1. THE CALCULUS OF SWEEDLER POWERS 7 diagram is commutative: J-ra X l n X lp pmjnxid -Lmn X lp Jmn,p (p(k)\*(j),T(i)) (p(k),(a-r)(l)) p-(cr-r) lp X ln X i m ' lp X -'ran id X^n,™ Pp,rnn In addition, the following diagram is also commutative: id X(pn,p T T Pm,np J-m X -*n X -*p -*ra X J-wp L mnp U j M H_). (p(k)\a(j),r(i)) ((p.a)(0,r(i)) (p-j)- -*P X % X ^77 (pP)Tlxid ~ ^ -*np X i m Vnp, l mnp Since the horizontal compositions in both diagrams are equal by Lemma 1.1, the associative law follows. For the discussion of the unit element, note that the diagrams Im X h " W n h X In V l , n " W n ( M ) ~ (l,r(i)) W ) , i ) h Xln -^ In In X h " W n commute, since (pn,i(h 1) = fc = £in(l,-i). This shows that id^- r = r and a • idi1 — a. • 1.3. Suppose now that if is a bialgebra with coproduct A and counit e. We define certain powers that will play an important role in the sequel: DEFINITION. For G e Sn and h e i7, we define the cr-th Sweedler power of h to be h° :=h (*(!)) • V(2)) • / i (a(n)) Here, we have used a variant of the so-called Sweedler notation (cf. [49], Sec. 1.2, p. 10) to denote the images under the iterated comultiplication A n : H — H®n as An(/i) = h{l) g h{2) (8)... (8) h(n) If n is the degree of G, we shall also say that ha is an n-th Sweedler power of h. This definition of Sweedler powers deviates slightly from the one in [23], [24], where the permutation G is always the identity. Of course, this permutation is not relevant if H is commutative or cocommutative, which is the case in the theory of algebraic groups, the setting in which Sweedler powers were first considered

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