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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsBibliography
E. Abe, Hopf Algebras (Cambridge University of Press, Cambridge, 1977)
S. Caenepeel, Brauer Groups, Hopf Algebras and Galois Theory. K-Monographs in Mathematics (Kluwer, Dordrecht, 1998)
S.U. Chase, M. Sweedler, Hopf Algebras and Galois Theory. Lecture Notes in Mathematics, vol. 97 (Springer, Berlin, 1969)
L.N. Childs, Taming wild extensions with Hopf algebras. Trans. Am. Math. Soc. 304, 111–140 (1987)
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)
T. Crespo, A. Rio, M. Vela, From Galois to Hopf Galois: theory and practice. Contemp. Math. (to appear). arXiv:1403.6300, (2014)
T. Crespo, A. Rio, M. Vela, On the Galois correspondence theorem in separable Hopf Galois theory. arXiv:1405.0881, (2014)
T. Crespo, A. Rio, M. Vela, The Hopf Galois property in subfield lattices. arXiv:1309.5754, (2014)
E. Dade, Group-graded rings and modules. Math. Z. 174, 241–262 (1980)
V.G. Drinfeld, Quantum groups, in Proceedings of International Congress of Mathematics, Berkeley, CA, vol. 1, 1986, pp. 789–820
V.G. Drinfeld, On almost commutative Hopf algebras. Leningrad Math. J. 1, 321–342 (1990)
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)
K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, 2nd edn. (Springer, New York, 1990)
S. Montgomery, Hopf Algebras and Their Actions on Rings. CBMS Regional Conference Series in Mathematics, vol. 82 (American Mathematical Society, Providence, RI, 1993)
J. Neukirch, Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften, vol. 322 (Springer, Berlin, 1999)
F. Nichita, Introduction to the Yang-Baxter equation with open problems. Axioms 1, 33–37 (2012)
D.E. Radford, Minimal quasitriangular Hopf algebras. J. Algebra 157, 285–315 (1993)
R. Underwood, An Introduction to Hopf Algebras (Springer, New York/Dordrecht/ Heidelberg/London, 2011)
W. Waterhouse, Introduction to Affine Group Schemes (Springer, New York, 1979)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Underwood, R.G. (2015). Applications of Hopf Algebras. In: Fundamentals of Hopf Algebras. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-18991-8_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-18991-8_4
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18990-1
Online ISBN: 978-3-319-18991-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)