AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
On finite GK-dimensional Nichols algebras over abelian groups
About this Title
Nicolás Andruskiewitsch, Iván Angiono and István Heckenberger
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 271, Number 1329
ISBNs: 978-1-4704-4830-1 (print); 978-1-4704-6636-7 (online)
DOI: https://doi.org/10.1090/memo/1329
Published electronically: August 11, 2021
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. Yetter-Drinfeld modules of dimension 2
- 4. Yetter-Drinfeld modules of dimension 3
- 5. One block and several points
- 6. Two blocks
- 7. Several blocks, several points
- 8. Appendix
Abstract
We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, $\operatorname {GKdim}$ for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces over $\mathbb {Z}$ with the generator acting as a single Jordan block and show that the corresponding Nichols algebra has finite $\operatorname {GKdim}$ if and only if the size of the block is 2 and the eigenvalue is $\pm 1$; when this is 1, we recover the quantum Jordan plane. We consider next a class of braided vector spaces that are direct sums of blocks and points that contains those of diagonal type. We conjecture that a Nichols algebra of diagonal type has finite $\operatorname {GKdim}$ if and only if the corresponding generalized root system is finite. Assuming the validity of this conjecture, we classify all braided vector spaces in the mentioned class whose Nichols algebra has finite $\operatorname {GKdim}$. Consequently we present several new examples of Nichols algebras with finite $\operatorname {GKdim}$, including two not in the class alluded to above. We determine which among these Nichols algebras are domains.- Nicolás Andruskiewitsch and Iván Ezequiel Angiono, On Nichols algebras with generic braiding, Modules and comodules, Trends Math., Birkhäuser Verlag, Basel, 2008, pp. 47–64. MR 2742620, DOI 10.1007/978-3-7643-8742-6_{3}
- Nicolás Andruskiewitsch, Iván Angiono, and István Heckenberger, Liftings of Jordan and super Jordan planes, Proc. Edinb. Math. Soc. (2) 61 (2018), no. 3, 661–672. MR 3834727, DOI 10.1017/s0013091517000402
- Nicolás Andruskiewitsch, Iván Angiono, and István Heckenberger, On finite GK-dimensional Nichols algebras of diagonal type, Tensor categories and Hopf algebras, Contemp. Math., vol. 728, Amer. Math. Soc., [Providence], RI, [2019] ©2019, pp. 1–23. MR 3943743, DOI 10.1090/conm/728/14653
- Nicolás Andruskiewitsch, Juan Cuadra, and Pavel Etingof, On two finiteness conditions for Hopf algebras with nonzero integral, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14 (2015), no. 2, 401–440. MR 3410615
- Nicolás Andruskiewitsch and Sorin Dăscălescu, Co-Frobenius Hopf algebras and the coradical filtration, Math. Z. 243 (2003), no. 1, 145–154. MR 1953053, DOI 10.1007/s00209-002-0456-0
- Nicolás Andruskiewitsch, István Heckenberger, and Hans-Jürgen Schneider, The Nichols algebra of a semisimple Yetter-Drinfeld module, Amer. J. Math. 132 (2010), no. 6, 1493–1547. MR 2766176
- N. Andruskiewitsch, D. Radford and H.-J. Schneider, The Nichols algebra of a semisimple Yetter-Drinfeld module. J. Algebra 324 (2010), 2932–2970.
