On Higher Frobenius-schur Indicators

by Kashina, Yevgenia; Sommerhauser, Yorck; Zhu, Yongchang

• ISBN13: 9780821838860
• ISBN10: 0821838865
• Format: Paperback
• Copyright: 2006-04-30
• Publisher: Amer Mathematical Society

Summary

We study the higher Frobenius-Schur indicators of modules over semisimple Hopf algebras, and relate them to other invariants as the exponent, the order, and the index. We prove various divisibility and integrality results for these invariants. In particular, we prove a version of Cauchy's theorem for semisimple Hopf algebras. Furthermore, we give some examples that illustrate the general theory.

Table of Contents

Introduction	1	(4)
The Calculus of Sweedler Powers	5	(8)
Monotone maps	5	(1)
The union of the symmetric groups	6	(1)
Bialgebras	7	(1)
A monoid	8	(1)
Permutations from sequences	9	(1)
Sweedler powers	10	(3)
Frobenius-Schur Indicators	13	(8)
Central Sweedler powers	13	(1)
The coproduct of the Sweedler powers	14	(1)
The first formula for the Frobenius-Schur indicators	15	(2)
The Frobenius-Schur theorem	17	(2)
Frobenius-Schur indicators of the regular representation	19	(2)
The Exponent	21	(8)
The exponent	21	(2)
The second formula for the Frobenius-Schur indicators	23	(1)
Sweedler powers of the integral	24	(1)
Cauchy's theorem	25	(4)
The Order	29	(6)
Order and multiplicity	29	(1)
The divisibility theorem	29	(2)
An example	31	(2)
The dimension of the simple modules	33	(2)
The Index	35	(8)
Indecomposable matrices	35	(2)
The normal form	37	(1)
The Perron-Frobenius theorem	38	(1)
The index formula	39	(4)
The Drinfel'd Double	43	(6)
The Drinfel'd double	43	(1)
Factorizability	43	(2)
The center of the character ring	45	(2)
The third formula for the Frobenius-Schur indicators	47	(2)
Examples	49	(10)
A class of extensions	49	(1)
The coefficients	50	(3)
Sweedler powers of the integral	53	(1)
The simple modules	54	(1)
Nonintegral indicators	55	(1)
Noncocommutative Sweedler powers	56	(1)
Noncentral Sweedler powers	56	(3)
Bibliography	59	(2)
Subject Index	61	(4)
Symbol Index	65