Cocommutative Hopf Algebras with Antipode by ... - Semantic Scholar
J-
Cocommutative Hopf Algebras with Antipode by Moss Eisenberg Sweedler
B.S.,
Massachusetts Institute of Technology
(1963) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY
August, 1965
Signature of Author .
.-.
. .
Department of Mathematics, August 31, 1965
Certified by ....-..
.
Thesis Supervisqr
Accepted by .......... 0................................... Chairman, Departmental Committee on Graduate Students v/I
2.
Cocommutative Hopf Algebras
19
with Antipode by Moss Eisenberg Sweedler Submitted to the Department of Mathematics on August 31, 1965, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Abstract In the first chapter the preliminaries of the theory of Hopf algebras are presented. The notion and properties of the antipode are developed. An important filtration is induced in the Hopf algebra by its dual when the Hopf algebra is split. It is shown conilpotence and an algebraically closed field insure a Hopf algebra is split. The monoid of grouplike elements is studied. In the second chapter conditions for an algebra A -which is a comodule for a Hopf algebra H --to be of the form A 'E B ® H (linear isomorphism) are given. The dual situation is studied. The graded Hopf algebra associated with a split Hopf algebra decomposes in the above manner. Chapter III contains the cohomology theory of a commutative algebra which is a module for a cocommutative Hopf algebra. There is extension theory and specialization to the situation the Hopf algebra is a group algebra. Chapter IV is dual to chapter III. Chapter V is devoted to coconnected cocommutative Hopf algebras, mostly in characteristic p > 0 . There, the notion of divided powers is developed and shown to characterize the coalgebra structure of a class of Hopf algebras. The Hopf algebras are shown to be extensions of certain sub Hopf algebras by their primitive elements. Thesis Supervisor: Title:
Bertram Kostant Professor of Mathematics
3.
Contents
Introduction
4
............................................
7
Chapter I.
Preliminaries .......................... ..
Chapter II.
Decompositions ......................... . 36
Chapter III.
Cohomology ....................----.
. 47
Chapter IV.
Cohomology ..........................
. 87
Chapter V.
Cocommutative Coconnected Hopf Algebras. . 93
Bibliography . ......................... Index ........ ......................... Biography ....
... *0..........................-.....-
...... 0...... 000............
162
166
I
Introduction
A Hopf algebra as considered herein is simultaneously an algebra and a coalgebra where the algebra structure morphisms are morphisms of the coalgebra structure, (or vice versa).
This differs from the graded Hopf algebras
of [2] Milnor and Moore, except in characteristic 2.
The
problem is to determine the structure of cocommutative Hopf algebras. Our first approach lies in a cohomology theory.
We
have constructed abelian cohomology groups Hi(HA) , where
H
"C.H.A."
H(CH)i
is a Hopf algebra, H-module,
H-comodule.
C
0 < i E Z
,
A
an algebra which is a left
a coalgebra which is a right "C.H.A."
We then determine the structures of algebras
(coalgebras) which are extensions of
H
(C) by
B
(H)
This theory applies to the algebra structure and the coalgebra structure--separately--of coconnected cocommutative Hopf algebras.
We hope to develop an extension theory
where the extension is a Hopf algebra and is an extension of one Hopf algebra by another. Our cohomology theory gives the familiar group cohomology in case G .
If
Ar
H =p-(G)
the group algebra of the group
is the group of regular (invertible) elements
5.
of
A
then Hi(H,A)
Furthermore, if underlying field
A
Hi(G,Ar)
=
is a finite Galois extension of the
k and
G
is the Galois group of
then the isomorphism classes of extensions of by
A
A/k
H =F(G)
form a subgroup of the Brauer Group.
Kostant has shown--the results are unpublished--that a split cocommutative Hopf algebra with antipode is a smash product of a group algebra and a coconnected cocommutative Hoof algebra; and that in characteristic zero a coconnected cocommutative Hopf algebra is a universal enveloping algebra of the Lie algebra of primitive elements.
We present
proofs of these results and study coconnected cocommutative Hopf algebras in characteristic- p > 0
.
For a cer-
tain class (including all where the restricted Lie algebra of primitive elements is finite dimensional) of coconnected cocommutative Hopf algebras we are able to determine the coalgebra structure.
The coalgebra structure is described
in a generalization of the Poincare-Birkhoff-Witt theorem. We now outline the generalization. In characteristic zero let for the Lie algebra algebra,
L ,
a Hopf algebra. I=
LC U
(x
be an ordered basis
its universal enveloping
If
i = 0,1,2,...
6. then dx(
=
z i=o
x2=0n
The Poincare-Birkhoff-Witt theorem is equivalent to:
a i)
x
-
a 0 , we show the Hopf algebra obtained
by factoring out the ideal generated by the primitive elements is isomorphic to a sub Hopf algebra of the original Hopf algebra, when the vector space structure on the quotient is altered.
We also show the original Hopf algebra
is an extension--as an algebra and coalgebra--of the quotient by the primitively generated sub Hopf algebra.
Chapter I
The study of Hopf algebras is a self-dual theory. For this reason diagram notation is useful, as it makes dual definitions and proofs evident. For all time we fix the field for all vector spaces. over
k ,
n
If
k
which is the base
X1 ,...,Xn
are vector spaces
the permutation group on
a e
n-letters
we consider
a:
Often
a
Xe
-
X
->
x1@
-0 -@
x0
->x.
will be written
i ,...1
X
(1 ,...
...
ex . (& x,
in)
where
1,2,...n) ; in this case X@--
111n
OX
'''
0y
... e
X
1 If
X1
X2 =
n ,
@Xn
.
n is a left G-
module. If
X, Y
are vector spaces
a linear map from
X
to
"f: X -Y"
means
f
is
Y
Once we define a "right" object such as module or comodule, we consider the "left" object to be defined with the mirror definition.
Similarly for "left" objects.
8.
Algebras (A,m,-q) space over
k
is
an algebra (over k) where
,
m: A @ A -+ A
T:
A
is a vector
k -+ A , if
the follow-
ing diagrams are commutative.
Igm A®A A
3 A @A
-
1S
m®I
I) AA
II) -m
A A (1k
A AA
is equivalent to associativity.
II) k
) A
A®1@A
A
m
m
I)
Ti@I k(DA~-
is equivalent to
TI(1),
a
is an algebra where
(rI)
is a unit.
is the identity,
m
is the usual
multiplication. An algebra
A
AO A
is commutative if
m
(2,1)
A
is commutative.
Ae A If then
A
is an algebra, -+
A
x
-+
l@ f(x)
If where
A
and
B
vector spaces
f: X ->Y
1 E A
always denotes
'QA(l)
Y
kD @f: X
where for an algebra
X, Y
A ,
are algebras
A& B
is
an algebra
9.
(A
B) q
(A @ B)
(>
A
A
BOB
m A mB mA® B and
nA e B = k @ 11B f: A ->B
A(
B
AD k .
is a morphism of algebras if the following
diagrams are commutative:
A
A
A
ff
ff
f
k B
B
B
4
B -$A
k
A
If space
A M
will often be written
is an algebra a left
with a map
?P: A®D M -. *M
k
k
A
A-module is a vector satisfying:
k@I A (A
M
M
)A
M
> A (M
M A0 M
If
M, N
phism of left
'f M
are left
A-modules
A-modules if
f: M -+ N
is a mor-
I 10.
M
M-)
A
is commutative .
jf SI@ f N > N A@N & An augmentation eA : A -+ k
morphism
.
A
of an algebra
EA
is an algebra
An augmented algebra is
an algebra
with a fixed augmentation. If
A
right)
is ar augmented algebra
k
is
a left (or
A-module by: EI
N @A M
m 3 k
k
k
-+
N
A-module and
is a left
M
If
I
-
A0 k
a right
A-module
is a vector space such that *N
NO A(
I
M
-
(9 ?M ) NO M
-
N (AM
_+ 0
is an exact sequence of vector spaces. If left
is an augmented algebra,
A
A-module, then
A+ = Ker e A,
k ®AM = Coker (A+O
) A0
M
i@I or if
A+ - M denotes
Im(A+
) A
M
M
a M) , ?P
M - M)
i®I k
A M = M/(A+ - M)
Coalgebras (C,d,E) space over
is a coalgebra over
k ,
k
where
C
is a vector
11.
d: C
-+
k
s: C -
CtC
if the following diagrams are commutative: k
d
C
I
II)
II)
d C SC 0C
I
C (k
I)
C
d
dO I
d C®C
C
C O----C
C I)
C '
C@ C
E
is equivalent to coassociativity E
is equivalent to k
is an augmentation of a coalgebra. d = k® I
is a coalgebra where
and
e = the
identity.
A coalgebra is cocommutative if C@ C C
(2,1) C OC If
C
is commutative.
d X, Y
is a coalgebra and
are vector spaces
f: X - Y , then E
C D X If
C, D
f:
c
C®X
k
Y
are coalgebras
-
Y
® C @) D
aYc is a coalgebra where
12.
dC(
dD
>C(C(DeD
CQD
2D 4) (1,3,
dC (
D (CO D) 0
f: C -* D
C
eD
'C
ECOD
(CO D)
is a morphism of coalgebras if the diagrams,
CO C
C f
C
f
dD D'-
D
k
f
f
D
D
6D
are commutative. If space
C M
is a coalgebra, a right $: M -. M @ C
with a map
C-comodule is a vector satisfying: M
M
$C MO C
>MP C %
I
>MD C
()®d C I (
If
M, N
are right
morphism of right
C-comodules
C-comodules if
E
f: M -* N
is a
13.
M-)
m® C
If
N
,
fI
N
is commutative.
C
A unit of a coalgebra
C
is a coalgebra morphism
k - C. A coalgebra with unit is a coalgebra with a fixed unit. If left)
C
is a coalgebra with unit
If
O
is a right (or
C-comodule where k
M DN
k
N
N
-%
d
kk
is a left
I
TIC
C-comodule
M a right
C-comodule,
is a vector space such that
-+ M 0
N
-+ M(DN
)
M
A
N
is an exact sequence of vector spaces. If 7: C -+ J
C
is a coalgebra with unit then
M D
k = Ker (M
-
JC = Coker M
C
iC
'
) M®J
)
We shall freely use facts such as kernels and cokernels of comodule morphisms are subcomodules or quotient comodules, when the "dual " fact is well known. If
X, Y
are vector spaces
Hom(X,Y) = ff: X -+Y.
14.
Hom(X,Y)
is a vector space.
Often
Hom(X,Y)
additional structure, for example if and
A
an algebra then
Hom(C,A)
C
carries
is a coalgebra
has a natural algebra
structure as follows Hom(C,A)
C,A)
) Hom(C D
Hom(C,A)
®
a
Hom(CA))
where
Hom(C,A)
g) = mA o f
a(f
g
d: C -+ C & C .
Multiplication in
denoted by
thus
in
*
Hom(C,A)
f*
-+
.
Hom(C,A)
X*
g = mA o f(& g o d .
(T)A 0
-C
'
X
is a coalgebra
-c. (X
0
X)*
a as above. a is injective for if n SAxi* y * s @ ,We can assume i=1 linearly independent. .
X*
Consider X*@0 X*
> =
The unit
Hom(Xk) , as usual we consider
denote
We have just shown if
is an algebra.
= (dim X) 2 = dim(X® X)*
is an isomorphism.
a
case
Xj0
is injective.
a
which shows
Y,*) i*
induce transpose mappings:
A* -+ k* = k , d = tim: A* -+ (A
a finite dimensional coalgebra
A*
A)*
= A*@
A*
A
is
Similarly if
is a coalgebra.
