Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T20:31:10.525Z Has data issue: false hasContentIssue false
  • English
  • Français

GLOBALIZATION OF TWISTED PARTIAL HOPF ACTIONS

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

Published online by Cambridge University Press:  26 February 2016

MARCELO M. S. ALVES ,
ELIEZER BATISTA ,
MICHAEL DOKUCHAEV  and
ANTONIO PAQUES
MARCELO M. S. ALVES
Affiliation:
Departamento de Matemática, Universidade Federal do Paraná, 81531-980 Curitiba, PR, Brazil email [email protected]
ELIEZER BATISTA
Affiliation:
Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, SC, Brazil email [email protected]
MICHAEL DOKUCHAEV
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 1010, 05508-090 São Paulo, SP, Brazil email [email protected]
ANTONIO PAQUES*
Affiliation:
Instituto de Matemática, Universidade Federal do Rio Grande do Sul, 91509-900 Porto Alegre, RS, Brazil email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this work, we review some properties of twisted partial actions of Hopf algebras on unital algebras and give necessary and sufficient conditions for a twisted partial action to have a globalization. We also elaborate a series of examples.

Keywords

partial group actionpartial Hopf actioncrossed producttwisted partial action

MSC classification

Primary: 16T05: Hopf algebras and their applications
Secondary: 16S40: Smash products of general Hopf actions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 101 , Issue 1 , August 2016 , pp. 1 - 28
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Abadie, F., ‘Sobre ações parciais, fibrados de Fell e grupóides’, PhD Thesis, Universidade de São Paulo, 1999.Google Scholar
Abadie, F., ‘Enveloping actions and Takai duality for partial actions’, J. Funct. Anal. 197(1) (2003), 14–67.Google Scholar
Akemann, C. A., Pedersen, G. K. and Tomiyama, J., ‘Multipliers of C ∗ -algebras’, J. Funct. Anal. 13 (1973), 277–301.Google Scholar
Alvares, E. R., Alves, M. M. S. and Batista, E., ‘Partial Hopf module categories’, J. Pure Appl. Algebra 217 (2013), 1517–1534.Google Scholar
Alves, M. M. S. and Batista, E., ‘Partial Hopf actions, partial invariants and a Morita context’, J. Algebra Discrete Math. 3 (2009), 1–19.Google Scholar
Alves, M. M. S. and Batista, E., ‘Enveloping actions for partial Hopf actions’, Comm. Algebra 38 (2010), 2872–2902.Google Scholar
Alves, M. M. S. and Batista, E., ‘Globalization theorems for partial Hopf (co)actions and some of their applications’, Contemp. Math. 537 (2011), 13–30.Google Scholar
Alves, M. M. S., Batista, E., Dokuchaev, M. and Paques, A., ‘Twisted partial actions of Hopf algebras’, Israel J. Math. 197 (2013), 263–308.Google Scholar
Alves, M. M. S., Batista, E. and Vercruysse, J., ‘Partial representations of Hopf algebras’, J. Algebra 426(15) (2015), 137–187.Google Scholar
Bagio, D., Cortes, W., Ferrero, M. and Paques, A., ‘Actions of inverse semigroups on algebras’, Comm. Algebra 35(12) (2007), 3865–3874.Google Scholar
Bagio, D. and Paques, A., ‘Partial groupoid actions: globalization, Morita theory and Galois theory’, Comm. Algebra 40 (2012), 3658–3678.Google Scholar
Bemm, L., ‘Ações parciais de grupos sobre anéis semiprimos’, PhD Thesis, Porto Alegre, 2011.Google Scholar
Bemm, L., Cortes, W., Ferrero, M. and Flora, S. S. D., ‘Partial crossed products and Goldie rings’, Comm. Algebra 43(9) (2015), 3705–3724.Google Scholar
Bemm, L. and Ferrero, M., ‘Globalization of partial actions on semiprime rings’, J. Algebra Appl. 12(4) (2013), 1250202.Google Scholar
Caenepeel, S. and Groot, E. D., ‘Galois corings applied to partial Galois theory’, Proc. Int. Conf. on Mathematics and its Applications (ICMA2004) (Kuwait University, 2005), 117–134.Google Scholar
