Society, Providence, RI, 1993)
[Ne99]
J. Neukirch, Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften, vol. 322 (Springer, Berlin, 1999)
[Ni12]
F. Nichita, Introduction to the Yang-Baxter equation with open problems. Axioms 1, 33–37 (2012)MATHCrossRef
[Ra93]
[Un11]
R. Underwood, An Introduction to Hopf Algebras (Springer, New York/Dordrecht/ Heidelberg/London, 2011)MATHCrossRef
[Wat79]
Metadata
Title
Applications of Hopf Algebras
Author
Robert G. Underwood
Copyright Year
2015
Book
Fundamentals of Hopf Algebras

Print ISBN: 978-3-319-18990-1

Electronic ISBN: 978-3-319-18991-8

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-18991-8_4

Premium Partner

    Image Credits