- Nicolás Andruskiewitsch and Hans-Jürgen Schneider, Pointed Hopf algebras, New directions in Hopf algebras, Math. Sci. Res. Inst. Publ., vol. 43, Cambridge Univ. Press, Cambridge, 2002, pp. 1–68. MR 1913436, DOI 10.2977/prims/1199403805
- Nicolás Andruskiewitsch and Hans-Jürgen Schneider, A characterization of quantum groups, J. Reine Angew. Math. 577 (2004), 81–104. MR 2108213, DOI 10.1515/crll.2004.2004.577.81
- Iván Ezequiel Angiono, A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 10, 2643–2671. MR 3420518, DOI 10.4171/JEMS/567
- Iván Angiono, On Nichols algebras of diagonal type, J. Reine Angew. Math. 683 (2013), 189–251. MR 3181554, DOI 10.1515/crelle-2011-0008
- Michael Artin and William F. Schelter, Graded algebras of global dimension $3$, Adv. in Math. 66 (1987), no. 2, 171–216. MR 917738, DOI 10.1016/0001-8708(87)90034-X
- Kenneth A. Brown, Representation theory of Noetherian Hopf algebras satisfying a polynomial identity, Trends in the representation theory of finite-dimensional algebras (Seattle, WA, 1997) Contemp. Math., vol. 229, Amer. Math. Soc., Providence, RI, 1998, pp. 49–79. MR 1676211, DOI 10.1090/conm/229/03310
- Kenneth A. Brown, Noetherian Hopf algebras, Turkish J. Math. 31 (2007), no. suppl., 7–23. MR 2368080
- Kenneth A. Brown and Paul Gilmartin, Hopf algebras under finiteness conditions, Palest. J. Math. 3 (2014), no. Special issue, 356–365. MR 3274613
- Mini-Workshop: infinite dimensional Hopf algebras, Oberwolfach Rep. 11 (2014), no. 2, 1111–1137. Abstracts from the mini-workshop held April 13–19, 2014; Organized by Ken Brown, Ken Goodearl, Tom Lenagan, and James Zhang. MR 3379364, DOI 10.4171/OWR/2014/20
- K. A. Brown and J. J. Zhang, Prime regular Hopf algebras of GK-dimension one, Proc. Lond. Math. Soc. (3) 101 (2010), no. 1, 260–302. MR 2661247, DOI 10.1112/plms/pdp060
- Lisa Carbone, Sjuvon Chung, Leigh Cobbs, Robert McRae, Debajyoti Nandi, Yusra Naqvi, and Diego Penta, Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits, J. Phys. A 43 (2010), no. 15, 155209, 30. MR 2608277, DOI 10.1088/1751-8113/43/15/155209
- E. E. Demidov, Yu. I. Manin, E. E. Mukhin, and D. V. Zhdanovich, Nonstandard quantum deformations of $\textrm {GL}(n)$ and constant solutions of the Yang-Baxter equation, Progr. Theoret. Phys. Suppl. 102 (1990), 203–218 (1991). Common trends in mathematics and quantum field theories (Kyoto, 1990). MR 1182166, DOI 10.1143/PTPS.102.203
- V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
- Pavel Etingof and Shlomo Gelaki, Quasisymmetric and unipotent tensor categories, Math. Res. Lett. 15 (2008), no. 5, 857–866. MR 2443987, DOI 10.4310/MRL.2008.v15.n5.a3
- Delia Flores de Chela and James A. Green, Quantum symmetric algebras, Algebr. Represent. Theory 4 (2001), no. 1, 55–76. Special issue dedicated to Klaus Roggenkamp on the occasion of his 60th birthday. MR 1825807, DOI 10.1023/A:1009953611721
- K. R. Goodearl, Noetherian Hopf algebras, Glasg. Math. J. 55 (2013), no. A, 75–87. MR 3110805, DOI 10.1017/S0017089513000517
- K. R. Goodearl and J. J. Zhang, Noetherian Hopf algebra domains of Gelfand-Kirillov dimension two, J. Algebra 324 (2010), no. 11, 3131–3168. MR 2732991, DOI 10.1016/j.jalgebra.2009.11.001
- Matías Graña, A freeness theorem for Nichols algebras, J. Algebra 231 (2000), no. 1, 235–257. MR 1779599, DOI 10.1006/jabr.2000.8363
- Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
- M. Graña and I. Heckenberger, On a factorization of graded Hopf algebras using Lyndon words, J. Algebra 314 (2007), no. 1, 324–343. MR 2331765, DOI 10.1016/j.jalgebra.2007.02.046
