1. DEFINITION S AN D EXAMPLE S 3 V an d V* determin e a non-degenerat e bilinea r for m { , ) : V* (g) V k via (/ , v) f(v)\ w e writ e i t a s a for m sinc e w e frequentl y wis h t o thin k of V a s actin g o n V*. I f f : V W i s k-linear , the n th e transpose o f ^ i s £ * : Wm - I/* , give n b y (1.2.1) f(f)(v) = f(f(v)), for a l l / W , u V . 1.2.2 LEMMA . 7/ C is a coalgebra, then C* is an algebra, with multiplication m = A * and unit u = £* . 7/ C i s cocommutative, then C* is commutative. The Lemm a i s proved simpl y b y dualizin g th e diagrams on e need s onl y the additiona l observatio n tha t sinc e C * ® C* C ( C ® C)*, we may restric t A* to get a map m : C* ® C*—• C* . Explicitly , m i s given b y m(f (&g)(c) = A*(/® /)(c ) = {f®g)Ac, fo r al l f,g C ' , c 6 C . If we begin wit h a n algebr a A, however , difficulties arise . For , i f A i s not finite-dimensional, A* ® A* i s a prope r subspac e o f (A ® A)* an d thu s th e image o f m * : A * (A ® A)* ma y no t li e i n A * ® A*. O f cours e i f A i s finite-dimensional, al l is well, and A* is a coalgebra. Fo r the general case , we require a definition . 1.2.3 DEFINITION . Le t A be a k-algebra. The finite dual of A i s A 0 = { / £ A* | /(/) = 0 , fo r som e ideal / o f A suc h tha t di m A/1 oo}. 1.2.4 PROPOSITION . If A is an algebra, then A 0 is a coalgebra, with co- multiplication A = m* and counit e u*. If A is commutative, then A 0 is cocommutative. Explicity, A/( a ® b) = m*/( a ® &) = /(a6) , for al l f e A°,a,b£ A. We wil l prov e 1.2. 4 i n Propositio n 9.1.2 . Som e additiona l characteriza - tions of A 0 wil l also be discussed in Chapter 9 . I n particular A 0 is the larges t subspace V o f A* such tha t m*(V) C V ® V. §1.3 Bialgebra s Now w e combine th e notion s o f algebr a an d coalgebra . 1.3.1 DEFINITION . A k-spac e B i s a bialgebra if (fl,m,u ) i s a n algebra , (J5, A,£) i s a coalgebra , an d eithe r o f th e followin g (equivalent ) condition s
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