Chapter 1
Definitions an d Example s
§1.1 Algebra s an d coalgebra s
Throughout w e le t k b e a field, althoug h muc h o f wha t w e d o i s vali d
over an y commutativ e ring . Tenso r product s ar e assume d t o b e ove r k unles s
stated otherwise .
We first expres s th e associativ e an d uni t propertie s o f an algebr a vi a map s
so tha t w e ma y dualiz e them .
1.1.1 DEFINITION . A k-algebra (wit h unit ) i s a k-vecto r spac e A togethe r
with tw o k-linea r maps , multiplicatio n rn : A ® A A an d uni t u : k A,
such tha t th e followin g diagram s ar e commutative :
a) associativit y b ) uni t
A
u
(g id

A
m
® id
•A® A
id
® m
k®A

A
id® u
A®k
The tw o lowe r map s i n b ) ar e give n b y scala r multiplication . 1.1.1)b,
gives th e usua l identit y elemen t i n A b y settin g 1& = u(lk) -
1.1.2 DEFINITION . Fo r an y k-space s V an d W, th e twist map r : W -
W ® V i s give n b y T(V ®W) = W®V.
Note tha t A i s commutativ e = rn o r = m o n A.
We no w dualiz e th e notio n o f algebra .
1.1.3 DEFINITION . A k-coalgebra (wit h counit) i s a k-vecto r spac e C to -
gether wit h tw o k-linea r maps , comultiplication A : C C ® C an d counit
e : C k , suc h tha t th e followin g diagram s ar e commutative :
l
http://dx.doi.org/10.1090/cbms/082/01
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