A*
With these maps
@
is an algebra; however,
this is the same algebra structure as deduced above. If
X, Y
are vector spaces -+
a: X* ® Y*
Y)*
(x
x E X, y E Y
=
<x*,x> x* E X*, y* E Y*
a
is injective.
same proof works.) a
in
(X @ Y)*
coalgebra then
(We proved this, above, for Identify If
M
X*
@
is a left
Y*
Y = X , the
with its image under C-comodule,
C
a
16.
C*@ M* ,
) (C@ M)*
IPM*
defines left
4'M* ,
and this gives
C*-module.
algebra,
M*
Similarly if,
M a left
A
the structure of a is a finite dimensional
A-module by
AQ M then
t M
-M
tPM: M* -+ (A
M
M)*
comodule structure on
A*Q M*
defines a left
M*
Hopf Algebras
(H,m,i3,d,E)
is
1)
(H,m,TI)
is an algebra with augmentation
2)
(H,d,e)
is a coalgebra with unit
3)
the diagram
H
a Hopf algebra when
m -
H
d H
)H
d)d HOH
is commutative.
L
H
Tj
H mDm
H
(1,3,2,4)1 Hf
HO H@ H
e ,
17.
3)
and
"
is equivalent to
3)
and
(H,m,Tj)"
"E
is a unit for the coalgebra "d:H -*H ® H
(H,d,E)"
is an algebra morphism".
is an augmentation for the algebra
is equivalent to
"m: H
DH
-+ H
is a coalgebra
morphism".
Since a Hopf algebra Hom(H,H)
H
is a coalgebra and algebra
has the structure of an algebra.
algebra structure and if
H
H* has an
is finite dimensional
a coalgebra structure, as well.
In this case
H*
has
H* is a
Hopf algebra. In
Hom(H,H)
consider
T
o s
k = k - 1 C H
is the unit.
so
e
We shall often
will denote the unit in
Hom(H,H) .
The identity
I: H -+H
Hom(H,H) .
S E Hom(H,H)
is an antipode for
I * S = E = S * I , that is
S
is an element of H
if
is the inverse to
I.
Example:
Let
G
be a monoid (group without inverses).
the "monoid" algebra =
n
7(G)
L
Q) k - g g E G
m
n,m
j=1
i,j=l
is an algebra where
where
F(G)
=
18.
-ri: k
F(G)
a Hopf algebra where
is
F(G)
d:
-+
g s: F(G)
'(G) @ F(G)
-+
g@g
-
k
+1
g
r(G)
(G)
-+
has an antipode if G
Suppose
. and only if
G
S: F(G)
is a group define
is
F(G)
-
-1
g then
S
is an antipode.
If
S
a group.
is an antipode,
n e = 1 = I - S(g) S(g) = g~
implies
gS(g)
=
,
,
hence
We now show that if
H
X g1 which i=l has inverses and is a group.
where G
S(g)
d o S = (2,1) o S @ S o d
2)
S o m = m o S @ S o (2,1)
3)
E o S = S o
4)
If
=s
is commutative (as an algebra) or co-
commutative (as a coalgebra) then We prove
1)
inverse to
52
=
S o S
=
as follows:
(2,1) o S 0 S o d e Hom(H, H@ H)
which is
d e Hom(H, HO H)
is a left
d o S , an algebra.
S , then
has an antipode,
1)
H
=
d
and
we show
(2,1) o S 0 S o d
d o S
is a right inverse
19.
to
d
This proves they are equal.
.
following: f: C -+ D
If
C, D
are coalgebras,
a coalgebra morphism,
phism and (h o g
We shall use the
g1 , g 2 E Hom(D,A)
A, B
h: A -+ B
are algebras, an algebra mor-
then
o f) * (h o g 2 o f)
= h o (g1 * g 2 ) 0 f
The proof is clear from the definitions. Consider
(d o S)* d = d o(S * I)= d o e = k (d o e = d We let
because
m's
.
d(l) = 1 0 1 ) .
mn =mo m@lo---om@
by associativity
mn
e E
O ...
T
n -1 is independent of the order of the
Similarly we let
dn = d
T
o---o o d
T o d
n -1 dn
is independent of the order of the
tivity.
If
h e H
Z hi'@ h "
d's
by coassocia-
we often let - --
hi(n+l) = dnh
Then
)
d * [(2,1) o S Q S o d](h) = Z h 'S(h
hi "S(h '") I*s
=Z h
'S (h
'")
@
E)
=
20.
Z hi'S(hi")@ 1 = E(h) i
1
I * S= which verifies For
2)
1) . we show
m o S @ S o (2,1)
S o m
is left inverse to
m * [m a S® S o (2,1)](h&
Z h
For
and
4)
g) = Zh
") =
E(g)S(h
= e
E = I *
For
Then
3)
a o I * S
=
'g'
H
m a S ® S
=
S(g")S(h ")
(e o I) * (E a
o E = (I o E) * (S o e)
suppose
E .
E(g)hi' S(h") =
=
S)
(g)E(h) S o 5
=
*E
=
o S
* S o E = S
a
is commutative so m o S ( S o (2,1)
=
S o m
S * S2 (h) = Z S(h i)S2(h"') = z S(hi' S(hi"))
i
i =
which shows H
and
is right inverse.
(S o m)* M = (S * I) a m = E O m = 6
=
m
S2
is
is cocommutative
right inverse to
S
®S
o d
S * 32 (h) = Z S(hi')S 2 (hi") i = m o I
m o I®
S (Z [S(h)]
j
S ,
so
(2,1) o S
= =
S o e(h) = e(h) S2
®S
.
If
o d = d o S
S(h ')@ S(hi")
S( i
[S(h)]J")
.
21.
I * S o S(h) = s o S(h) = s(h) which shows
S2
is right inverse to
H
We have
of
H*
on
as an
S
H*-module
H** ; under this action
H
is a submodule.
We then concern ourselves with the action of H**
a left
h**= h e H,
Then if
1
H** D H , we define a left action
H , H*
Consider
S2
hence
on
H
H*-module by
dh =Z hi'
hi"
h =Z h ' e H C H**
b*.
i
-
This shows
H
the action of module.
is a submodule of H*
on
H
.
H**
We consider
(We could have begun with
=
Again
H*H*H
would be a submodule where h - a* = Z 2) 1=1 where g E G are distinct ki . By the minimality of
n
a* E H*
g 2 'g3 ' 'n where
are linearly independent.
= 1
,
= 0
Choose i=3,.. .n
26.
n Then
X1 -
-
iA =-2
1=2
n
n
0 = a* - 0 = a* -(
%g) X
g
=
1=1i=
= -A2 g1 + XA 2 92 = This implies
X2 = 0
minimality of
n
X2 (g2
g1 = g2
or
d(gh) = (dg)(dh) = (g ) g)(h
which contradicts the
g, h e G
Observe if
.
g1 )
-
QDh)
= gh(® gh
Thus
.
G
a monoid and since it is a linearly independent set in the space spanned by
G
T(G)
as an algebra.
,
is
H
We consider
T(G) C H . If
H
dS(g) = (2,1) o S 0 S o dg gS(g) so
S(g)
S , g e G
has an antipode
=
I
=
S(g)(D S(g)
*
S(g) = E(g) = 1
is the inverse to
g
in
Filtration of
G
H
is split.
so that
and
G
S(g) E G
; is a group.
H
We exhibit a natural filtration on exhaustive when
then
H
which is
The multiplication and dia-
gonal map respect the filtration as does an antipode when present.
The filtration leads to a graded Hopf algebra.
It is also an invaluable basis for induction. Within
H*
let
J = F(G)
.
It is a straight
forward verification that for any subcoalgebra
C C H
27.
Thus
C4
C C H @H)
(do CC® J
is a
We assume
il= (J1+1L
1)
H = UHi
2)
H H
3)
dH
let
c
Z
D: then
.
.
J i+l
a**b* E J +
b*,
x>
b*
x = 0
let
x
x> = 0
= 0
;
and we are done. H* 0
H*
O+ z E H 0 H
is
"dense " in
there is
In fact there is + 0 prcve
(H
z* E H*
a*@ b* E H* .
H)*
in
H*
that if
where
$ 0 where
Thus to
d(a*
.
h) = Z h i'
a* - h "
we need
28.
only show
d(a* - h)> = B
B T1
The left diagram means
B -+ B
m: B
morphism; the right diagram means H-module morphism.
B
H
A
Regn (C,A)
Regn(C,A) B
0 < n E Z
For a coalgebra
C n n n Hom (C,A) = Hom(C ® '-.-C,A)
B -.B
Regn (H,B)
Homo(C,A) = Hom(k,A) = A
Now we assume
@
let
,
H-module
regular elements of
s: H
is a commutative algebra
is a multiplicative subgroup of
We let
is a left
regular (invertible) elements of
=
is a left C.H.A.
group.
B
a cocommutative Hopf algebra.
and algebra Let
k -+ B
I:
H-module
is called commutative Hopf algebra
H-module (C.H.A. H-module) if and
is a left
A B
is a left C.H.A.
.
If
is an abelian and
Rego(C,A) =
H-module where
Regn(H,B)
Define
,n: Regn(HB)
Homn(C, A)
= Ar
is the module action.
is an abelian group.
Homn(C,A).
-+ Homn+1(HB)
0 < n E Z
49.
n
f(-)
on7
o (I)
---
I 0I -.m
)
~1=1 i - 1
n - i
(-1)n+l
where
n > 0 ; for
f E Regn(HB) h
b E Br
H
E
We shall let
6 0 b(h) = (h-b)b~
n = 0
F[a]
denote
a F
Q)
---
unit.
In
Q)F
n > 0 ,
Regn(H,B)
e .
This is often denoted
and given
to
morphism.
6 01 (h) = (h - 1)1
h - 1 = e(h)
H.A.
Regn+1(H,B)
6n
6n so
E(h)
is a group homo601 = E .
is one of the conditions for
B
Note
to be an
H-module, we now use the other:
6 0 (bc) (h) = (h - (bc))c~Ab~A =
Z (hi 'I - b)b~
(hi
=Z (hi i
For the case ,n(E)
- c)(c
1b 1
i
= [6 0 (b) * 60(c)](h)
.
n > 0 n T
?P 0
E ()i
n E *
E *
i=1
k
as a map from
and that =
on 6 n() = 6 n(f) * ,n(g)
* g)
These imply that we can consider Regn(H, B)
the
We shall show
6 n (f
f,g e Regn(H,B)
is
E[n]
E =
E
o
[i
[n-i
E
)
50.
Note
o I
&
e = e
because
-
and
(f
(1)~
* g
o
bc(-1)us [i-] 1 because
I
is an H.A.
H-module.
?P o I()(f * g)= (7P o I
This also guarantees (f * g)(~l)
B
g(-l)i
*
m
I
is clear
~ (f * g)(l)
and thus
6n(f *
g) =
coogb (-)
[1-1 p
=
*
(f(l)n+ll
I n- ]
m
is a coalgebra morphism.
E
6 n(f)
I[n-i]
[n-i] I
mi1 M0in-il I) 1
It
(-1)n+1
n(g)
We have the sequence of abelian groups and homomorphisms 0 Reg (H,1B)
60 ) Reg (H,B)
1
2
-' Reg (H,B) -
62
)
We shall now show 6 n+1 6 n
= E: Regn (H,B) -+Regn± (HB)
We do this by expanding -0
* term *
---
41 *
term *
6 n+1 6 nf
.
