der Universitat Munchen Set 11
Prof. Dr. B. Pareigis
Problem set for
Quantum Groups and Noncommutative Geometry
(41) Let (M
;ev) be a left dual for M. Show that there is a unique morphism
db:I !M M
satisfyingthe conditions
(M db1
! M M
M 1ev
!M)=1
M
(M
1db
!M
M M
ev1
! M
)=1
M :
(Uniquenessof the dual basis.)
(42) LetBbethebialgebraKhx;yi=I,whereIisgeneratedbyx 2
;xy+yx,withthe
diagonal(y)=yy,(x)=x1+yx,andthecounit"(y)=1;"(x)=0.
A chain complex has the form
M =(:::
@3
!M
2
@2
!M
1
@1
!M
0 )
with @
n 1
@
n
=0.
Show that the category K-Comp of chain complexes is equivalent to the
categoryB-Comod of B-comodules.
Use the following construction. If M is a chain complex then dene a B-
comodule onM =
i2N M
i
withthe structure map Æ :M !BM, Æ(m):=
y i
m+xy i 1
@
i
(m) for allm 2M
i
and for alli2 N resp. Æ(m):=1m
for m 2 M
0
. Conversely if M;Æ : M ! B M is a B-comodule then we
deneK-modules M
i
:=fm2Mj9m 0
2M[Æ(m)=y i
m+xy i 1
m 0
]g and
K-linear maps@
i :M
i
!M
i 1 by@
i
(m):=m 0
forÆ(m)=y i
m+xy i 1
m 0
.
Check that this denes an equivalenceof categories.
(Hint: Let m 2 M 2 B-Comod. Since y i
;xy i
form a basis of B we have
Æ(m)= P
i y
i
m
i +
P
i xy
i
m 0
i
. We apply tothis the equation (1Æ)Æ=
(1)Æ and compare coeÆcients to get
Æ(m
i )=y
i
m
i +xy
i 1
m 0
i 1
; Æ(m 0
i )=y
i
m 0
i
foralli2N
0
(with m 0
1
=0). Consequently foreachm
i 2M
i
thereisexactly
one @(m
i )=m
0
i 1
2M such that
Æ(m )=y i
m +xy i 1
@(m):
Since Æ(m 0
i 1 ) = y
i 1
m 0
i 1
for all i 2 N we see that @(m
i ) 2 M
i 1
. So we
have dened @ : M
i
! M
i 1
. Furthermore we see from this equation that
@ 2
(m
i
) = 0 for all i 2 N. Hence we have obtained a chain complex from
(M;Æ).
If we apply (1)Æ(m) =m then we get m = P
m
i
with m
i 2 M
i
hence
M = L
i2N M
i
. This together with the inverse construction leads to the
requiredequivalence.)
(43) FindanexampleofanobjectM inamonoidalcategoryC thathas aleftdual
but noright dual.
(44) (a) In the category of N-graded vector spaces determine all objects M that
have aleft dual.
(b) InthecategoryofchaincomplexesK-Comp determineallobjectsM that
have aleft dual.
(c) In thecategoryof cochain complexesK-Cocomp determineallobjectsM
that have a leftdual.
