Abstract
In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a group, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Benkart and Witherspoon (Algebr. Represent. Theory 7(3) ? BC) who showed that two-parameter quantum groups are Drinfeld doubles. It is possible to recover a Lie algebra from these doubles in the case where the group is free abelian and the parameters are generic. The Lie algebras arising are generated by Lie subalgebras isomorphic to \(\mathfrak {sl}_{2}\).
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Andruskiewitsch, N., Carnovale, G., García, G.A.: Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type I. Non-semisimple classes in PSL(n,q). J. Algebra 442, 36–65 (2015). MR3395052
Andruskiewitsch, N., Carnovale, G., García, G.A.: Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type II. Unipotent classes in symplectic groups, (2014). Available at arXiv:1412.7397
Andruskiewitsch, N., Fantino, F., Graña, M., Vendramin, L.: Finite-dimensional pointed Hopf algebras with alternating groups are trivial. Ann. Mat. Pura Appl. (4) 190(2), 225–245 (2011). MR2786171 (2012c:16095)
García, G.A.: Multiparameter quantum groups, bosonizations and cocycle deformations, (2014). Available at arXiv:http://arXiv.org/abs/1406.2561
Andruskiewitsch, N.: On finite-dimensional Hopf algebras. Proceedings of the ICM Seoul 2014. Vol II, pp. 117–141 (2014)
Andruskiewitsch, N., Radford, D., Schneider, H.: Complete reducibility theorems for modules over pointed Hopf algebras. J. Algebra 324(11), 2932–2970 (2010). MR2732981 (2012b:16080)
Andruskiewitsch, N., Schneider, H.: Finite quantum groups and Cartan matrices. Adv. Math. 154(1), 1–45 (2000). MR1780094 (2001g:16070)
Andruskiewitsch, N., Schneider, H.-J.: Pointed Hopf algebras, New directions in Hopf algebras, 2002, pp. 1–68. MR1913436 (2003e:16043)
Andruskiewitsch, N., Schneider, H.-J.: A characterization of quantum groups. J. Reine Angew. Math. 577, 81–104 (2004). MR2108213 (2005i:16083)
Andruskiewitsch, N., Schneider, H.-J.: On the classification of finite-dimensional pointed Hopf algebras. Ann. Math. (2) 171(1), 375–417 (2010). MR2630042 (2011j:16058)
Artin, M., Schelter, W., Tate, J.: Quantum deformations of GLn. Comm. Pure Appl. Math. 44(8–9), 879–895 (1991). MR1127037 (92i:17014)
Bazlov, Y., Berenstein, A.: Braided doubles and rational Cherednik algebras. Adv. Math. 220(5), 1466–1530 (2009). MR2493618 (2010e:16046)
Brown, K.A., Goodearl, K.R.: Lectures on algebraic quantum groups, Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel (2002). MR1898492 (2003f:16067)
Benkart, G., Witherspoon, S.: Two-parameter quantum groups and Drinfel’d doubles. Algebr. Represent. Theory 7(3) (2004). 261.286. MR2070408 (2005g:17028)
Chin, W., Musson, I.M.: Multiparameter quantum enveloping algebras. J. Pure Appl. Algebra 107(2–3), 171?-191 (1996). Contact Franco-Belge en Alg‘ebre (Diepenbeek, 1993). MR1383171 (97c:17016)
Chari, V., Pressley, A.: A guide to quantum groups. Cambridge University Press, Cambridge (1995). Corrected reprint of the 1994 original. MR1358358 (96h:17014)
De Concini, C., Kac, V.G. Representations of quantum groups at roots of 1, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), 1990, pp. 471–506. MR1103601 (92g:17012)
Drinfel’d, V.G.: Quantum groups. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155 (1986). no. Differentsialnaya Geometriya, Gruppy Li i Mekh. VIII, 18–49, 193. MR869575 (88f:17017)
Faddeev, L.D., Reshetikhin, N.Y., Takhtajan, L.A.: Quantization of Lie groups and Lie algebras. Algebraic Anal. 1, 129–139 (1988). MR992450 (91d:17017)
Gelaki, S.: Pointed Hopf algebras and Kaplansky’s 10th conjecture. J. Algebra 209(2), 635–657 (1998). MR1659891 (99j:16024)
Laugwitz, R.: Braided Drinfeld and Heisenberg doubles. J. Pure Appl. Algebra 219(10), 4541–4596 (2015). MR3346505
Majid, S.: Doubles of quasitriangular Hopf algebras. Comm. Algebra 19(11), 3061–3073 (1991). MR1132774 (92k:16052)
Majid, S.: Foundations of quantum group theory. Cambridge University Press, Cambridge (1995). MR1381692 (97g:17016)
Majid, S.: Double-bosonization of braided groups and the construction of Uq(g). Math. Proc. Camb. Philos. Soc. 1, 151–192 (1999). MR1645545 (2000a:17017)
Milnor, J.W., Moore, J.C.: On the structure of Hopf algebras. Ann. Math. (2) 81, 211–264 (1965). MR0174052 (30 #4259)
Montgomery, S.: Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1993). MR1243637 (94i:16019)
Montgomery, S.: Indecomposable coalgebras, simple comodules, and pointed Hopf algebras. Proc. Amer. Math. Soc. 123(8), 2343–2351 (1995). MR1257119 (95j:16046)
Pei, Y., Hu, N., Rosso, M. Multi-parameter quantum groups and quantum shuffles. I, Quantum affine algebras, extended affine Lie algebras, and their applications, 2010, pp. 145–171. MR2642565 (2011c:17034)
Radford, D.E. On Kauffman’s knot invariants arising from finite-dimensional Hopf algebras, Advances in Hopf algebras (Chicago, IL, 1992), 1994, pp. 205?266. MR1289427 (96g:57013)
Reshetikhin, N.Y.: Multiparameter quantum groups and twisted quasitriangular Hopf algebras. Lett. Math. Phys. 20(4), 331–335 (1990). MR1077966 (91k:17012)
Rosso, M.: Quantum groups and quantum shuffles. Invent. Math. 133(2), 399–416 (1998). MR1632802 (2000a:17021)
Sudbery, A.: Consistent multiparameter quantisation of GL(n). J. Phys. A 23 (15), L697–L704 (1990). MR1068228 (91m:17022)
Sweedler, M.E.: Hopf algebras, Mathematics Lecture Note Series. W. A. Benjamin, Inc., New York (1969). MR0252485 (40 #5705)
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Presented by Vyjayanthi Chari.
Supported by EPSRC grant EP/I033343/1.
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Laugwitz, R. Pointed Hopf Algebras with Triangular Decomposition. Algebr Represent Theor 19, 547–578 (2016). https://doi.org/10.1007/s10468-015-9588-x
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DOI: https://doi.org/10.1007/s10468-015-9588-x
Keywords
- Multiparameter quantum groups
- Pointed Hopf algebras
- Nichols–Woronowicz algebras
- Braided doubles
- Drinfeld doubles