In our notation
expands to
42 term * term *
The explanations of the expansions follow the computation and a typical note is of the form: "texplanation".
g)
because the group is abelian
m
I
f) * (p o I
4 -+ 41 * 42 * 43:
51.
6 n+1 6 n
=
(where
n > 0)
1
o 1O(p
0
f-
r
1 @f
0(1[i-
®m
[1-i])
* f (_)
*
2
/"--, n+1
-----------------
i
n (
o I e f
*
o(I[i-e
f
m
1=1
j=1
I[n-l)
(-1) J f (-1)n+1
0
( 1 i]
0
m
0 1 [n+1-j])
3
"K * (*
o I
rf
0([1-110 Me
f 1=1
(-+1) _n2
*
11
=*0 I
1 n-i3)
12
(* o
) f)
o (11-ilomer i=1
13
*
4
I
f
-
))
52.
21
n+-1 j1:1
'i)
(*, oIOf(i
o -1 (I 0 M®@ I[n-1j])
22
-------------[1-]
*(-1)
[ -1
o (
1[J1]l(Dm®
I[n+1-j
*
I [n+l-i])
1D
H
PD=D-)D D
D
kE.
D
The diagram: P
P
I
E
T -- >D
B
IPD
T
)D DH
It can be rewritten
is commutative.
) D
T
m oP
k
P
PD H k.
JD
B
TID
H Since
Im P
Im TD = D
are left
DH k ,
D H~~
Clearly
EH k.
B-module morphisms
is a factoring
Q
m oP @ P
is surjective, it follows
I C Ker m o P (FP, IT C Ker m o P
,
T
Thus
T -+ D
D
@
since P ,
P,
P
or there
86.
(R@k B
)
-
T
D
)
B
is commutative. have shown
The horizontal composite is is
TrD
injective. 11D
B
VD
hence we
Thus
$D
D
D
-
)H
is a split exact sequence. The second condition for extensions is satisfied. Thus
D If
is an extension of
y: A -+ A, ,
H
by
y: A -+ A1
B
are equivalences of exten-
sions then: ,y() Y: A®() A
y y@ ~y|T
factors to
of extensions.
~yIT: T
A -T
yD: D-+ D
and
yD
is
an equivalence
Thus the method of producting extensions
gives the isomorphism classes of extensions a product struc-
f ture.
Direct calculation shows the product of
f
f B
H
is extension isomorphic to
B
B
H ,
f' H ,
so that
under the correspondence of theorem 3, the multiplicative structures correspond.
This shows the isomorphism classes
form a group and the correspondence is a group isomorphism.
pop--
87.
Chapter IV Cohomology
We treat the dual situation to the previous chapter. We outline the definition of abelian cohomology groups H(BH)i
i=0,1,2... , when
algebra,
B
H
is a commutative Hopf
a cocommutative coalgebra which is a right
H-comodule satisfying certain conditions.
We also outline
the dual extension theory, defining when a. coalgebra is an extension of
B
by
H
and when extensions are isomorphic.
The isomorphism classes of extension again correspond to H(O2 , we define the dual extension to B f) H of the H(B,H) previous chapter; this indicates the correspondence beH(B,H) 2
tween B
is
is
a right
and the isomorphism classes of extensions.
42.
and
d: B
-BO)B
s: B
->k , B
5: B B ®B
H.A.
is a right C.H.A.
H-comodule,
is a right H.A. H
B .
is a right
B
are comodule morphisms. B
a Hopf algebra
H-comodule structure for
H-comodule under
if
H
a coalgebra,
B
-* B(
H
is an
H-comodule if
H-comodule
is cocommutative
is commutative.
For a coalgebra
C
and an algebra
0 < n e Z
let
88.
Hom(C,A)n = Hom(C,A En)
Then C.H.A.
f
Reg(B,H)n
B
Let
B
B
is a
Reg(B,H)O = Reg(B,k).
be a right C.H.A.
Reg(BH)
E
Reg(C,A)n = (Hom(C,A)n)r
is an abelian group when
H-comodule.
Let
Let
.
H-comodule.
For
consider
4k@H= >B@H
H.
Define 60
Reg(B,H)
=
6 0f we are considering
-+ Reg(B,H) 1
f: B
-+
f * fI
k C H .
'I When
o
),
n > 0
define
6 n: Reg(B,H)n -+ Reg(B,H)n+l
6nf = k O f *
o fn+1 i *
I
d
I i=1
Io Then
6n+1an = E,
6
b
.
is a group homomorphism so we have
a complex 61
6n Reg(B,H) 0
Reg(B,H)
) -
and define the cohomology groups H(B,H)n
=
Ker 6 n/Im
H (B, H)O
=
Ker 6 0
6
n-1
n > 1
89.
As in the previous theory we have a normal complex: n > 0
= ff 6 Reg(B,H )n
Reg(B,H)
B
[n]
-
k
is commutative
Sn+ =
n|iReg(B,H)+n: Reg(B,H)+n -*Reg(B,H)+n+1
Reg(B,H)+ 0
Reg(B,H) 0
=
60+ = 60: Reg(B,H)+ 0
-+ Reg(B,H)+ 1
The cohomology computed by this complex is the same.
Extension Theory
H
is a Hopf algebra
A, B
H-module (as a coalgebra), structure.
called
A
a left
*: H(@ A -+ A , being the module
The sequence H(
is
coalgebras,
A
) A -- -B
,
right exact if
1)
r
2)
Coim r = k OH A
is a surjective coalgebra morphism
Ker Tr = H
i.e.
A
The sequence is called split exact when:
H
has an antipode,
90.
the sequence is right exact, and there is a regular left H-module morphism d A
)A
A
P: A -+ H
P g r -
H(
comodules and left give A
B
B
In this case
.
is an isomorphism of right
H-modules by theorem 2.
the structure of a right C.H.A.
is called an extension of
4
r
A
-+ B
B
by
H
H-comodule, then when:
HD @A
2)
the following diagram is commutative, d
A->
is a split exact sequence,
I@7r A@A
A@B
-
--4
d
A
$: B -+B
Let
1)
-+
B-
A
B
B
H
H
where the
A*
A ir~I Given two extensions
A, A'
of
B
by
split exact sequences are
4 H@ A
HwA'
-+
ri
Al -+
B
A'a
B ,
we say they are isomorphic if there is a coalgebra morphism 'y: A -+ A' ,
such that,
H
'1 91.
H@ 10
A
A
-
B
y
Y H@
)A'
Being isomorphic is an equivalence rela-
is commutative.
tion and we can form isomorphism classes of extensions. An example: Given
g e Reg(B,HH)
2
a
2-cocycle, form
H
®B
which
g H®@ B
is
H-module.
as a left
I
E
7r: H
B BB
'
g I@ P: H
E
B
0 H
.
g d@ g) g B
H
B
)H
$
I
B
B
g
(1,3,5,2,4,6,7) H 0H
H
H
B
H
B
H(
B
H
H
H
B
'I
m2 (
m H
(H
defines the coalgebra structure on
H
@ g
H
H
B
g
B
-H
g
B
B .
( B) g
Then
(H
@ B) g
92.
is split exact, the second condition for extensions is satisfied so
H a B is an extension of B by H . Two g such extensions are isomorphic if and only if the 2-cocycles are homologous.
There is such an extension in each isomor-
phism class of extensions. tive correspondence between classes of extensions.
This yields the natural bijecH(B,H) 2
and the isomorphism
93.
Chapter V Cocommutative, Coconnected Hopf Algebras
We have shown a split conilpotent Hopf algebra with antipode is a smash product
E
B (D H
where
Hopf algebra acting as automorphisms of split Hopf algebra where
G(B) = 1j.
B
H .
is a group B
is a
We shall study such
Hopf algebras which-are cocommutative.
In characteristic
zero we prove such a Hopf algebra is a universal enveloping algebra--u.e.a.--of a Lie algebra, in characteristic p > 0
it contains a restricted u.e.a.--r.u.e.a.--of a res-
tricted Lie algebra.
(We assume familiarity with univer-
sal enveloping algebras and restricted universal enveloping algebras as can be found in
[2]
Lie Algebras.)
We define divided powers as sequences of elements
1
=
I 0 '2l''''
N
n
n < N
din = Z and show that in a i=0 class of cocommutative split Hopf algebras H , where where for
G(H) =
[1
the coalgebra structure of
H
is characterized
by divided powers. A Hopf algebra G(H) =1j. phisms of
Since k
to
H G(H)
H
,
is coconnected if it is split and corresponds to the coalgebra mor-
coconnectivity is equivalent to:
94.
being split, and morphism of
k
TI:
k -+ H
is the unique coalgebra
H .
to
all coconnected Hopf algebras have antipodes.
By
All Hopf Algebras Henceforth Are Assumed To Be Coconnected Unless Otherwise Specified. A primitive element
x
in a Hopf algebra
H
is one
where dx Then
x E H1
=
xEl
+
1@ x
e(x) = 0 .
and
Let
under
[
,
] ; if characteristic =
dxp
(binomially expand
+
1@ x
L(H)
H .
space of primitive elements in
(dx)P), so
.
L
denote the
(=L)
forms a Lie algebra
p > 0 x
L
1
forms a restricted Lie
algebra. Suppose
x E H dx E H dx
I * E = I
E * E = E
I
=
,
E(x) = 0 . @ H1
+ H
10y
+
H0 , z
HO =
so
l E(y) +
imply
k
z
= X
+
E(z)
= x
E (y) +
E(Z)
=
y
E(x)
=
0
95.
dx = 1QD (y + e(z)) + (z + E(y))Y
Thus
and
x
is primitive.
U
and
L C U .
U
H
x&l,
= L(q H 0 = L(0 k
denote the u.e.a. of
and the r.u.e.a. of
E(Z)
This shows
Ker efH 1= L Let
+
l®x
=
1 - 19
L
in
L
in
char p > 0 .
.
char p =0 We consider
is a (coconnected) Hopf algebra where the co-
algebra structure is induced by specifying the elements of L
are primitive.
Then
L(U) = L .
By the defining property of
U
there is an algebra
morphism : U -H which is induced by the identification of L C H .
y
L C U
with
is also a coalgebra morphism; hence, is a mor-
phism of Hopf algebras. That
y is injective follows from the next lemma.
Lemma 4I:) bra, if
V
v: A -+ C
,
A
a Hopf algebra
a coalgebra morphism.
vjL(A) Proof:
v
C a coalge-
is injective if and only
is injective. Suppose
a coalgebra morphism
v|L(A)
is injective.
Since
V
is
96.
EC(V(L(A))) = 0 thus
vA 1
VIAn
is injective
is injective.
injective.
C(V(l)) = 1
,
By induction we can assume
a e A n+
Suppose
v(a) = 0
@A
VI An
is
then
da=l0a+a®l +Y, Suppose
V
n > 1 ; hence,
YEAn
A
.
then
®
0 = dv(a) = v @ voda = v(1)
v(a) + v(a)
) v(1) + (v
@
v)(y)
= (v&@v)(Y) . Since
v
VIA |A
n
is injective it follows
a E L(A) ; which contradicts
VIL
is injective.
The converse is obvious. This implies
'y: U -+ H
Q.E.D. is injective.
with its image which is the subalgebra of L .
(Thus
U
Y = 0,
is the subalgebra of
H
We identify H
U
generated by
generated by
L
both as algebra and coalgebra.) We shall show if
where
J = k
C H*
H
is cocommutative and
char p = 0
U = H ,
char p > 0
U = (H*p(J))£
,
p(J) =
the ideal generated by
p(J)
a*P|a* s J] .
and
(H*p(J))
We do this by developing
is
97.
a technique which "picks-out" subspaces of associate with subspaces of the polynomial algebra on (Note:
H , subspaces of
fxaJ a basis for
work with
f
k[ f xa
k[ I xJ ] L(H)
.
