Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T04:04:24.046Z Has data issue: false hasContentIssue false
  • English
  • Français

Centrification of algebras and Hopf algebras

Part of: Hopf algebras, quantum groups and related topics Rings and algebras arising under various constructions

Published online by Cambridge University Press:  08 April 2021

Dmitriy Rumynin  and
Matthew Westaway
Dmitriy Rumynin
Affiliation:
Department of Mathematics, University of Warwick, Coventry CV4 7AL, UK and Laboratory of Algebraic Geometry, National Research University Higher School of Economics, Moscow, Russia e-mail: [email protected]
Matthew Westaway*
Affiliation:
School of Mathematics, University of Birmingham, BirminghamB15 2TT, UK
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate a method of construction of central deformations of associative algebras, which we call centrification. We prove some general results in the case of Hopf algebras and provide several examples.

Keywords

Generators and relations for associative algebrasGröbner–Shirshov basisdeformationHopf algebra

MSC classification

Primary: 16S15: Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
Secondary: 16T05: Hopf algebras and their applications
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 1 , March 2022 , pp. 155 - 169
Check for updates
Copyright
© Canadian Mathematical Society 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arik, M. and Kayserilioglu, U., The anticommutator spin algebra, its representations and quantum group invariance. Int. J. Modern Phys. A 18(2003), 5039–5045. https://doi.org/10.1142/s0217751x03015933 CrossRefGoogle Scholar
Brown, K. and Goodearl, K., Lectures on algebraic quantum groups . Advanced Courses in Mathematics, CRM, Birkhäuser Verlag, Barcelona, Basel, 2002. https://doi.org/10.1007/978-3-0348-8205-7 CrossRefGoogle Scholar
Gorodnii, M. and Podkolzin, G., Irreducible representations of a graded Lie algebra. In: Spectral theory of operators and infinite-dimensional analysis, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1984, 66–77.Google Scholar
Kharchenko, V., Quantum Lie theory. A multilinear approach . Lecture Notes in Math., 2150, Springer, Berlin, Germany, 2015. https://doi.org/10.1007/978-3-319-22704-7 Google Scholar
Korovnichenko, A. and Zhedanov, A., Classical Leonard triples . In: Elliptic integrable systems (Kyoto, 2004), Rokko Lectures in Mathematics, 18, Kobe University, Kobe, Japan, 2005, pp. 71–84.Google Scholar
Montgomery, S., Hopf algebras and their actions on rings . CBMS Regional Conf. Ser. in Math., 82, American Mathematical Society, Providence, RI, 1993. https://doi.org/10.1090/cbms/082 CrossRefGoogle Scholar
Rumynin, D., Hopf-Galois extensions with central invariants and their geometric properties. Algebr. Represent. Theory 1(1998), 353–381. https://doi.org/10.1023/a:1009944607078 CrossRefGoogle Scholar
Schneider, H., Representation theory of Hopf Galois extensions. Israel J. Math. 72(1990), 196–231. https://doi.org/10.1007/BF02764620 CrossRefGoogle Scholar
Takeuchi, M., On the dimension of the space of integrals of Hopf algebras. J. Algebra 21(1972), 174–177. https://doi.org/10.1016/0021-8693(72)90015-4 CrossRefGoogle Scholar
Terwilliger, P., The universal Askey-Wilson algebra. SIGMA Symmetry Integrability Geom. Methods Appl. 7(2011), 24. https://doi.org/10.3842/sigma.2011.069 Google Scholar
Tsujimoto, S., Vinet, L., and Zhedanov, A., Dunkl shift operators and Bannai-Ito polynomials. Adv. Math. 229(2012), 2123–2158. https://doi.org/10.1016/j.aim.2011.12.020 CrossRefGoogle Scholar
Weibel, C., An introduction to homological algebra . Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994. https://doi.org/10.1017/cbo9781139644136 CrossRefGoogle Scholar
Zhedanov, A., “Hidden symmetry” of Askey-Wilson polynomials. Teoret. Mat. Fiz. 89(1991), 190–204. https://doi.org/10.1007/bf01015906 Google Scholar