2 1. DEFINITION S AN D EXAMPLE S a) coassociativit y c * — b) couni t c®c c®c id® A A ® id - c®c®c c Cgk 'id g ) e c®c The tw o uppe r map s i n 1.1.3.b) are given b y c *- 1 ® C and c H- C® 1 , for any c € C. W e say C is cocommutative i f r o A = A . Note tha t 1.1. 3 b) gives tha t A is injective, jus t a s 1.1.1 b) gives tha t m is surjective . 1.1.4 DEFINITION . Le t C and D b e coalgebras, wit h comultiplication s A c and A# , and counits sc an d £#, respectively. a) A map / : C — D is a coalgebra morphism i f A/) o / = ( / ® / ) Ac and i f £? = ^ D ° /• b) A subspace / C C is a coirfea / i f A / C / ® C + C ® 7 and if e(I) = 0. It is easy t o check that i f / i s a coideal, then th e k-space C/I i s a coalgebr a with comultiplicatio n induce d fro m A , and conversely . Finally, w e may also us e the twist ma p to dualize th e notion o f opposit e algebra. Fo r a give n algebr a A, recal l tha t A op i s the algebra obtaine d b y using A a s a vecto r space , bu t wit h ne w multiplication a 0 • 6° = (6a)° , for a0,6° G A op . I n term s o f map s thi s ne w multiplication i s give n b y m ' : A® A —* A, wher e ra' = rn o r. 1.1.5 DEFINITION . Le t C be a coalgebra. The n th e coopposite coalgebra C cop is given a s follows: C cop = C as a vector space , wit h ne w comultiplicatio n A ' given b y A' = r o A. It i s easy t o see that C cop i s also a coalgebra . §1.2. Dual s o f algebra s an d coalgebra s We shal l no w see that ther e i s a ver y clos e relationshi p betwee n algebra s and coalgebras , b y looking a t thei r dua l spaces . For an y k-space V , let V* = Homk(V , k) denot e th e linear dua l o f V.

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