Caenepeel, S. and Janssen, K., ‘Partial (co)actions of Hopf algebras and partial Hopf–Galois theory’, Comm. Algebra 36 (2008), 2923–2946.Google Scholar
Cortes, W. and Ferrero, M., ‘Globalizations of partial actions on semiprime rings’, Contemp. Math. 499 (2009), 27–35.Google Scholar
Cortes, W., Ferrero, M. and Marcos, E., ‘Partial actions on categories’, Comm. Algebra, to appear, arXiv:1107.3850.Google Scholar
Dokuchaev, M. and Exel, R., ‘Associativity of crossed products by partial actions, enveloping actions and partial representations’, Trans. Amer. Math. Soc. 357(5) (2005), 1931–1952.Google Scholar
Dokuchaev, M., Exel, R. and Piccione, P., ‘Partial representations and partial group algebras’, J. Algebra 226(1) (2000), 251–268.Google Scholar
Dokuchaev, M., Exel, R. and Simón, J. J., ‘Crossed products by twisted partial actions and graded algebras’, J. Algebra 320 (2008), 3278–3310.Google Scholar
Dokuchaev, M., Exel, R. and Simón, J. J., ‘Globalization of twisted partial actions’, Trans. Amer. Math. Soc. 362 (2010), 4137–4160.Google Scholar
Dokuchaev, M., Ferrero, M. and Paques, A., ‘Partial actions and Galois theory’, J. Pure Appl. Algebra 208(1) (2007), 77–87.Google Scholar
Dokuchaev, M., del Río, A. and Simón, J. J., ‘Globalizations of partial actions on non-unital rings’, Proc. Amer. Math. Soc. 135(2) (2007), 343–352.Google Scholar
Exel, R., ‘Circle actions on C ∗ -algebras, partial automorphisms and generalized Pimsner–Voiculescu exact sequences’, J. Funct. Anal. 122 (1994), 361–401.Google Scholar
Exel, R., ‘Twisted partial actions: a classification of regular C ∗ -algebraic bundles’, Proc. Lond. Math. Soc. (3) 74(3) (1997), 417–443.Google Scholar
Exel, R., ‘Partial actions of groups and actions of inverse semigroups’, Proc. Amer. Math. Soc. 126(12) (1998), 3481–3494.Google Scholar
Exel, R., Giordano, T. and Gonçalves, D., ‘Envelope algebras of partial actions as groupoid C ∗ -algebras’, J. Operator Theory 65 (2011), 197–210.Google Scholar
Ferrero, M., ‘Partial actions of groups on semiprime rings’, in: Groups, Rings and Group Rings, Lecture Notes in Pure and Applied Mathematics, 248 (Chapman & Hall/CRC, Boca Raton, FL, 2006), 155–162.Google Scholar
Freitas, D. and Paques, A., ‘On partial Galois Azumaya extensions’, Algebra Discrete Math. 11 (2011), 64–77.Google Scholar
Gilbert, N. D., ‘Actions and expansions of ordered groupoids’, J. Pure Appl. Algebra 198 (2005), 175–195.Google Scholar
Janssen, K. and Vercruysse, J., ‘Multiplier bi- and Hopf algebras’, J. Algebra Appl. 9(2) (2010), 275–303.Google Scholar
Kellendonk, J. and Lawson, M. V., ‘Partial actions of groups’, Internat. J. Algebra Comput. 14(1) (2004), 87–114.Google Scholar
Kuo, J.-M. and Szeto, G., ‘The structure of a partial Galois extension’, Monatsh Math. 175(4) (2014), 565–576.Google Scholar
McClanahan, K., ‘K-theory for partial crossed products by discrete groups’, J. Funct. Anal. 130(1) (1995), 77–117.Google Scholar
Megrelishvili, M. G. and Schröder, L., ‘Globalization of confluent partial actions on topological and metric spaces’, Topology Appl. 145 (2004), 119–145.Google Scholar
Montgomery, S., Hopf Algebras and Their Actions on Rings, CBMS Regional Conference Series in Mathematics, 82 (American Mathematical Society, Providence, RI, 1993).Google Scholar
Paques, A., Rodrigues, V. and Sant’Ana, A., ‘Galois correspondences for partial Galois Azumaya extensions’, J. Algebra Appl. 10(5) (2011), 835–847.CrossRefGoogle Scholar
Steinberg, B., ‘Inverse semigroup homomorphisms via partial group actions’, Bull. Aust. Math. Soc. 64(1) (2001), 157–168.Google Scholar
Steinberg, B., ‘Partial actions of groups on cell complexes’, Monatsh. Math. 138(2) (2003), 159–170.Google Scholar