Preface These lectur e note s ar e a n expande d versio n o f te n lecture s give n a t th e CBMS conferenc e o n Hop f Algebra s an d Thei r Action s o n Rings , whic h too k place a t DePau l Universit y i n Chicago , Augus t 10-14 , 1992 . It was a very good tim e to have such a conference, fo r severa l reasons. Th e most obviou s o f thes e i s th e curren t grea t interes t i n quantu m groups thes e are Hop f algebra s whic h aros e i n statistica l mechanic s an d no w hav e con - nections t o man y area s o f mathematics . Howeve r ther e hav e bee n a numbe r of significan t recen t development s withi n Hop f algebra s themselves . Severa l old conjecture s o f Kaplansk y hav e recentl y bee n solved , th e mos t strikin g of whic h i s a kin d o f Lagrange' s theore m fo r Hop f algebras . I n a differen t direction ther e ha s bee n a lo t o f wor k o n action s o f Hop f algebras , whic h unifies earlie r result s know n fo r grou p actions , action s o f Li e algebras , an d graded algebras . The objec t o f the meeting , an d o f thes e notes, wa s t o brin g togethe r man y of thes e recen t developments i n fac t ther e i s a grea t dea l o f interconnectio n between th e variou s directions . Th e poin t o f vie w throughout , however , i s the algebrai c structur e o f Hop f algebra s an d thei r action s an d coactions . Quantum group s ar e treate d a s a n importan t exampl e rathe r tha n a s a n en d in themselves never-the-les s th e reade r intereste d i n quantu m group s shoul d find muc h basi c materia l here . Most o f Chapter s 1 and 2 is old, an d i n fact appear s i n th e book s o n Hop f algebras b y Sweedle r [S ] an d Ab e [A] this i s als o tru e o f part s o f Chapter s 5 an d 9 . I hav e include d thi s materia l i n orde r t o b e a s self-containe d a s possible moreove r som e o f th e argument s ar e new . Th e res t o f thes e note s has no t previousl y appeare d i n boo k form . Althoug h man y o f th e proof s ar e only sketched , an d eve n occasionall y omitte d (wit h appropriat e reference s t o the literature) , enoug h detai l i s give n s o tha t thi s boo k coul d b e use d fo r a graduate leve l course . I n fac t thes e note s gre w ou t o f course s I gav e a t US C in 198 9 an d i n 1992 . A standar d first-yea r graduat e algebr a clas s shoul d b e a sufficien t prerequisite . There ar e man y peopl e I wis h t o thank . Firs t o f al l i s Jef f Bergen , wh o organized th e conferenc e an d mad e al l th e arrangements , an d secon d ar e xiii

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