Given a vector space 9-.module.
n > 0
SnX =
n = 0
S0 X
xa]
x
we have chosen to
,
]--which makes the basis clear--rather L .)
than the symmetric algebra on
Let
We
.
since the method of associating subspaces with sub-
spaces depends upon the basis
left
H
X
n
is a
x =®X
,
The symmetric tensors of degree Y E X[n]aj=
Y
Y
=
are:
J
aE
k .
be an ordered basis for k[
all
n
x
X ,
is a graded algebra.
]I
CO
k[
x a]
SX =
is isomorphic to
e
SnX
as follows:
n=0
a Let (i.e.
(x 1 )1
e 1 +--+
a Let Let
---
Y =
1el x
em
(tm) em =n,
=
-..
L
H
except
a
ES
This proves:
0
h E Hn - Hn-l
For
h e H
if
0 := E[n] o dn-lh E SnL .
let
- H
K(h) =P 0oE
Then we have
and only if
n]o dn-l(h)
Kn
H-J/yJVK h
-+ K(h)
which is not linear. morphism we have,
But by
and since
0
is an iso-
100.
if
h E H
, g E H
and
K(h) - K(g) = 0 h -g
For a subspace K(X)
let
the subspace of
K(H) I K .
In general
Xn = X
Recall for a subspace K(Xn) = K(X) n (Kn a
Lemma 5:)
- --
0 + K(h)
fh
C Xn ci
n - 1 .
? 1K(h 1 ) +---
then it is clear
K0 )
X, Y
We show by induction
then
=
($
1
n Hn
K(X) = K(Y)
n = 0 , suppose true for
7 1(p o
K
Given subspaces
X 0 =YO ; then
Proof:
0
spanned by
K
S X.
(K(x)|x
X C Y,
and
H n-1*
E
X C H =
g e Hn
then
of
H
where X = Y.
implies
X D Yn . Let
True for
h e Yn -n-l'
+ ?mK(hm) ,
where
A
E k ,
Thus [n] o d nl(h )) + +
oo E[n] o d n-(?h
which implies
g = N h
+
+ Am(P o E[n] o dn-l(hM + Nmhm)
-
+
mhm
E Xn - Xn-1
and
101.
K(g) x
K(h)
.
- g E Y n-
=h
=
h
Observe if
=
X, Y
By so that
C X
Q.E.D.
x + g e X .
are both subalgebras, subcoalgebras, XO =
ideals or coideals the condition
is automatically
satisfied.
0)
If
x e Hn
[
o dn-l(X)
p
Y
where
Hq (xa)
a
1
a E
]o d
f
. 0 ( x ' m)
1
(o
0)
emae-
2
2e
a
e 1 (x
m
2
1
] o dn
E
n - 1 , then if
---
B*
is a right
B-module where
= If
x E L(B)
then
x
b* e B* , a, b E B
acts as a derivation of
a basis of monomials in
, within
xi
basis (not necessarily a basis for basis for
B
x aim
a i*]
Let
.
so
(x M)
=69
e .
E
a a * 2)
105.
= 0
by induction on the length of the monomial;
U C (H*p(J))
hence,
UO = (H*p(J))0 Letting f xa) K = k[ rx"I]
Let
V
fal
xe
k .
be an ordered basis for
L
Q
then by
b tD K(U)
=
HO0
m = 0,1,...
am p. --
a
p
m
0Then
Let
a
CJ
be a
1 =1
a* = al*
m
e
H*p(J)
and
=
t
106.
Xt = 0
which implies
and
K(h) E V .
Thus we are done
by lemma 5. We can draw the corollary that in characteristic p > 0
if
U = H
if and only if for each
H
is cocommutative and coconnected then
follows because Since
U
H*p(J) = 0
a*
H
generated by the primitives in
We shall show
is
J
a*P = 0 .
This
p(J) = 0
if and only if
may not equal
we examine the ideal H .
=a*p Ia*
LH = HL = p(H*)
E
e
LH = p(H*)
H*] HL = p(H*)
the proof
the same (up to reflection). HL c p(H*)
:
Let
=
2
act on
H*
from the right by
a* e H* ,
Then for
.
=
=
E L , h E H ,
0
acts as a derivation.
If fx
is an ordered basis for
L
K = k[ { xl] ,
K D W = the space spanned by
e 1 ) a (x
m 00(a
(x
m
=
1,2,...
a1 < - -< am for some
then
K(p(H*)
) C W .
Since if
e. p &ei
h E p(H*)
107.
K(h) =
Z
A (x
i = t
Suppose for
a
a1 eii
plet
m
C J
, then letting
...pet 1
a*
eim
)
m x i m
be a dual basis to
i=1 et
et
a* = a
1
*--x
implies
t
=
am* 0
= At
and
K(LH) D W n K(H) .
Suppose
K(h) s W , we shall show there is
K(g) = K(h) .
which
K(p(H*) ) C W .
and
Next we show
where
m e p(H*)
g E LH
h s H
- H
where
Let ei
K(h)
=
a M I A (xa) i=1 0
a
a*(x) = .)
1 < n(i)
_
o
xa f
1
o ds-2
a
@ - - - @ xa s-
Y
H+[S-1]
E
f +.-+f s-1 We show
= Ker e)
(H
Ker(E @ E o d) E
= H
Y e L
H+[s-2]
Since
.
it suffices to show
E 0 I[s-2] o d (
I[s-2](Y) = 0
or it suffices to show a
E @ E @ I[s-2] o d 0I[s-2]( Z Xa f > 0 i f +- -± --
t
E ® E ® I[s-2] o d 0 I[s-2]( xaf f i>0 1 f +- --
+ Y)
s-1 =
s
)
t.
The left hand side (top) =
E
E0I
[s2
o d
I[s-2] o E[Sl1
(since =
Els]
E0E od oE =EOEod) a
o ds-1(Yas)
Xfi
f i> 0 f +- . - - fs= t
a
which is the right hand side, (bottom). ing
Y E H
* .*
o ds-2 (as
L
- --.@ H+
so
By similar reason-
Y e L[s-1]
127.
Thus
~1
E
o dss-2]
as=
e.
a1
a
m
+ 2 Aa(x
z
f > 0 f1 +- -.+f
where
1 _
s-1
= t
e 11+- o*+
= S - 1
e
a1 p
r
qr =1
by the choice of an extending basis,
n(1 )+1
xa
Without loss we assume
so we can choose a dual basis ai*)m C J
where we assume
a
E
so
Since
ya
E
1
m
(Vna )+L -
(Vn(a)+1)
(a*) na
]
x
to
=
Ker p
)+1 = 0
let Vn-ItI(()
S(a*n
and
z E H
=
y .
)-n+|t|+
where
Let
128. eF -pnda)-n+ t|+1 a*= a1 *
1
I.
e1
*
am m
*
Note for
b* a H* ,
h e H
- h)
Vn(pn(b*)
m
b* - V"(h)
since
pn(b*)
= Consider
0 =
n(a 1 )-n+It|+1 =
=
X
a e
> --1-
k If H
H
is the
is finite
is connected H
is coconnected.
is cocommutative.
SupH*
is finite dimensional, connected, commutative; k
is split and
is perfect, then
is coconnected and
H*
cocommutative and has an extending basis by finile dimensionThus by theorem 6:
ality.
H* ~
as a coalgebra.
a
(in the notation of the theorem) a
m
By finite dimensionality
char p > 0
and
n(ai) < 00 . H*
has a basis of monomials as described in theorem 6.
Let {[a T**] C (H*)* for
H*
.
(a
to (x a
m=1
Let ta ,
be a dual basis to the monomial basis **
m
C
a
*
be the dual basis
(the extending basis for
L(H*).)
144.
k[x 1 ]
Then
(H*)*
n-l
k[Y ]
n~a p(
-@-m
)+1n(am)+1 3
p
as an algebra where an isomorphism is induced by -a1 aY **
By finite dimensionality
1=1,...0
m
H** = H
.
.
Thus we have proved
a finite dimensional commutative, connected, Hopf algebra over a perfect field where the dual is split is of the form:
(as an algebra) k[Y ]1
k[Y ] em
X1 >-2mp If
k
is algebraically closed then by the results of the
first chapter we can drop the condition that the dual is split since it is automatically satisfied, also
k
is
automatically perfect. We now conclude by applying results of previous chapters.
We show a coconnected cocommutative Hopf algebra
over a perfect field is an extension--as an algebra and a coalgebra--of
H/LH
by
U
characteristic zero
H = U
is trivial.
k
Assume
when and
U
is commutative.
LH = H+
In
so, the result
is perfect of characteristic
p > 0.
--4
145.
Lemma 6: and
K = k[
Let
be an ordered basis for
{ xa? If
xa ]I
where
h e V1
e
e =
K(h)
1 1
(a
A
a
m
i g
then there is
H
£
where pei
1 pe
a K(g)
=
Z 1
''
m
xm
and V(g)
V(f) = h .
Say
show we can choose choose
f
where
f
#(f)
K(f)
=
I
> pn
is minimal then
..
Pt
Suppose for some
be a dual basis t
i
,
f
)
C
x
Pt
We first
.
Suppose not
f
x1
S pn
+00+fit
pIf
90*
t x kf
k=1
a* = a*
t
Let
it
ak]
f
fj /p *.
*
1 /j
=
= = H
Observe
H/LH Dl k .
U C H
By
theorem 1 i
H/LH
(H
a
k H/LH
O
m )H
(D H
--
)H
H/LH is a linear isomorphism.
U-
is a split exact sequence. dition for
H
Hence,
U = H
H
H
®
0
k , and
H/LH
This verifies the first con-
to be an extension of
H/LH
by
U .
161.
Bibliography
[1]
E. Artin, C. Nesbitt, and R. Thrall, Rings with Minimum Condition, University of Michigan Press, 1944.
[2]
N. Jacobson,
[3]
J. Milnor and J. Moore, On the Structure of Hopf Algebras, Princeton University Notes.
Lie Algebras, Interscience, 1962.
162.
Index
Adjoint representation, Algebra, 8 111
A ,
commutative, 8 Antipode, 17 Augmentation, 10, 11 Brauer Group, 80 Coalgebra, 10 C.,
111
cocommutative, 11 Coconnected, 93 Cohomology, 47, 87
6, 48 6n
,
88
H(B,H)1 , 88
H' (H,B), 56 Hom(C,A)n, 88 Homn(C,A), 48 normal complex, 58 Reg(C,A)n
88
Regn (C,A), 48
155
r
163.
Coideal,
43
Comodule, 12 (C.)H.A. comodule, 87 comodule as an algebra, 36
$2,
structure on
X © X, 36
Connected, 143 Divided power, 122 Duality, 14 Extension, 61, 62, 89,
90, 155
f B 0) H, 67 Brauer Group, 80 H 9) g
B,
91
isomorphism of extensions, 62, 90 left exact sequence, 61 product of extensions, 81 right exact sequence, 89 smash product, 63 split exact sequence, 61, 89 Filtration, 26
r(G),
17
Grouplike element, 24 Hopf algebra, 16 coconnected, 93 conilpotent,
22
164.
filtration, 26 24
G(H), Hg,
25
L(H),
94
split, 21
U
95
,
J , 26 K(x),
98
L , 121 Module,
9
(C.)H.A. H
as
module, H*,
48
21
module as a coalgebra, 43 e3' -module, n 2
7
structure on
Poincare-Birkhoff-Witt
X 0 X, 42 theorem, 140
Primitive element, 94
p
,
113
Restricted Universal Enveloping Algebra, 93, 95 Smash product, 63 m He Tr(G) -+ H , 66 Symmetric tensor, 97 Universal Enveloping Algebra,
93, 95
165.
