Chapter 1 Definitions an d Example s §1.1 Algebra s an d coalgebra s Throughout w e le t k b e a field, althoug h muc h o f wha t w e d o i s vali d over an y commutativ e ring . Tenso r product s ar e assume d t o b e ove r k unles s stated otherwise . We first expres s th e associativ e an d uni t propertie s o f an algebr a vi a map s so tha t w e ma y dualiz e them . 1.1.1 DEFINITION . A k-algebra (wit h unit ) i s a k-vecto r spac e A togethe r with tw o k-linea r maps , multiplicatio n rn : A ® A A an d uni t u : k A, such tha t th e followin g diagram s ar e commutative : a) associativit y b ) uni t A u (g id A m ® id •A® A id ® m k®A A id® u A®k The tw o lowe r map s i n b ) ar e give n b y scala r multiplication . 1.1.1 , b ) gives th e usua l identit y elemen t i n A b y settin g 1& = u(lk) - 1.1.2 DEFINITION . Fo r an y k-space s V an d W, th e twist map r : W - W ® V i s give n b y T(V ®W) = W®V. Note tha t A i s commutativ e = rn o r = m o n A. We no w dualiz e th e notio n o f algebra . 1.1.3 DEFINITION . A k-coalgebra (wit h counit) i s a k-vecto r spac e C to - gether wit h tw o k-linea r maps , comultiplication A : C C ® C an d counit e : C k , suc h tha t th e followin g diagram s ar e commutative : l http://dx.doi.org/10.1090/cbms/082/01
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