4 1. DEFINITION S AN D EXAMPLE S holds: 1) A an d e ar e algebr a morphism s 2) m an d u ar e coalgebr a morphisms . As expected , a ma p / : B * B/ o f bialgebra s i s calle d a bialgebra mor- phism i f i t i s bot h a n algebr a an d a coalgebr a morphism , an d a subspac e / C B i s a biideal i f i t i s bot h a n idea l an d a coideal . Th e quotien t B/I i s a bialgebra precisel y whe n / i s a biidea l o f B. 1.3.2 EXAMPLE . Le t G b e an y grou p an d le t B = kG b e it s grou p algebra . Then B i s a bialgebr a vi a Ag = g ® g an d e(g) = 1 , fo r al l g G. 1.3.3 EXAMPLE . Le t g b e an y k-Li e algebr a an d le t B = (7(g ) b e it s universal envelopin g algebra . The n B become s a bialgebr a b y definin g A x = x ® 1 + 1 ® x an d e{x) = 0 , fo r al l x G g . Note tha t example s 1.3. 2 an d 1.3. 3 ar e cocommutative . In an y coalgebra , element s whos e A i s a s i n 1.3. 2 o r 1.3. 3 ar e ver y impor - tant thu s w e giv e the m a name . 1.3.4 DEFINITION . Le t C b e an y coalgebra , an d le t ceC. a) c i s calle d group-like i f A c = c ® c an d i f e(c) = 1 . Th e se t o f group-lik e elements i n C i s denote d b y G(C). b) Fo r g y h G G(C),c i s called g, h-primitive i f A c = c®g+h®c. Th e se t o f all g, /i-primitive s i s denote d b y P 9y h(C). I f C B i s a bialgebr a an d g = h 1, the n th e element s o f P{B) = Pi i i(B) ar e simpl y calle d th e primitive element s o f B. It i s no t difficul t t o prov e tha t i n an y coalgebra , distinc t group-lik e ele - ments ar e k-independen t [S , 3.2.1], [A , 2.1.2]. A s a consequence , i f B = kG , then G(B) = G , th e origina l group . If B = f/(g ) an d cha r k = 0 , the n P(B) = g , th e origina l Li e algebra . However i f cha r k = p ^ 0 , the n P{B) i s th e spa n o f al l x p , f c 0 , x G g i t is a restricte d p-Li e algebra . Se e §5.5 . As anothe r exampl e o f group-lik e elements , le t A b e an y algebr a an d define (1.3.5) Alg(i4,k ) = { / e A* | / i s a n algebr a ma p } .
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