Unit, 8,
V V,
,
13
115 119
Vector space, 7 X , 110
166.
Biography
Moss Eisenberg Sweedler was born in Brooklyn, New York, April 29, 1942.
He was graduated from the Massachusetts
Institute of Technology with a B.S. in June 1963. tinued study there as a graduate student.
He con-
Cocommutative Hopf Algebras with Antipode by Moss Eisenberg Sweedler
B.S.,
Massachusetts Institute of Technology
(1963) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY
August, 1965
Signature of Author .
.-.
. .
Department of Mathematics, August 31, 1965
Certified by ....-..
.
Thesis Supervisqr
Accepted by .......... 0................................... Chairman, Departmental Committee on Graduate Students v/I
2.
Cocommutative Hopf Algebras
19
with Antipode by Moss Eisenberg Sweedler Submitted to the Department of Mathematics on August 31, 1965, in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Abstract In the first chapter the preliminaries of the theory of Hopf algebras are presented. The notion and properties of the antipode are developed. An important filtration is induced in the Hopf algebra by its dual when the Hopf algebra is split. It is shown conilpotence and an algebraically closed field insure a Hopf algebra is split. The monoid of grouplike elements is studied. In the second chapter conditions for an algebra A -which is a comodule for a Hopf algebra H --to be of the form A 'E B ® H (linear isomorphism) are given. The dual situation is studied. The graded Hopf algebra associated with a split Hopf algebra decomposes in the above manner. Chapter III contains the cohomology theory of a commutative algebra which is a module for a cocommutative Hopf algebra. There is extension theory and specialization to the situation the Hopf algebra is a group algebra. Chapter IV is dual to chapter III. Chapter V is devoted to coconnected cocommutative Hopf algebras, mostly in characteristic p > 0 . There, the notion of divided powers is developed and shown to characterize the coalgebra structure of a class of Hopf algebras. The Hopf algebras are shown to be extensions of certain sub Hopf algebras by their primitive elements. Thesis Supervisor: Title:
Bertram Kostant Professor of Mathematics
3.
Contents
Introduction
4
............................................
7
Chapter I.
Preliminaries .......................... ..
Chapter II.
Decompositions ......................... . 36
Chapter III.
Cohomology ....................----.
. 47
Chapter IV.
Cohomology ..........................
. 87
Chapter V.
Cocommutative Coconnected Hopf Algebras. . 93
Bibliography . ......................... Index ........ ......................... Biography ....
... *0..........................-.....-
...... 0...... 000............
162
166
I
Introduction
A Hopf algebra as considered herein is simultaneously an algebra and a coalgebra where the algebra structure morphisms are morphisms of the coalgebra structure, (or vice versa).
This differs from the graded Hopf algebras
of [2] Milnor and Moore, except in characteristic 2.
The
problem is to determine the structure of cocommutative Hopf algebras. Our first approach lies in a cohomology theory.
We
have constructed abelian cohomology groups Hi(HA) , where
H
"C.H.A."
H(CH)i
is a Hopf algebra, H-module,
H-comodule.
C
0 < i E Z
,
A
an algebra which is a left
a coalgebra which is a right "C.H.A."
We then determine the structures of algebras
(coalgebras) which are extensions of
H
(C) by
B
(H)
This theory applies to the algebra structure and the coalgebra structure--separately--of coconnected cocommutative Hopf algebras.
We hope to develop an extension theory
where the extension is a Hopf algebra and is an extension of one Hopf algebra by another. Our cohomology theory gives the familiar group cohomology in case G .
If
Ar
H =p-(G)
the group algebra of the group
is the group of regular (invertible) elements
5.
of
A
then Hi(H,A)
Furthermore, if underlying field
A
Hi(G,Ar)
=
is a finite Galois extension of the
k and
G
is the Galois group of
then the isomorphism classes of extensions of by
A
A/k
H =F(G)
form a subgroup of the Brauer Group.
Kostant has shown--the results are unpublished--that a split cocommutative Hopf algebra with antipode is a smash product of a group algebra and a coconnected cocommutative Hoof algebra; and that in characteristic zero a coconnected cocommutative Hopf algebra is a universal enveloping algebra of the Lie algebra of primitive elements.
We present
proofs of these results and study coconnected cocommutative Hopf algebras in characteristic- p > 0
.
For a cer-
tain class (including all where the restricted Lie algebra of primitive elements is finite dimensional) of coconnected cocommutative Hopf algebras we are able to determine the coalgebra structure.
The coalgebra structure is described
in a generalization of the Poincare-Birkhoff-Witt theorem. We now outline the generalization. In characteristic zero let for the Lie algebra algebra,
L ,
a Hopf algebra. I=
LC U
(x
be an ordered basis
its universal enveloping
If
i = 0,1,2,...
6. then dx(
=
z i=o
x2=0n
The Poincare-Birkhoff-Witt theorem is equivalent to:
a i)
x
-
a 0 , we show the Hopf algebra obtained
by factoring out the ideal generated by the primitive elements is isomorphic to a sub Hopf algebra of the original Hopf algebra, when the vector space structure on the quotient is altered.
We also show the original Hopf algebra
is an extension--as an algebra and coalgebra--of the quotient by the primitively generated sub Hopf algebra.
Chapter I
The study of Hopf algebras is a self-dual theory. For this reason diagram notation is useful, as it makes dual definitions and proofs evident. For all time we fix the field for all vector spaces. over
k ,
n
If
k
which is the base
X1 ,...,Xn
are vector spaces
the permutation group on
a e
n-letters
we consider
a:
Often
a
Xe
-
X
->
x1@
-0 -@
x0
->x.
will be written
i ,...1
X
(1 ,...
...
ex . (& x,
in)
where
1,2,...n) ; in this case X@--
111n
OX
'''
0y
... e
X
1 If
X1
X2 =
n ,
@Xn
.
n is a left G-
module. If
X, Y
are vector spaces
a linear map from
X
to
"f: X -Y"
means
f
is
Y
Once we define a "right" object such as module or comodule, we consider the "left" object to be defined with the mirror definition.
Similarly for "left" objects.
8.
Algebras (A,m,-q) space over
k
is
an algebra (over k) where
,
m: A @ A -+ A
T:
A
is a vector
k -+ A , if
the follow-
ing diagrams are commutative.
Igm A®A A
3 A @A
-
1S
m®I
I) AA
II) -m
A A (1k
A AA
is equivalent to associativity.
II) k
) A
A®1@A
A
m
m
I)
Ti@I k(DA~-
is equivalent to
TI(1),
a
is an algebra where
(rI)
is a unit.
is the identity,
m
is the usual
multiplication. An algebra
A
AO A
is commutative if
m
(2,1)
A
is commutative.
Ae A If then
A
is an algebra, -+
A
x
-+
l@ f(x)
If where
A
and
B
vector spaces
f: X ->Y
1 E A
always denotes
'QA(l)
Y
kD @f: X
where for an algebra
X, Y
A ,
are algebras
A& B
is
an algebra
9.
(A
B) q
(A @ B)
(>
A
A
BOB
m A mB mA® B and
nA e B = k @ 11B f: A ->B
A(
B
AD k .
is a morphism of algebras if the following
diagrams are commutative:
A
A
A
ff
ff
f
k B
B
B
4
B -$A
k
A
If space
A M
will often be written
is an algebra a left
with a map
?P: A®D M -. *M
k
k
A
A-module is a vector satisfying:
k@I A (A
M
M
)A
M
> A (M
M A0 M
If
M, N
phism of left
'f M
are left
A-modules
A-modules if
f: M -+ N
is a mor-
I 10.
M
M-)
A
is commutative .
jf SI@ f N > N A@N & An augmentation eA : A -+ k
morphism
.
A
of an algebra
EA
is an algebra
An augmented algebra is
an algebra
with a fixed augmentation. If
A
right)
is ar augmented algebra
k
is
a left (or
A-module by: EI
N @A M
m 3 k
k
k
-+
N
A-module and
is a left
M
If
I
-
A0 k
a right
A-module
is a vector space such that *N
NO A(
I
M
-
(9 ?M ) NO M
-
N (AM
_+ 0
is an exact sequence of vector spaces. If left
is an augmented algebra,
A
A-module, then
A+ = Ker e A,
k ®AM = Coker (A+O
) A0
M
i@I or if
A+ - M denotes
Im(A+
) A
M
M
a M) , ?P
M - M)
i®I k
A M = M/(A+ - M)
Coalgebras (C,d,E) space over
is a coalgebra over
k ,
k
where
C
is a vector
11.
d: C
-+
k
s: C -
CtC
if the following diagrams are commutative: k
d
C
I
II)
II)
d C SC 0C
I
C (k
I)
C
d
dO I
d C®C
C
C O----C
C I)
C '
C@ C
E
is equivalent to coassociativity E
is equivalent to k
is an augmentation of a coalgebra. d = k® I
is a coalgebra where
and
e = the
identity.
A coalgebra is cocommutative if C@ C C
(2,1) C OC If
C
is commutative.
d X, Y
is a coalgebra and
are vector spaces
f: X - Y , then E
C D X If
C, D
f:
c
C®X
k
Y
are coalgebras
-
Y
® C @) D
aYc is a coalgebra where
12.
dC(
dD
>C(C(DeD
CQD
2D 4) (1,3,
dC (
D (CO D) 0
f: C -* D
C
eD
'C
ECOD
(CO D)
is a morphism of coalgebras if the diagrams,
CO C
C f
C
f
dD D'-
D
k
f
f
D
D
6D
are commutative. If space
C M
is a coalgebra, a right $: M -. M @ C
with a map
C-comodule is a vector satisfying: M
M
$C MO C
>MP C %
I
>MD C
()®d C I (
If
M, N
are right
morphism of right
C-comodules
C-comodules if
E
f: M -* N
is a
13.
M-)
m® C
If
N
,
fI
N
is commutative.
C
A unit of a coalgebra
C
is a coalgebra morphism
k - C. A coalgebra with unit is a coalgebra with a fixed unit. If left)
C
is a coalgebra with unit
If
O
is a right (or
C-comodule where k
M DN
k
N
N
-%
d
kk
is a left
I
TIC
C-comodule
M a right
C-comodule,
is a vector space such that
-+ M 0
N
-+ M(DN
)
M
A
N
is an exact sequence of vector spaces. If 7: C -+ J
C
is a coalgebra with unit then
M D
k = Ker (M
-
JC = Coker M
C
iC
'
) M®J
)
We shall freely use facts such as kernels and cokernels of comodule morphisms are subcomodules or quotient comodules, when the "dual " fact is well known. If
X, Y
are vector spaces
Hom(X,Y) = ff: X -+Y.
14.
Hom(X,Y)
is a vector space.
Often
Hom(X,Y)
additional structure, for example if and
A
an algebra then
Hom(C,A)
C
carries
is a coalgebra
has a natural algebra
structure as follows Hom(C,A)
C,A)
) Hom(C D
Hom(C,A)
®
a
Hom(CA))
where
Hom(C,A)
g) = mA o f
a(f
g
d: C -+ C & C .
Multiplication in
denoted by
thus
in
*
Hom(C,A)
f*
-+
.
Hom(C,A)
X*
g = mA o f(& g o d .