- Ralf Günther, Crossed products for pointed Hopf algebras, Comm. Algebra 27 (1999), no. 9, 4389–4410. MR 1705876, DOI 10.1080/00927879908826704
- D. I. Gurevich, The Yang-Baxter equation and the generalization of formal Lie theory, Dokl. Akad. Nauk SSSR 288 (1986), no. 4, 797–801 (Russian). MR 852270
- I. Heckenberger, The Weyl groupoid of a Nichols algebra of diagonal type, Invent. Math. 164 (2006), no. 1, 175–188. MR 2207786, DOI 10.1007/s00222-005-0474-8
- I. Heckenberger, Classification of arithmetic root systems, Adv. Math. 220 (2009), no. 1, 59–124. MR 2462836, DOI 10.1016/j.aim.2008.08.005
- I. Heckenberger and H.-J. Schneider, Yetter-Drinfeld modules over bosonizations of dually paired Hopf algebras, Adv. Math. 244 (2013), 354–394. MR 3077876, DOI 10.1016/j.aim.2013.05.009
- Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
- Günter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Revised edition, Graduate Studies in Mathematics, vol. 22, American Mathematical Society, Providence, RI, 2000. MR 1721834, DOI 10.1090/gsm/022
- A. Yu. Lazarev and M. V. Movshev, Quantization of some Lie groups and algebras, Uspekhi Mat. Nauk 46 (1991), no. 6(282), 215–216 (Russian); English transl., Russian Math. Surveys 46 (1991), no. 6, 225–226. MR 1164208, DOI 10.1070/RM1991v046n06ABEH002868
- Gongxiang Liu, On Noetherian affine prime regular Hopf algebras of Gelfand-Kirillov dimension 1, Proc. Amer. Math. Soc. 137 (2009), no. 3, 777–785. MR 2457414, DOI 10.1090/S0002-9939-08-09034-5
- D.-M. Lu, Q.-S. Wu, and J. J. Zhang, Homological integral of Hopf algebras, Trans. Amer. Math. Soc. 359 (2007), no. 10, 4945–4975. MR 2320655, DOI 10.1090/S0002-9947-07-04159-1
- George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
- Ch. Ohn, A $*$-product on $\textrm {SL}(2)$ and the corresponding nonstandard quantum-$U({\mathfrak {s}}{\mathfrak {l}}(2))$, Lett. Math. Phys. 25 (1992), no. 2, 85–88. MR 1182027, DOI 10.1007/BF00398304
- Sebastián Reca and Andrea Solotar, Homological invariants relating the super Jordan plane to the Virasoro algebra, J. Algebra 507 (2018), 120–185. MR 3807045, DOI 10.1016/j.jalgebra.2018.04.008
- Marc Rosso, Groupes quantiques et algèbres de battage quantiques, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 2, 145–148 (French, with English and French summaries). MR 1320345
- Marc Rosso, Quantum groups and quantum shuffles, Invent. Math. 133 (1998), no. 2, 399–416. MR 1632802, DOI 10.1007/s002220050249
- Peter Schauenburg, A characterization of the Borel-like subalgebras of quantum enveloping algebras, Comm. Algebra 24 (1996), no. 9, 2811–2823. MR 1396857, DOI 10.1080/00927879608825714
- Hans-Jürgen Schneider, Normal basis and transitivity of crossed products for Hopf algebras, J. Algebra 152 (1992), no. 2, 289–312. MR 1194305, DOI 10.1016/0021-8693(92)90034-J
- Stefan Ufer, PBW bases for a class of braided Hopf algebras, J. Algebra 280 (2004), no. 1, 84–119. MR 2081922, DOI 10.1016/j.jalgebra.2004.06.017
- Hiroyuki Yamane, Representations of a $\Bbb Z/3\Bbb Z$-quantum group, Publ. Res. Inst. Math. Sci. 43 (2007), no. 1, 75–93. MR 2317113
- D.-G. Wang, J. J. Zhang, and G. Zhuang, Hopf algebras of GK-dimension two with vanishing Ext-group, J. Algebra 388 (2013), 219–247. MR 3061686, DOI 10.1016/j.jalgebra.2013.03.032
- S. Zakrzewski, A Hopf star-algebra of polynomials on the quantum $\textrm {SL}(2,\textbf {R})$ for a “unitary” $R$-matrix, Lett. Math. Phys. 22 (1991), no. 4, 287–289. MR 1131752, DOI 10.1007/BF00405603
- Guangbin Zhuang, Properties of pointed and connected Hopf algebras of finite Gelfand-Kirillov dimension, J. Lond. Math. Soc. (2) 87 (2013), no. 3, 877–898. MR 3073681, DOI 10.1112/jlms/jds079