A class of finite-by-cocommutative Hopf algebras
Annals of Representation Theory, Volume 1 (2024) no. 1, pp. 73-94.

We present a rich source of Hopf algebras starting from a cofinite central extension of a Noetherian Hopf algebra and a subgroup of the algebraic group of characters of the central Hopf subalgebra. The construction is transparent from a Tannakian perspective. We determine when the new Hopf algebras are co-Frobenius, or cosemisimple, or Noetherian, or regular, or have finite Gelfand-Kirillov dimension.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/art.5
Classification: 16T05, 17B37, 18M05
Andruskiewitsch, Nicolás 1; Natale, Sonia Luján 2; Torrecillas, Blas 3

1 FaMAF-CIEM (CONICET), Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, Argentina
2 Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba. CIEM-CONICET, Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, Argentina
3 Department of Mathematics, Universidad de Almería, Ctra. Sacramento s/n, 04071 Almería, Spain
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{ART_2024__1_1_73_0,
     author = {Andruskiewitsch, Nicol\'as and Natale, Sonia Luj\'an and Torrecillas, Blas},
     title = {A class of finite-by-cocommutative {Hopf} algebras},
     journal = {Annals of Representation Theory},
     pages = {73--94},
     publisher = {The Publishers of ART},
     volume = {1},
     number = {1},
     year = {2024},
     doi = {10.5802/art.5},
     language = {en},
     url = {https://art.centre-mersenne.org/articles/10.5802/art.5/}
}
TY  - JOUR
AU  - Andruskiewitsch, Nicolás
AU  - Natale, Sonia Luján
AU  - Torrecillas, Blas
TI  - A class of finite-by-cocommutative Hopf algebras
JO  - Annals of Representation Theory
PY  - 2024
SP  - 73
EP  - 94
VL  - 1
IS  - 1
PB  - The Publishers of ART
UR  - https://art.centre-mersenne.org/articles/10.5802/art.5/
DO  - 10.5802/art.5
LA  - en
ID  - ART_2024__1_1_73_0
ER  - 
%0 Journal Article
%A Andruskiewitsch, Nicolás
%A Natale, Sonia Luján
%A Torrecillas, Blas
%T A class of finite-by-cocommutative Hopf algebras
%J Annals of Representation Theory
%D 2024
%P 73-94
%V 1
%N 1
%I The Publishers of ART
%U https://art.centre-mersenne.org/articles/10.5802/art.5/
%R 10.5802/art.5
%G en
%F ART_2024__1_1_73_0
Andruskiewitsch, Nicolás; Natale, Sonia Luján; Torrecillas, Blas. A class of finite-by-cocommutative Hopf algebras. Annals of Representation Theory, Volume 1 (2024) no. 1, pp. 73-94. doi : 10.5802/art.5. https://art.centre-mersenne.org/articles/10.5802/art.5/

[1] Andruskiewitsch, N. Notes on extensions of Hopf algebras, Can. J. Math., Volume 48 (1996) no. 1, pp. 3-42 | Zbl

[2] Andruskiewitsch, N. An Introduction to Nichols Algebras, Quantization, Geometry and Noncommutative Structures in Mathematics and Physics (Mathematical Physics Studies), Springer, 2017, pp. 135-195 | Zbl

[3] Andruskiewitsch, N.; Angiono, I. On finite dimensional Nichols algebras of diagonal type, Bull. Math. Sci., Volume 7 (2017) no. 3, pp. 353-573 | Zbl

[4] Andruskiewitsch, N.; Angiono, I.; Yakimov, M. Poisson orders on large quantum groups, Adv. Math., Volume 428 (2023), Paper no. 109134 | DOI

[5] Andruskiewitsch, N.; Cuadra, J. On the structure of (co-Frobenius) Hopf algebras, J. Noncommut. Geom., Volume 7 (2013), pp. 83-104 | Zbl

[6] Andruskiewitsch, N.; Cuadra, J.; Etingof, P. On two finiteness conditions for Hopf algebras with nonzero integral, Ann. Sc. Norm. Super. Pisa, Volume 14 (2015), pp. 401-440 | Zbl

[7] Andruskiewitsch, N.; Dǎscǎlescu, S. Co-Frobenius Hopf algebras and the coradical filtration, Math. Z., Volume 243 (2003), pp. 145-154 | Zbl