(T)A 0
-C
'
X
is a coalgebra
-c. (X
0
X)*
a as above. a is injective for if n SAxi* y * s @ ,We can assume i=1 linearly independent. .
X*
Consider X*@0 X*
> =
The unit
Hom(Xk) , as usual we consider
denote
We have just shown if
is an algebra.
= (dim X) 2 = dim(X® X)*
is an isomorphism.
a
case
Xj0
is injective.
a
which shows
Y,*) i*
induce transpose mappings:
A* -+ k* = k , d = tim: A* -+ (A
a finite dimensional coalgebra
A*
A)*
= A*@
A*
A
is
Similarly if
is a coalgebra.
A*
With these maps
@
is an algebra; however,
this is the same algebra structure as deduced above. If
X, Y
are vector spaces -+
a: X* ® Y*
Y)*
(x
x E X, y E Y
=
<x*,x> x* E X*, y* E Y*
a
is injective.
same proof works.) a
in
(X @ Y)*
coalgebra then
(We proved this, above, for Identify If
M
X*
@
is a left
Y*
Y = X , the
with its image under C-comodule,
C
a
16.
C*@ M* ,
) (C@ M)*
IPM*
defines left
4'M* ,
and this gives
C*-module.
algebra,
M*
Similarly if,
M a left
A
the structure of a is a finite dimensional
A-module by
AQ M then
t M
-M
tPM: M* -+ (A
M
M)*
comodule structure on
A*Q M*
defines a left
M*
Hopf Algebras
(H,m,i3,d,E)
is
1)
(H,m,TI)
is an algebra with augmentation
2)
(H,d,e)
is a coalgebra with unit
3)
the diagram
H
a Hopf algebra when
m -
H
d H
)H
d)d HOH
is commutative.
L
H
Tj
H mDm
H
(1,3,2,4)1 Hf
HO H@ H
e ,
17.
3)
and
"
is equivalent to
3)
and
(H,m,Tj)"
"E
is a unit for the coalgebra "d:H -*H ® H
(H,d,E)"
is an algebra morphism".
is an augmentation for the algebra
is equivalent to
"m: H
DH
-+ H
is a coalgebra
morphism".
Since a Hopf algebra Hom(H,H)
H
is a coalgebra and algebra
has the structure of an algebra.
algebra structure and if
H
H* has an
is finite dimensional
a coalgebra structure, as well.
In this case
H*
has
H* is a
Hopf algebra. In
Hom(H,H)
consider
T
o s
k = k - 1 C H
is the unit.
so
e
We shall often
will denote the unit in
Hom(H,H) .
The identity
I: H -+H
Hom(H,H) .
S E Hom(H,H)
is an antipode for
I * S = E = S * I , that is
S
is an element of H
if
is the inverse to
I.
Example:
Let
G
be a monoid (group without inverses).
the "monoid" algebra =
n
7(G)
L
Q) k - g g E G
m
n,m
j=1
i,j=l
is an algebra where
where
F(G)
=
18.
-ri: k
F(G)
a Hopf algebra where
is
F(G)
d:
-+
g s: F(G)
'(G) @ F(G)
-+
g@g
-
k
+1
g
r(G)
(G)
-+
has an antipode if G
Suppose
. and only if
G
S: F(G)
is a group define
is
F(G)
-
-1
g then
S
is an antipode.
If
S
a group.
is an antipode,
n e = 1 = I - S(g) S(g) = g~
implies
gS(g)
=
,
,
hence
We now show that if
H
X g1 which i=l has inverses and is a group.
where G
S(g)
d o S = (2,1) o S @ S o d
2)
S o m = m o S @ S o (2,1)
3)
E o S = S o
4)
If
=s
is commutative (as an algebra) or co-
commutative (as a coalgebra) then We prove
1)
inverse to
52
=
S o S
=
as follows:
(2,1) o S 0 S o d e Hom(H, H@ H)
which is
d e Hom(H, HO H)
is a left
d o S , an algebra.
S , then
has an antipode,
1)
H
=
d
and
we show
(2,1) o S 0 S o d
d o S
is a right inverse
19.
to
d
This proves they are equal.
.
following: f: C -+ D
If
C, D
are coalgebras,
a coalgebra morphism,
phism and (h o g
We shall use the
g1 , g 2 E Hom(D,A)
A, B
h: A -+ B
are algebras, an algebra mor-
then
o f) * (h o g 2 o f)
= h o (g1 * g 2 ) 0 f
The proof is clear from the definitions. Consider
(d o S)* d = d o(S * I)= d o e = k (d o e = d We let
because
m's
.
d(l) = 1 0 1 ) .
mn =mo m@lo---om@
by associativity
mn
e E
O ...
T
n -1 is independent of the order of the
Similarly we let
dn = d
T
o---o o d
T o d
n -1 dn
is independent of the order of the
tivity.
If
h e H
Z hi'@ h "
d's
by coassocia-
we often let - --
hi(n+l) = dnh
Then
)
d * [(2,1) o S Q S o d](h) = Z h 'S(h
hi "S(h '") I*s
=Z h
'S (h
'")
@
E)
=
20.
Z hi'S(hi")@ 1 = E(h) i
1
I * S= which verifies For
2)
1) . we show
m o S @ S o (2,1)
S o m
is left inverse to
m * [m a S® S o (2,1)](h&
Z h
For
and
4)
g) = Zh
") =
E(g)S(h
= e
E = I *
For
Then
3)
a o I * S
=
'g'
H
m a S ® S
=
S(g")S(h ")
(e o I) * (E a
o E = (I o E) * (S o e)
suppose
E .
E(g)hi' S(h") =
=
S)
(g)E(h) S o 5
=
*E
=
o S
* S o E = S
a
is commutative so m o S ( S o (2,1)
=
S o m
S * S2 (h) = Z S(h i)S2(h"') = z S(hi' S(hi"))
i
i =
which shows H
and
is right inverse.
(S o m)* M = (S * I) a m = E O m = 6
=
m
S2
is
is cocommutative
right inverse to
S
®S
o d
S * 32 (h) = Z S(hi')S 2 (hi") i = m o I
m o I®
S (Z [S(h)]
j
S ,
so
(2,1) o S
= =
S o e(h) = e(h) S2
®S
.
If
o d = d o S
S(h ')@ S(hi")
S( i
[S(h)]J")
.
21.
I * S o S(h) = s o S(h) = s(h) which shows
S2
is right inverse to
H
We have
of
H*
on
as an
S
H*-module
H** ; under this action
H
is a submodule.
We then concern ourselves with the action of H**
a left
h**= h e H,
Then if
1
H** D H , we define a left action
H , H*
Consider
S2
hence
on
H
H*-module by
dh =Z hi'
hi"
h =Z h ' e H C H**
b*.
i
-
This shows
H
the action of module.
is a submodule of H*
on
H
.
H**
We consider
(We could have begun with
=
Again
H*H*H
would be a submodule where h - a* = Z 2) 1=1 where g E G are distinct ki . By the minimality of
n
a* E H*
g 2 'g3 ' 'n where
are linearly independent.
= 1
,
= 0
Choose i=3,.. .n
26.
n Then
X1 -
-
iA =-2
1=2
n
n
0 = a* - 0 = a* -(
%g) X
g
=
1=1i=
= -A2 g1 + XA 2 92 = This implies
X2 = 0
minimality of
n
X2 (g2
g1 = g2
or
d(gh) = (dg)(dh) = (g ) g)(h
which contradicts the
g, h e G
Observe if
.
g1 )
-
QDh)
= gh(® gh
Thus
.
G
a monoid and since it is a linearly independent set in the space spanned by
G
T(G)
as an algebra.
,
is
H
We consider
T(G) C H . If
H
dS(g) = (2,1) o S 0 S o dg gS(g) so
S(g)
S , g e G
has an antipode
=
I
=
S(g)(D S(g)
*
S(g) = E(g) = 1
is the inverse to
g
in
Filtration of
G
H
is split.
so that
and
G
S(g) E G
; is a group.
H
We exhibit a natural filtration on exhaustive when
then
H
which is
The multiplication and dia-
gonal map respect the filtration as does an antipode when present.
The filtration leads to a graded Hopf algebra.
It is also an invaluable basis for induction. Within
H*
let
J = F(G)
.
It is a straight
forward verification that for any subcoalgebra
C C H
27.
Thus
C4
C C H @H)
(do CC® J
is a
We assume
il= (J1+1L
1)
H = UHi
2)
H H
3)
dH
let
c
Z
D: then
.
.
J i+l
a**b* E J +
b*,
x>
b*
x = 0
let
x
x> = 0
= 0
;
and we are done. H* 0
H*
O+ z E H 0 H
is
"dense " in
there is
In fact there is + 0 prcve
(H
z* E H*
a*@ b* E H* .
H)*
in
H*
that if
where
$ 0 where
Thus to
d(a*
.
h) = Z h i'
a* - h "
we need
28.
only show
d(a* - h)> = B
B T1
The left diagram means
B -+ B
m: B
morphism; the right diagram means H-module morphism.
B
H
A
Regn (C,A)
Regn(C,A) B
0 < n E Z
For a coalgebra
C n n n Hom (C,A) = Hom(C ® '-.-C,A)
B -.B
Regn (H,B)
Homo(C,A) = Hom(k,A) = A
Now we assume
@
let
,
H-module
regular elements of
s: H
is a commutative algebra
is a multiplicative subgroup of
We let
is a left
regular (invertible) elements of
=
is a left C.H.A.
group.
B
a cocommutative Hopf algebra.
and algebra Let
k -+ B
I:
H-module
is called commutative Hopf algebra
H-module (C.H.A. H-module) if and
is a left
A B
is a left C.H.A.
.
If
is an abelian and
Rego(C,A) =
H-module where
Regn(H,B)
Define
,n: Regn(HB)
Homn(C, A)
= Ar
is the module action.
is an abelian group.
Homn(C,A).
-+ Homn+1(HB)
0 < n E Z
49.
n
f(-)
on7
o (I)
---
I 0I -.m
)
~1=1 i - 1
n - i
(-1)n+l
where
n > 0 ; for
f E Regn(HB) h
b E Br
H
E
We shall let
6 0 b(h) = (h-b)b~
n = 0
F[a]
denote
a F
Q)
---
unit.
In
Q)F
n > 0 ,
Regn(H,B)
e .
This is often denoted
and given
to
morphism.
6 01 (h) = (h - 1)1
h - 1 = e(h)
H.A.
Regn+1(H,B)
6n
6n so
E(h)
is a group homo601 = E .
is one of the conditions for
B
Note
to be an
H-module, we now use the other:
6 0 (bc) (h) = (h - (bc))c~Ab~A =
Z (hi 'I - b)b~
(hi
=Z (hi i
For the case ,n(E)
- c)(c
1b 1
i
= [6 0 (b) * 60(c)](h)
.
n > 0 n T
?P 0
E ()i
n E *
E *
i=1
k
as a map from
and that =
on 6 n() = 6 n(f) * ,n(g)
* g)
These imply that we can consider Regn(H, B)
the
We shall show
6 n (f
f,g e Regn(H,B)
is
E[n]
E =
E
o
[i
[n-i
E
)
50.
Note
o I
&
e = e
because
-
and
(f
(1)~
* g
o
bc(-1)us [i-] 1 because
I
is an H.A.
H-module.
?P o I()(f * g)= (7P o I
This also guarantees (f * g)(~l)
B
g(-l)i
*
m
I
is clear
~ (f * g)(l)
and thus
6n(f *
g) =
coogb (-)
[1-1 p
=
*
(f(l)n+ll
I n- ]
m
is a coalgebra morphism.
