2
1. DEFINITION S AN D EXAMPLE S
a) coassociativit y
c *
b) couni t
c®c
c®c
id® A
A ® id
- c®c®c
c
Cgk
'id
g )
e
c®c
The tw o uppe r map s i n 1.1.3.b) are given b y c *- 1 ® C and c H- 1, for
any c C. W e say C is cocommutative i f r o A = A .
Note tha t 1.1. 3 b) gives tha t A is injective, jus t a s 1.1.1 b) gives tha t m
is surjective .
1.1.4 DEFINITION . Le t C and D b e coalgebras, wit h comultiplication s A c
and A# , and counits sc an d £#, respectively.
a) A map / : C D is a coalgebra morphism i f A/) o / = ( / ® / ) Ac
and i f £? = ^ D ° /•
b) A subspace / C C is a coirfea / i f A / C / ® C + C ® 7 and if e(I) = 0.
It is easy t o check that i f / i s a coideal, then th e k-space C/I i s a coalgebr a
with comultiplicatio n induce d fro m A , and conversely .
Finally, w e may also us e the twist ma p to dualize th e notion o f opposit e
algebra. Fo r a give n algebr a A, recal l tha t A op i s the algebra obtaine d b y
using A a s a vecto r space , bu t wit h ne w multiplication a
0
= (6a)° , for
a0,6°
G A
op.
I n term s o f map s thi s ne w multiplication i s give n b y m ' :
A —* A, wher e ra' = rn o r.
1.1.5 DEFINITION . Le t C be a coalgebra. The n th e coopposite coalgebra C
cop
is given a s follows: C
cop
= C as a vector space , wit h ne w comultiplicatio n A '
given b y A' = r o A.
It i s easy t o see that C cop i s also a coalgebra .
§1.2. Dual s o f algebra s an d coalgebra s
We shal l no w see that ther e i s a ver y clos e relationshi p betwee n algebra s
and coalgebras , b y looking a t thei r dua l spaces .
For an y k-space V , let V* = Homk(V , k) denot e th e linear dua l o f V.
Previous Page Next Page