Probabilistically nilpotent Hopf algebras

Cohen, Miriam; Westreich, Sara

doi:10.1090/tran/6462

Probabilistically nilpotent Hopf algebras
HTML articles powered by AMS MathViewer

by Miriam Cohen and Sara Westreich PDF

Trans. Amer. Math. Soc. 368 (2016), 4295-4314 Request permission

Abstract:

In this paper we investigate nilpotenct and probabilistically nilpotent Hopf algebras. We define nilpotency via a descending chain of commutators and give a criterion for nilpotency via a family of central invertible elements. These elements can be obtained from a commutator matrix $A$ which depends only on the Grothendieck ring of $H.$ When $H$ is almost cocommutative we introduce a probabilistic method. We prove that every semisimple quasitriangular Hopf algebra is probabilistically nilpotent. In a sense we thereby answer the title of our paper Are we counting or measuring anything? by Yes, we are.

References

Nicolás Andruskiewitsch, Notes on extensions of Hopf algebras, Canad. J. Math. 48 (1996), no. 1, 3–42. MR 1382474, DOI 10.4153/CJM-1996-001-8
Alon Amit and Uzi Vishne, Characters and solutions to equations in finite groups, J. Algebra Appl. 10 (2011), no. 4, 675–686. MR 2834108, DOI 10.1142/S0219498811004690
Sebastian Burciu, Kernels of representations and coideal subalgebras of Hopf algebras, Glasg. Math. J. 54 (2012), no. 1, 107–119. MR 2862388, DOI 10.1017/S0017089511000395
Miriam Cohen and Sara Westreich, Structure constants related to symmetric Hopf algebras, J. Algebra 324 (2010), no. 11, 3219–3240. MR 2732995, DOI 10.1016/j.jalgebra.2010.07.003
Miriam Cohen and Sara Westreich, Conjugacy classes, class sums and character tables for Hopf algebras, Comm. Algebra 39 (2011), no. 12, 4618–4633. MR 2863455, DOI 10.1080/00927872.2011.617265
Miriam Cohen and Sara Westreich, Higman ideals and Verlinde-type formulas for Hopf algebras, Ring and module theory, Trends Math., Birkhäuser/Springer Basel AG, Basel, 2010, pp. 91–114. MR 2744044, DOI 10.1007/978-3-0346-0007-1_{5}
Miriam Cohen and Sara Westreich, Hit-invariants and commutators for Hopf algebras, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 56(104) (2013), no. 3, 299–313. MR 3114476
Miriam Cohen and Sara Westreich, Are we counting or measuring something?, J. Algebra 398 (2014), 111–130. MR 3123755, DOI 10.1016/j.jalgebra.2013.09.015
Shlomo Gelaki and Dmitri Nikshych, Nilpotent fusion categories, Adv. Math. 217 (2008), no. 3, 1053–1071. MR 2383894, DOI 10.1016/j.aim.2007.08.001
V. Linchenko and S. Montgomery, A Frobenius-Schur theorem for Hopf algebras, Algebr. Represent. Theory 3 (2000), no. 4, 347–355. Special issue dedicated to Klaus Roggenkamp on the occasion of his 60th birthday. MR 1808131, DOI 10.1023/A:1009949909889
Hans-Jürgen Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72 (1990), no. 1-2, 167–195. Hopf algebras. MR 1098988, DOI 10.1007/BF02764619
Aner Shalev, Word maps, conjugacy classes, and a noncommutative Waring-type theorem, Ann. of Math. (2) 170 (2009), no. 3, 1383–1416. MR 2600876, DOI 10.4007/annals.2009.170.1383
Mitsuhiro Takeuchi, Quotient spaces for Hopf algebras, Comm. Algebra 22 (1994), no. 7, 2503–2523. MR 1271619, DOI 10.1080/00927879408824973