E
6 n(f)
I[n-i]
[n-i] I
mi1 M0in-il I) 1
It
(-1)n+1
n(g)
We have the sequence of abelian groups and homomorphisms 0 Reg (H,1B)
60 ) Reg (H,B)
1
2
-' Reg (H,B) -
62
)
We shall now show 6 n+1 6 n
= E: Regn (H,B) -+Regn± (HB)
We do this by expanding -0
* term *
---
41 *
term *
6 n+1 6 nf
.
In our notation
expands to
42 term * term *
The explanations of the expansions follow the computation and a typical note is of the form: "texplanation".
g)
because the group is abelian
m
I
f) * (p o I
4 -+ 41 * 42 * 43:
51.
6 n+1 6 n
=
(where
n > 0)
1
o 1O(p
0
f-
r
1 @f
0(1[i-
®m
[1-i])
* f (_)
*
2
/"--, n+1
-----------------
i
n (
o I e f
*
o(I[i-e
f
m
1=1
j=1
I[n-l)
(-1) J f (-1)n+1
0
( 1 i]
0
m
0 1 [n+1-j])
3
"K * (*
o I
rf
0([1-110 Me
f 1=1
(-+1) _n2
*
11
=*0 I
1 n-i3)
12
(* o
) f)
o (11-ilomer i=1
13
*
4
I
f
-
))
52.
21
n+-1 j1:1
'i)
(*, oIOf(i
o -1 (I 0 M®@ I[n-1j])
22
-------------[1-]
*(-1)
[ -1
o (
1[J1]l(Dm®
I[n+1-j
*
I [n+l-i])
1D
H
PD=D-)D D
D
kE.
D
The diagram: P
P
I
E
T -- >D
B
IPD
T
)D DH
It can be rewritten
is commutative.
) D
T
m oP
k
P
PD H k.
JD
B
TID
H Since
Im P
Im TD = D
are left
DH k ,
D H~~
Clearly
EH k.
B-module morphisms
is a factoring
Q
m oP @ P
is surjective, it follows
I C Ker m o P (FP, IT C Ker m o P
,
T
Thus
T -+ D
D
@
since P ,
P,
P
or there
86.
(R@k B
)
-
T
D
)
B
is commutative. have shown
The horizontal composite is is
TrD
injective. 11D
B
VD
hence we
Thus
$D
D
D
-
)H
is a split exact sequence. The second condition for extensions is satisfied. Thus
D If
is an extension of
y: A -+ A, ,
H
by
y: A -+ A1
B
are equivalences of exten-
sions then: ,y() Y: A®() A
y y@ ~y|T
factors to
of extensions.
~yIT: T
A -T
yD: D-+ D
and
yD
is
an equivalence
Thus the method of producting extensions
gives the isomorphism classes of extensions a product struc-
f ture.
Direct calculation shows the product of
f
f B
H
is extension isomorphic to
B
B
H ,
f' H ,
so that
under the correspondence of theorem 3, the multiplicative structures correspond.
This shows the isomorphism classes
form a group and the correspondence is a group isomorphism.
pop--
87.
Chapter IV Cohomology
We treat the dual situation to the previous chapter. We outline the definition of abelian cohomology groups H(BH)i
i=0,1,2... , when
algebra,
B
H
is a commutative Hopf
a cocommutative coalgebra which is a right
H-comodule satisfying certain conditions.
We also outline
the dual extension theory, defining when a. coalgebra is an extension of
B
by
H
and when extensions are isomorphic.
The isomorphism classes of extension again correspond to H(O2 , we define the dual extension to B f) H of the H(B,H) previous chapter; this indicates the correspondence beH(B,H) 2
tween B
is
is
a right
and the isomorphism classes of extensions.
42.
and
d: B
-BO)B
s: B
->k , B
5: B B ®B
H.A.
is a right C.H.A.
H-comodule,
is a right H.A. H
B .
is a right
B
are comodule morphisms. B
a Hopf algebra
H-comodule structure for
H-comodule under
if
H
a coalgebra,
B
-* B(
H
is an
H-comodule if
H-comodule
is cocommutative
is commutative.
For a coalgebra
C
and an algebra
0 < n e Z
let
88.
Hom(C,A)n = Hom(C,A En)
Then C.H.A.
f
Reg(B,H)n
B
Let
B
B
is a
Reg(B,H)O = Reg(B,k).
be a right C.H.A.
Reg(BH)
E
Reg(C,A)n = (Hom(C,A)n)r
is an abelian group when
H-comodule.
Let
Let
.
H-comodule.
For
consider
4k@H= >B@H
H.
Define 60
Reg(B,H)
=
6 0f we are considering
-+ Reg(B,H) 1
f: B
-+
f * fI
k C H .
'I When
o
),
n > 0
define
6 n: Reg(B,H)n -+ Reg(B,H)n+l
6nf = k O f *
o fn+1 i *
I
d
I i=1
Io Then
6n+1an = E,
6
b
.
is a group homomorphism so we have
a complex 61
6n Reg(B,H) 0
Reg(B,H)
) -
and define the cohomology groups H(B,H)n
=
Ker 6 n/Im
H (B, H)O
=
Ker 6 0
6
n-1
n > 1
89.
As in the previous theory we have a normal complex: n > 0
= ff 6 Reg(B,H )n
Reg(B,H)
B
[n]
-
k
is commutative
Sn+ =
n|iReg(B,H)+n: Reg(B,H)+n -*Reg(B,H)+n+1
Reg(B,H)+ 0
Reg(B,H) 0
=
60+ = 60: Reg(B,H)+ 0
-+ Reg(B,H)+ 1
The cohomology computed by this complex is the same.
Extension Theory
H
is a Hopf algebra
A, B
H-module (as a coalgebra), structure.
called
A
a left
*: H(@ A -+ A , being the module
The sequence H(
is
coalgebras,
A
) A -- -B
,
right exact if
1)
r
2)
Coim r = k OH A
is a surjective coalgebra morphism
Ker Tr = H
i.e.
A
The sequence is called split exact when:
H
has an antipode,
90.
the sequence is right exact, and there is a regular left H-module morphism d A
)A
A
P: A -+ H
P g r -
H(
comodules and left give A
B
B
In this case
.
is an isomorphism of right
H-modules by theorem 2.
the structure of a right C.H.A.
is called an extension of
4
r
A
-+ B
B
by
H
H-comodule, then when:
HD @A
2)
the following diagram is commutative, d
A->
is a split exact sequence,
I@7r A@A
A@B
-
--4
d
A
$: B -+B
Let
1)
-+
B-
A
B
B
H
H
where the
A*
A ir~I Given two extensions
A, A'
of
B
by
split exact sequences are
4 H@ A
HwA'
-+
ri
Al -+
B
A'a
B ,
we say they are isomorphic if there is a coalgebra morphism 'y: A -+ A' ,
such that,
H
'1 91.
H@ 10
A
A
-
B
y
Y H@
)A'
Being isomorphic is an equivalence rela-
is commutative.
tion and we can form isomorphism classes of extensions. An example: Given
g e Reg(B,HH)
2
a
2-cocycle, form
H
®B
which
g H®@ B
is
H-module.
as a left
I
E
7r: H
B BB
'
g I@ P: H
E
B
0 H
.
g d@ g) g B
H
B
)H
$
I
B
B
g
(1,3,5,2,4,6,7) H 0H
H
H
B
H
B
H(
B
H
H
H
B
'I
m2 (
m H
(H
defines the coalgebra structure on
H
@ g
H
H
B
g
B
-H
g
B
B .
( B) g
Then
(H
@ B) g
92.
is split exact, the second condition for extensions is satisfied so
H a B is an extension of B by H . Two g such extensions are isomorphic if and only if the 2-cocycles are homologous.
There is such an extension in each isomor-
phism class of extensions. tive correspondence between classes of extensions.
This yields the natural bijecH(B,H) 2
and the isomorphism
93.
Chapter V Cocommutative, Coconnected Hopf Algebras
We have shown a split conilpotent Hopf algebra with antipode is a smash product
E
B (D H
where
Hopf algebra acting as automorphisms of split Hopf algebra where
G(B) = 1j.
B
H .
is a group B
is a
We shall study such
Hopf algebras which-are cocommutative.
In characteristic
zero we prove such a Hopf algebra is a universal enveloping algebra--u.e.a.--of a Lie algebra, in characteristic p > 0
it contains a restricted u.e.a.--r.u.e.a.--of a res-
tricted Lie algebra.
(We assume familiarity with univer-
sal enveloping algebras and restricted universal enveloping algebras as can be found in
[2]
Lie Algebras.)
We define divided powers as sequences of elements
1
=
I 0 '2l''''
N
n
n < N
din = Z and show that in a i=0 class of cocommutative split Hopf algebras H , where where for
G(H) =
[1
the coalgebra structure of
H
is characterized
by divided powers. A Hopf algebra G(H) =1j. phisms of
Since k
to
H G(H)
H
,
is coconnected if it is split and corresponds to the coalgebra mor-
coconnectivity is equivalent to:
94.
being split, and morphism of
k
TI:
k -+ H
is the unique coalgebra
H .
to
all coconnected Hopf algebras have antipodes.
By
All Hopf Algebras Henceforth Are Assumed To Be Coconnected Unless Otherwise Specified. A primitive element
x
in a Hopf algebra
H
is one
where dx Then
x E H1
=
xEl
+
1@ x
e(x) = 0 .
and
Let
under
[
,
] ; if characteristic =
dxp
(binomially expand
+
1@ x
L(H)
H .
space of primitive elements in
(dx)P), so
.
L
denote the
(=L)
forms a Lie algebra
p > 0 x
L
1
forms a restricted Lie
algebra. Suppose
x E H dx E H dx
I * E = I
E * E = E
I
=
,
E(x) = 0 . @ H1
+ H
10y
+
H0 , z
HO =
so
l E(y) +
imply
k
z
= X
+
E(z)
= x
E (y) +
E(Z)
=
y
E(x)
=
0
95.
dx = 1QD (y + e(z)) + (z + E(y))Y
Thus
and
x
is primitive.
U
and
L C U .
U
H
x&l,
= L(q H 0 = L(0 k
denote the u.e.a. of
and the r.u.e.a. of
E(Z)
This shows
Ker efH 1= L Let
+
l®x
=
1 - 19
L
in
L
in
char p > 0 .
.
char p =0 We consider
is a (coconnected) Hopf algebra where the co-
algebra structure is induced by specifying the elements of L
are primitive.
Then
L(U) = L .
By the defining property of
U
there is an algebra
morphism : U -H which is induced by the identification of L C H .
y
L C U
with
is also a coalgebra morphism; hence, is a mor-
phism of Hopf algebras. That
y is injective follows from the next lemma.
Lemma 4I:) bra, if
V
v: A -+ C
,
A
a Hopf algebra
a coalgebra morphism.
vjL(A) Proof:
v
C a coalge-
is injective if and only
is injective. Suppose
a coalgebra morphism
v|L(A)
is injective.
Since
V
is
96.
EC(V(L(A))) = 0 thus
vA 1
VIAn
is injective
is injective.
injective.
C(V(l)) = 1
,
By induction we can assume
a e A n+
Suppose
v(a) = 0
@A
VI An
is
then
da=l0a+a®l +Y, Suppose
V
n > 1 ; hence,
YEAn
A
.
then
®
0 = dv(a) = v @ voda = v(1)
v(a) + v(a)
) v(1) + (v
@
v)(y)
= (v&@v)(Y) . Since
v
VIA |A
n
is injective it follows
a E L(A) ; which contradicts
VIL
is injective.
