Structure of a Class of Semisimple Hopf Algebras

doi:10.12386/A2011sxxb0030

Acta Mathematica Sinica, Chinese Series 鈥衡�� 2011, Vol. 54 鈥衡�� Issue (2): 293-300.DOI: 10.12386/A2011sxxb0030

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Structure of a Class of Semisimple Hopf Algebras

Jing Cheng DONG^1,2

1. School of Mathematics, Yangzhou University, Yangzhou 225002, P. R. China;
2. College of Engineering, Nanjing Agricultural University, Nanjing 210031, P. R. China

Received:2009-12-09 Revised:2010-10-22 Online:2011-03-15 Published:2011-03-15

涓�绫诲崐鍗旽opf浠ｆ暟鐨勭粨鏋�

钁ｄ簳鎴�^1,2

1. 鎵窞澶у鏁板绉戝瀛﹂櫌鎵窞 225002;
2. 鍗椾含鍐滀笟澶у宸ュ闄� 鍗椾含 210031

鍩洪噾璧勫姪:
鍥藉鑷劧绉戝鍩洪噾璧勫姪椤圭洰(10771183);鏁欒偛閮ㄥ崥澹偣鍩洪噾璧勫姪椤圭洰(200811170001)

Abstract

Abstract:

Let k be an algebraically closed field of characteristic zero, H a semisimple Hopf algebra of dimension pq² of Frobenius type, where p, q are distinct prime numbers. This paper proves that if p > q and H^* is also of Frobenius type then H is a biproduct of a group algebra A of dimension q² and a Yetter-Drinfeld Hopf algebra R over A of dimension p. That is H 鈮� R#A. As an example, this paper then proves that every semisimple Hopf algebra of dimension 63 or 68 is of Frobenius type.

Key words: semisimple Hopf algebra, Kaplansky’s sixth conjecture, biproduct

鎽樿锛�

璁�k鏄壒寰佷负闆剁殑浠ｆ暟闂煙, H鏄�k涓婄殑pq²缁� Frobenius鍨嬪崐鍗� Hopf浠ｆ暟, 鍏朵腑p,q涓轰笉鍚岀殑绱犳暟. 鏈枃璇佹槑浜�,濡傛灉p>q涓�H^*涔熸槸Frobenius鍨婬opf浠ｆ暟, 鍒�H鏄�q²缁寸兢浠ｆ暟A涓�A涓�p缁� Yetter--Drinfeld Hopf浠ｆ暟R鐨勫弻绉�, 鍗�H 鈮� R # A. 浣滀负渚嬪瓙, 鏈枃杩樿瘉鏄庝簡浠绘剰63缁存垨68缁寸殑鍗婂崟Hopf浠ｆ暟鍧囦负Frobenius鍨婬opf浠ｆ暟.

鍏抽敭璇�: 鍗婂崟Hopf浠ｆ暟, Kaplansky绗叚鐚滄兂, 鍙岀Н

CLC Number:

O153.3

Jing Cheng DONG. Structure of a Class of Semisimple Hopf Algebras[J]. Acta Mathematica Sinica, Chinese Series, 2011, 54(2): 293-300.

钁ｄ簳鎴�. 涓�绫诲崐鍗旽opf浠ｆ暟鐨勭粨鏋刐J]. 鏁板瀛︽姤, 2011, 54(2): 293-300.

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Structure of a Class of Semisimple Hopf Algebras

涓�绫诲崐鍗旽opf浠ｆ暟鐨勭粨鏋�

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