[9] Andruskiewitsch, N.; García, G. A. Finite subgroups of a simple quantum group, Compos. Math., Volume 145 (2009), pp. 476-500 | Zbl

[10] Andruskiewitsch, N.; Schneider, H.-J. Pointed Hopf algebras, Recent developments in Hopf algebras Theory (Mathematical Sciences Research Institute Publications), Volume 43, Cambridge University Press, 2002, pp. 1-68 | Zbl

[11] Angiono, I. Distinguished pre-Nichols algebras, Transform. Groups, Volume 21 (2016), pp. 1-33 | Zbl

[12] Auslander, L. On a problem of Philip Hall, Ann. Math., Volume 86 (1967), pp. 112-116 | Zbl

[13] Blattner, R. J.; Cohen, M.; Montgomery, S. Crossed products and inner actions of Hopf algebras, Trans. Am. Math. Soc., Volume 298 (1986), pp. 671-711 | Zbl

[14] Brown, K. A.; Couto, M. Affine commutative-by-finite Hopf algebras, J. Algebra, Volume 573 (2021), pp. 56-94 | Zbl

[15] Brown, K. A.; Couto, M.; Jahn, A. The finite dual of commutative-by-finite Hopf algebras, Glasg. Math. J., Volume 65 (2000), pp. 62-89 | DOI | Zbl

[16] Byott, N. P. Cleft extensions of Hopf algebras, J. Algebra, Volume 57 (1993) no. 2, pp. 405-429 | Zbl

[17] Byott, N. P. Cleft extensions of Hopf algebras II, Proc. Lond. Math. Soc., Volume 67 (1993), pp. 277-304 | Zbl

[18] Dăscălescu, S.; Năstăsescu, C.; Torrecillas, B. Homological dimension of coalgebras and crossed coproducts, K-Theory, Volume 23 (2001), pp. 53-65 | Zbl

[19] De Concini, C.; Kac, V. Representations of quantum groups at roots of 1, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Progress in Mathematics), Volume 92, Birkhäuser, 1990, pp. 471-506 | Zbl

[20] De Concini, C.; Kac, V.; Procesi, C. Quantum coadjoint action, J. Am. Math. Soc., Volume 5 (1992), pp. 151-189 | Zbl

[21] De Concini, C.; Lyubashenko, V. Quantum function algebra at roots of 1, Adv. Math., Volume 108 (1994) no. 2, pp. 205-262 | Zbl

[22] De Concini, C.; Procesi, C. Quantum groups, D-modules, representation theory and quantum groups (Lecture Notes in Mathematics), Volume 1565, Springer, 1993, pp. 31-140 | Zbl

[23] Doi, Y. On the structure of relative Hopf modules, Commun. Algebra, Volume 11 (1983), pp. 243-255 | Zbl

[24] Doi, Y. Braided bialgebras and quadratic bialgebras, Commun. Algebra, Volume 21 (1993), pp. 1731-1749 | Zbl

[25] Doi, Y.; Takeuchi, M. Cleft comodule algebras for a bialgebra, Commun. Algebra, Volume 14 (1986), pp. 801-818 | Zbl

[26] Etingof, P.; Gelaki, S.; Nikshych, D.; Ostrik, V. Tensor categories, Mathematical Surveys and Monographs, 205, American Mathematical Society, 2015 | Zbl

[27] Formanek, E.; Jategaonkar, A. V. Subrings of Noetherian rings, Proc. Am. Math. Soc., Volume 46 (1974), pp. 181-186 | Zbl

[28] Gromov, M. Groups of polynomial growth and expanding maps. Appendix by Jacques Tits, Publ. Math., Inst. Hautes Étud. Sci., Volume 53 (1981), pp. 53-78 | Zbl

[29] Hall, P. Finiteness conditions for soluble groups, Proc. Lond. Math. Soc., Volume 4 (1954), pp. 419-436 | Zbl

[30] Heckenberger, I. Classification of arithmetic root systems, Adv. Math., Volume 220 (2009), pp. 59-124 | Zbl

[31] Hofstetter, I. Extensions of Hopf algebras and their cohomological description, J. Algebra, Volume 164 (1994), pp. 264-298 | Zbl

[32] Ivanov, S. V. Group rings of Noetherian groups, Math. Notes, Volume 46 (1989), pp. 929-933 | Zbl

