1. DEFINITION S AN D EXAMPLE S 5 In 9.1. 4 w e wil l se e tha t Alg(A, k ) = G(A°), th e se t o f group-lik e element s i n the coalgebr a A°. We continu e wit h ou r example s o f bialgebras . 1.3.6 EXAMPLE . I f B i s any bialgebra , the n i s a bialgebra thi s i s prove d in §9.1 . I n particular , w e conside r th e specia l cas e whe n B kG. I n thi s case i s calle d th e se t o f representative functions R^(G) o n G. I t ca n als o be describe d a s follows : = R k (G) = { / (kG)* | dimk spa n {G /} oo} , where G act s o n (kG) * vi a (x f)(y) = f{yx), fo r al l x,y £ G , / £ (kG)* . The algebr a structur e o n (o r o n B*) i s give n b y (fg)(x) = A*( / ® /)(*) = ( / ® g)(x ® x) = f{x)g(x), all £ £ G , /, # 5* tha t is , i t i s th e usua l pointwis e multiplication . Th e coalgebra structur e i s given , a s fo r an y bialgebr a B, b y Af(x g ) y) = m*f(x ® y) = /(xj/) , all x,y £ B,f B°. However , thi s doe s no t giv e a n explici t formul a fo r A / as a n elemen t o f g ) 5 °. Whe n 5 i s finite-dimensional , tha t i s \G\ oo , we ca n giv e suc h a description , a s follows : Let {p x | x 6 G } b e a basi s o f (kG) * dua l t o th e basi s o f grou p element s in kG tha t i s p x (y) = 8 x%y , al l a:, y £ G . The n (1.3.7) Ap r = ^ p u ®p v . uv~x 1.3.8 EXAMPLE . Le t B = 0(M n (k)) = k[X 0 -|l z, i n] , the polynomia l functions o n n x n matrices . A s a n algebra , B i s simpl y th e commutativ e polynomial rin g i n th e n z indeterminate s {Xij}. Fo r th e coalgebr a structure , think o f X^ a s th e coordinat e functio n o n th e ij th entr y o f th e rin g M n (k) of n x n matrices . The n A i s th e dua l o f matri x multiplication tha t i s AXij ^21=1 Xik ® Xkj. B y settin g e(X lJ ) = 8 l3 , B become s a bialgebra . If w e le t X = [X^], th e n x n matri x wit h ij th entr y Xij, the n on e ma y check tha t de t X £ G{B).
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