Download EXAMPLE SHEET 3 1. Let A be a k-linear category, for a

EXAMPLE SHEET 3
TOPICS IN CATEGORY THEORY
PART III – LENT TERM 2016
1. Let A be a k-linear category, for a commutative ring k. If V is a k-module and X P A,
define the tensor product of X by V as an object V ˚ X P A with a k-linear isomorphism
ApV ˚ X, Y q – Homk pV, ApX, Y qq, natural in Y .
(a) Prove that k ˚ X – X, and that if A has finite colimits, then the tensor products
V ˚ X exist for all finitely presentable k-modules V .
(b) Deduce that tensoring with V extends to a k-linear functor pV ˚ ´q : A Ñ A.
(c) Exhibit canonical isomorphisms V ˚ pW ˚ Xq – pV b W q ˚ X.
2. (a) Suppose that V is a monoidal category and X P V has a left dual _ X with
evaluation e : _ X b X Ñ I. If n, n1 : I Ñ X b _ X is are coevaluations that make
p_ X, X, n, eq and p_ X, X, n1 , eq dual pairs, then n “ n1 .
(b) Let V be finite-dimensional a k-vector space, and denote by V ˚ the usual linear
˚
˚
dual of V . If te1 , . . . , en u is a basis
řnof V˚, denote ei P ˚V the linear functional that
˚
satisfies ei pej q “ δij . Prove that i“1 ei b ei P V b V is independent of the choice
of the basis of V .
3. Let k be a field and Mn pkq the algebra of n ˆ n matrices with entries in k, and denote
by OpMn pkqq be the free commutative algebra on the variables tXij : 1 ď i, j ď nu (ie
the plynomial algebra krXij : 1 ď i, j ď ns). (One may think of Xij as the function that
sends a matrix to is pi, jq coefficient.)
(a) Use the matrix multiplication to define a bialgebra structure on OpMn pkqq.
Let GLn pkq be the subalgebra of Mn pkq consisting of the invertible matrices, and OpGLn pkqq
the quotient of the polynomial algebra krY, Xij : 1 ď i, j ď ns by the ideal Y det “ 1
(ie OpGLn pkqq is the localization of OpMn pkqq at the element det). Here det is the
polynomial of the variables Xij that is the usual determinant.
(b) Show OpGLn pkqq has a bialgebra structure that makes the canonical algebra morphism OpMn pkqq Ñ OpGLn pkqq a morphism of bialgebras.
(c) Show that OpGLn pkqq is a Hopf algebra.
4. Let H be a bialgebra over a field k and S : H Ñ H be an anti-morphism of algebras.
(a) Show that for each left H-module M the formula
ÿ
px ¨ f qpmq “
x1 ¨ f pSpx2 q ¨ mq
x P H, f P Homk pM, kq, m P M
defines an H-module structure on Homk pM, kq.
(b) Show that the evaluation e : Homk pM, kq b M Ñ k and the coevaluation n : k Ñ
M b Homk pM, kq are morphisms of H-modules iff S is an antipode for H.
Send corrections and comments to [email protected].
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TOPICS IN CT – PART III 2016
5. Let C be a comonoid and B a bimonoid in a braided category V. Prove that if f P
VpC, Bq is a morphism of comonoids, then f is invertible with respect to the convolution
product. State and prove the dual statement.
6. Let H and K be two Hopf monoids in a braided category, with respective antipodes
sH and sK . Prove that, if f : H Ñ K is a morphism of bimonoids (= morphism of
monoids and of comonoids) then f ¨ sH “ sK ¨ f .
7. Prove that the forgetful functor Bialg Ñ Coalg has a left adjoint given by C ÞÑ T pCq.
8. Let V be a braided monoidal category. Show that the monoidal category MonpVq is
braided and its forgetful functor into V is braided iff the braiding of V is a symmetry.
Deduce a similar result for ComonpVq.
Extra exercises
The next series of exercises are a guided proof of the fact that the antipode of a Hopf
algebra need not be invertible (with respect to composition). You may solve them if you
have the time.
9. (Free Hopf algebras over coalgebras) All the vector spaces will be defined over a field
k. Let C be a coalgebra, and define inductively coalgebras
8
ÿ
cop
V0 “ C,
Vi`1 “ Ci ,
V “
Vi .
i“0
Let Ŝ : V Ñ V
cop
be the coalgebra map given by
px0 , x1 , x2 , . . . , xn , 0, 0 . . . q ÞÝÑ p0, x0 , x1 , x2 , . . . , xn , 0, 0 . . . q.
(a) Prove that Ŝ induces a bialgebra map S̃ : T pV q Ñ T pV qop .
(b) If I is the 2-sided ideal of T pV q generated by
Aÿ
E
ÿ
x1 S̃px2 q ´ εpxq1, S̃px1 qx2 ´ εpxq1 : x P V
show that HpCq – T pV q{I has a coalgebra structure that makes the quotient map
T pV q HpCq a bialgebra map.
(c) Show that S̃ descends to a bialgebra map S : HpCq Ñ HpCqcop,op that is an antipode
for HpCq.
(d) If ηC is the the composition C “ V0 V T pV q HpCq, prove that
BialgpHpCq, Hq ÝÑ CoalgpC, Hq
f ÞÑ f ¨ ηC
is a bijection, for any Hopf algebra H. [Hint: if f : C Ñ H is a coalgebra map,
consider morphisms f0 “ f : V0 Ñ H and fi`1 “ fi´1 P VpVi , Hq].
10. Let C be a coalgebra and A an algebra over a field k. Denote by LpC, Aq the set of
sequences pfi qiě0 of elements of Homk pC, Aq which satisfy fi`1 “ fi´1 in Homk pC, Aq if
i is even, and fi`1 “ fi´1 in Homk pC cop , Aq if i is odd. Prove that
AlgpHpCq, Aq ÝÑ LpC, Aq
f ÞÑ pf ¨ S j ¨ ηC qjě0
is a bijection.
11. Use the previous exercise to prove that the antipode of HpCq is a bijection iff for
any algebra A and pfi qiě0 P LpC, Aq, the map f0 is invertible in Homk pC cop , Aq.
EXAMPLE SHEET 3
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12. (a) If C is a finite-dimensional coalgebra, show that LpC, Aq is the set of sequences
pxi qiě0 in the algebra C ˚ b A which satisfy
xi`1 “ x´1
in C ˚ b A
i
if i even,
xi`1 “ x´1
in C ˚ b pAop q if i odd.
i
(b) If Mm pkq is the algebra of m ˆ m matrices with entries in k, then LpMm pkq˚ , Aq is
the set of sequences pXi qiě0 of elements of Mm pAq that satisfy
Xi`1 “ Xi´1
if i even,
t
Xi`1
“ pXit q´1 if i odd.
[Hint: use Mm pkq b A – Mm pAq.]
13. Suppose that k is a field of characteristic zero.
(a) Show that there is a non-zero
ř k-algebra A with elements y ‰ z P A that satisfy
yz ´ zy “ 1. [Eg, A “ Endp 8
i“´8 kei q and ypei q “ ei´1 , zpei q “ iei`1 .]
(b) Define scalars and matrices in M2 pAq by
1 ¨ 3 ¨ . . . p2n ´ 1q
2 ¨ 4 ¨ . . . 2n
2 ¨ 4 ¨ . . . 2n
a1 “ 1,
a2n`1 “
1 ¨ 3 ¨ . . . p2n ` 1q
ˆ
˙
ˆ
˙
1
y
2n ` 1 ` yz ´y
X2n “ a2n
and X2n`1 “ a2n`1
.
z 2n ` yz
´z
1
Show that pXi qiě0 defines an element of LpM2 pkq˚ , Aq, where M2 pkq˚ is the coalgebra dual to the finite dimensional algebra M2 pkq.
(c) Deduce that the antipode of HpM2 pkqq is not invertible.
a0 “ 1,
a2n “