[33] Kac, G. Extensions of groups to ring groups, Math. USSR, Sb., Volume 5 (1968), pp. 451-474 translation from Mat. Sb., N. Ser. 76no. 118, 473-496 (1968) | Zbl

[34] Krause, G.; Lenagan, T. Growth of algebras and Gelfand-Kirillov dimension, Graduate Studies in Mathematics, 22, American Mathematical Society, 2000 | Zbl

[35] Larson, R. G.; Radford, D. Finite dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple, J. Algebra, Volume 117 (1988), pp. 267-289 | Zbl

[36] Li, K.; Liu, G. The Finite Duals of Affine Prime Regular Hopf Algebras of GK-Dimension One, AIMS Math., Volume 8 (2023) no. 3, pp. 6829-6879 | DOI

[37] Lorenz, M.; Lorenz, M. On crossed products of Hopf algebras, Proc. Am. Math. Soc., Volume 123 (1995), pp. 33-38 | Zbl

[38] Majid, S. Crossed products by braided groups and bosonization, J. Algebra, Volume 163 (1994), pp. 165-190 | Zbl

[39] Masuoka, A. On Hopf algebras with cocommutative coradicals, J. Algebra, Volume 144 (1991), pp. 415-466 | Zbl

[40] McConnell, J. C.; Robson, J. C. Noncommutative Noetherian rings, Graduate Studies in Mathematics, 30, American Mathematical Society, 2001 (with the cooperation of L. W. Small. Reprinted with corrections from the 1987 original) | Zbl

[41] Molnar, R. K. A commutative Noetherian Hopf algebra over a field is finitely generated, Proc. Am. Math. Soc., Volume 51 (1975), pp. 501-502 | Zbl

[42] Montgomery, S. Hopf Algebras and their Actions on Rings. Expanded version of ten lectures given at the CBMS Conference on Hopf algebras and their actions on rings, which took place at DePaul University in Chicago, USA, August 10-14, 1992, Regional Conference Series in Mathematics, 82, American Mathematical Society, 1993 | Zbl

[43] Nastasescu, C.; Van Oysteyen, F. On strongly graded rings and crossed products, Commun. Algebra, Volume 10 (1982), pp. 2085-2106 | Zbl

[44] Nichols, W. D.; Zöller, M. B. A Hopf algebra freeness Theorem, Am. J. Math., Volume 111 (1989), pp. 381-385 | Zbl

[45] Radford, D. E. Hopf algebras, Series on Knots and Everything, 49, World Scientific, 2012 | Zbl

[46] Schauenburg, P.; Schneider, H.-J. Galois type extensions and Hopf algebras (2004) (preprint)

[47] Schneider, H.-J. Zerlegbare Erweiterungen affiner Gruppen, J. Algebra, Volume 66 (1980), pp. 569-593 | Zbl

[48] Schneider, H.-J. Zerlegbare Untergruppen affiner Gruppen, Math. Ann., Volume 255 (1981), pp. 139-158 | Zbl

[49] Schneider, H.-J. Some remarks on exact sequences of quantum groups, Commun. Algebra, Volume 21 (1993) no. 9, pp. 3337-3358 | Zbl

[50] Singer, W. Extension theory for connected Hopf algebras, J. Algebra, Volume 21 (1972), pp. 1-16 | Zbl

[51] Swan, R. G. Representations of polycyclic groups, Proc. Am. Math. Soc., Volume 18 (1967), pp. 573-574 | Zbl

[52] Takeuchi, M. Matched pair of groups and bismash product of Hopf algebras, Commun. Algebra, Volume 9 (1981), pp. 841-882 | Zbl

[53] Takeuchi, M. Quotient spaces for Hopf algebras, Commun. Algebra, Volume 22 (1994), pp. 2503-2523 | Zbl

[54] Tits, J. Free subgroups in linear groups, J. Algebra, Volume 20 (1972), pp. 250-270 | Zbl

[55] Ulbrich, K. H. Vollgraduierte Algebren, Abh. Math. Semin. Univ. Hamb., Volume 51 (1981), pp. 136-148 | Zbl

[56] Wang, X.; Yu, X.; Zhang, Y. Calabi-Yau property under monoidal Morita–Takeuchi equivalence, Pac. J. Math., Volume 290 (2017), pp. 481-510 | Zbl

Cited by Sources: