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L : Let k be any field. Let (A,M,L) be a k-algebra and let (B, A, ) be a k-coalgebra. Now let Hom(C, A), that is, L is the k-vector space of k-linear maps C + A. We define a multiplication * on L as follows. For T1, T2 E L, the product Ti*Tis the k-linear map defined by T1 *T(c) = T1(c)T2(c) for every c E C, where A(c) = ci C2. In other words, the * operation is defined by the composition c4cocTher, A A A A 2. Let (H, , 6, A, E, S) be a Hopf algebra. Set C := HH, so that (C,023 0 (AOA), 70 (EE)) is a coalgebra. (Here n : kok + k is the canonical isomorphism.) Also set A := H, so that (A, M, e) is an algebra. Finally, let L := Hom(C, A) be equipped with the * operation defined above. (a) Define TEL as follows: T1( ab):= S(ab) for all a, b e H. Prove that Ti *= 10. (b) Define T2 E L as follows: T2( ab) := S(6)S(a) for all a, b e H. Prove that ui* T2 = 10. (c) of H. Using (a) and (b) prove that T1 = T2. This proves that S is an anti-automorphism L : Let k be any field. Let (A,M,L) be a k-algebra and let (B, A, ) be a k-coalgebra. Now let Hom(C, A), that is, L is the k-vector space of k-linear maps C + A. We define a multiplication * on L as follows. For T1, T2 E L, the product Ti*Tis the k-linear map defined by T1 *T(c) = T1(c)T2(c) for every c E C, where A(c) = ci C2. In other words, the * operation is defined by the composition c4cocTher, A A A A 2. Let (H, , 6, A, E, S) be a Hopf algebra. Set C := HH, so that (C,023 0 (AOA), 70 (EE)) is a coalgebra. (Here n : kok + k is the canonical isomorphism.) Also set A := H, so that (A, M, e) is an algebra. Finally, let L := Hom(C, A) be equipped with the * operation defined above. (a) Define TEL as follows: T1( ab):= S(ab) for all a, b e H. Prove that Ti *= 10. (b) Define T2 E L as follows: T2( ab) := S(6)S(a) for all a, b e H. Prove that ui* T2 = 10. (c) of H. Using (a) and (b) prove that T1 = T2. This proves that S is an anti-automorphism

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