The converse is obvious. This implies
'y: U -+ H
Q.E.D. is injective.
with its image which is the subalgebra of L .
(Thus
U
Y = 0,
is the subalgebra of
H
We identify H
U
generated by
generated by
L
both as algebra and coalgebra.) We shall show if
where
J = k
C H*
H
is cocommutative and
char p = 0
U = H ,
char p > 0
U = (H*p(J))£
,
p(J) =
the ideal generated by
p(J)
a*P|a* s J] .
and
(H*p(J))
We do this by developing
is
97.
a technique which "picks-out" subspaces of associate with subspaces of the polynomial algebra on (Note:
H , subspaces of
fxaJ a basis for
work with
f
k[ f xa
k[ I xJ ] L(H)
.
Given a vector space 9-.module.
n > 0
SnX =
n = 0
S0 X
xa]
x
we have chosen to
,
]--which makes the basis clear--rather L .)
than the symmetric algebra on
Let
We
.
since the method of associating subspaces with sub-
spaces depends upon the basis
left
H
X
n
is a
x =®X
,
The symmetric tensors of degree Y E X[n]aj=
Y
Y
=
are:
J
aE
k .
be an ordered basis for k[
all
n
x
X ,
is a graded algebra.
]I
CO
k[
x a]
SX =
is isomorphic to
e
SnX
as follows:
n=0
a Let (i.e.
(x 1 )1
e 1 +--+
a Let Let
---
Y =
1el x
em
(tm) em =n,
=
-..
L
H
except
a
ES
This proves:
0
h E Hn - Hn-l
For
h e H
if
0 := E[n] o dn-lh E SnL .
let
- H
K(h) =P 0oE
Then we have
and only if
n]o dn-l(h)
Kn
H-J/yJVK h
-+ K(h)
which is not linear. morphism we have,
But by
and since
0
is an iso-
100.
if
h E H
, g E H
and
K(h) - K(g) = 0 h -g
For a subspace K(X)
let
the subspace of
K(H) I K .
In general
Xn = X
Recall for a subspace K(Xn) = K(X) n (Kn a
Lemma 5:)
- --
0 + K(h)
fh
C Xn ci
n - 1 .
? 1K(h 1 ) +---
then it is clear
K0 )
X, Y
We show by induction
then
=
($
1
n Hn
K(X) = K(Y)
n = 0 , suppose true for
7 1(p o
K
Given subspaces
X 0 =YO ; then
Proof:
0
spanned by
K
S X.
(K(x)|x
X C Y,
and
H n-1*
E
X C H =
g e Hn
then
of
H
where X = Y.
implies
X D Yn . Let
True for
h e Yn -n-l'
+ ?mK(hm) ,
where
A
E k ,
Thus [n] o d nl(h )) + +
oo E[n] o d n-(?h
which implies
g = N h
+
+ Am(P o E[n] o dn-l(hM + Nmhm)
-
+
mhm
E Xn - Xn-1
and
101.
K(g) x
K(h)
.
- g E Y n-
=h
=
h
Observe if
=
X, Y
By so that
C X
Q.E.D.
x + g e X .
are both subalgebras, subcoalgebras, XO =
ideals or coideals the condition
is automatically
satisfied.
0)
If
x e Hn
[
o dn-l(X)
p
Y
where
Hq (xa)
a
1
a E
]o d
f
. 0 ( x ' m)
1
(o
0)
emae-
2
2e
a
e 1 (x
m
2
1
] o dn
E
n - 1 , then if
---
B*
is a right
B-module where
= If
x E L(B)
then
x
b* e B* , a, b E B
acts as a derivation of
a basis of monomials in
, within
xi
basis (not necessarily a basis for basis for
B
x aim
a i*]
Let
.
so
(x M)
=69
e .
E
a a * 2)
105.
= 0
by induction on the length of the monomial;
U C (H*p(J))
hence,
UO = (H*p(J))0 Letting f xa) K = k[ rx"I]
Let
V
fal
xe
k .
be an ordered basis for
L
Q
then by
b tD K(U)
=
HO0
m = 0,1,...
am p. --
a
p
m
0Then
Let
a
CJ
be a
1 =1
a* = al*
m
e
H*p(J)
and
=
t
106.
Xt = 0
which implies
and
K(h) E V .
Thus we are done
by lemma 5. We can draw the corollary that in characteristic p > 0
if
U = H
if and only if for each
H
is cocommutative and coconnected then
follows because Since
U
H*p(J) = 0
a*
H
generated by the primitives in
We shall show
is
J
a*P = 0 .
This
p(J) = 0
if and only if
may not equal
we examine the ideal H .
=a*p Ia*
LH = HL = p(H*)
E
e
LH = p(H*)
H*] HL = p(H*)
the proof
the same (up to reflection). HL c p(H*)
:
Let
=
2
act on
H*
from the right by
a* e H* ,
Then for
.
=
=
E L , h E H ,
0
acts as a derivation.
If fx
is an ordered basis for
L
K = k[ { xl] ,
K D W = the space spanned by
e 1 ) a (x
m 00(a
(x
m
=
1,2,...
a1 < - -< am for some
then
K(p(H*)
) C W .
Since if
e. p &ei
h E p(H*)
107.
K(h) =
Z
A (x
i = t
Suppose for
a
a1 eii
plet
m
C J
, then letting
...pet 1
a*
eim
)
m x i m
be a dual basis to
i=1 et
et
a* = a
1
*--x
implies
t
=
am* 0
= At
and
K(LH) D W n K(H) .
Suppose
K(h) s W , we shall show there is
K(g) = K(h) .
which
K(p(H*) ) C W .
and
Next we show
where
m e p(H*)
g E LH
h s H
- H
where
Let ei
K(h)
=
a M I A (xa) i=1 0
a
a*(x) = .)
1 < n(i)
_
o
xa f
1
o ds-2
a
@ - - - @ xa s-
Y
H+[S-1]
E
f +.-+f s-1 We show
= Ker e)
(H
Ker(E @ E o d) E
= H
Y e L
H+[s-2]
Since
.
it suffices to show
E 0 I[s-2] o d (
I[s-2](Y) = 0
or it suffices to show a
E @ E @ I[s-2] o d 0I[s-2]( Z Xa f > 0 i f +- -± --
t
E ® E ® I[s-2] o d 0 I[s-2]( xaf f i>0 1 f +- --
+ Y)
s-1 =
s
)
t.
The left hand side (top) =
E
E0I
[s2
o d
I[s-2] o E[Sl1
(since =
Els]
E0E od oE =EOEod) a
o ds-1(Yas)
Xfi
f i> 0 f +- . - - fs= t
a
which is the right hand side, (bottom). ing
Y E H
* .*
o ds-2 (as
L
- --.@ H+
so
By similar reason-
Y e L[s-1]
127.
Thus
~1
E
o dss-2]
as=
e.
a1
a
m
+ 2 Aa(x
z
f > 0 f1 +- -.+f
where
1 _
s-1
= t
e 11+- o*+
= S - 1
e
a1 p
r
qr =1
by the choice of an extending basis,
n(1 )+1
xa
Without loss we assume
so we can choose a dual basis ai*)m C J
where we assume
a
E
so
Since
ya
E
1
m
(Vna )+L -
(Vn(a)+1)
(a*) na
]
x
to
=
Ker p
)+1 = 0
let Vn-ItI(()
S(a*n
and
z E H
=
y .
)-n+|t|+
where
Let
128. eF -pnda)-n+ t|+1 a*= a1 *
1
I.
e1
*
am m
*
Note for
b* a H* ,
h e H
- h)
Vn(pn(b*)
m
b* - V"(h)
since
pn(b*)
= Consider
0 =
n(a 1 )-n+It|+1 =
=
X
a e
> --1-
k If H
H
is the
is finite
is connected H
is coconnected.
is cocommutative.
SupH*
is finite dimensional, connected, commutative; k
is split and
is perfect, then
is coconnected and
H*
cocommutative and has an extending basis by finile dimensionThus by theorem 6:
ality.
H* ~
as a coalgebra.
a
(in the notation of the theorem) a
m
By finite dimensionality
char p > 0
and
n(ai) < 00 . H*
has a basis of monomials as described in theorem 6.
Let {[a T**] C (H*)* for
H*
.
(a
to (x a
m=1
Let ta ,
be a dual basis to the monomial basis **
m
C
a
*
be the dual basis
(the extending basis for
L(H*).)
144.
k[x 1 ]
Then
(H*)*
n-l
k[Y ]
n~a p(
-@-m
)+1n(am)+1 3
p
as an algebra where an isomorphism is induced by -a1 aY **
By finite dimensionality
1=1,...0
m
H** = H
.
.
Thus we have proved
a finite dimensional commutative, connected, Hopf algebra over a perfect field where the dual is split is of the form:
(as an algebra) k[Y ]1
k[Y ] em
X1 >-2mp If
k
is algebraically closed then by the results of the
first chapter we can drop the condition that the dual is split since it is automatically satisfied, also
k
is
automatically perfect. We now conclude by applying results of previous chapters.
We show a coconnected cocommutative Hopf algebra
over a perfect field is an extension--as an algebra and a coalgebra--of
H/LH
by
U
characteristic zero
H = U
is trivial.
k
Assume
when and
U
is commutative.
LH = H+
In
so, the result
is perfect of characteristic
p > 0.
--4
145.
Lemma 6: and
K = k[
Let
be an ordered basis for
{ xa? If
xa ]I
where
h e V1
e
e =
K(h)
1 1
(a
A
a
m
i g
then there is
H
£
where pei
1 pe
a K(g)
=
Z 1
''
m
xm
and V(g)
V(f) = h .
Say
show we can choose choose
f
where
f
#(f)
K(f)
=
I
> pn
is minimal then
..
Pt
Suppose for some
be a dual basis t
i
,
f
)
C
x
Pt
We first
.
Suppose not
f
x1
S pn
+00+fit
pIf
90*
t x kf
k=1
a* = a*
t
Let
it
ak]
f
fj /p *.
*
1 /j
=
= = H
Observe
H/LH Dl k .
U C H
By
theorem 1 i
H/LH
(H
a
k H/LH
O
m )H
(D H
--
)H
H/LH is a linear isomorphism.
U-
is a split exact sequence. dition for
H
Hence,
U = H
H
H
®
0
k , and
H/LH
This verifies the first con-
to be an extension of
H/LH
by
U .
161.
Bibliography
[1]
E. Artin, C. Nesbitt, and R. Thrall, Rings with Minimum Condition, University of Michigan Press, 1944.
[2]
N. Jacobson,
[3]
J. Milnor and J. Moore, On the Structure of Hopf Algebras, Princeton University Notes.
Lie Algebras, Interscience, 1962.
162.
Index
Adjoint representation, Algebra, 8 111
A ,
commutative, 8 Antipode, 17 Augmentation, 10, 11 Brauer Group, 80 Coalgebra, 10 C.,
111
cocommutative, 11 Coconnected, 93 Cohomology, 47, 87
6, 48 6n
,
88
H(B,H)1 , 88
H' (H,B), 56 Hom(C,A)n, 88 Homn(C,A), 48 normal complex, 58 Reg(C,A)n
88
Regn (C,A), 48
155
r
163.
Coideal,
43
Comodule, 12 (C.)H.A. comodule, 87 comodule as an algebra, 36
$2,
structure on
X © X, 36
Connected, 143 Divided power, 122 Duality, 14 Extension, 61, 62, 89,
90, 155
f B 0) H, 67 Brauer Group, 80 H 9) g
B,
91
isomorphism of extensions, 62, 90 left exact sequence, 61 product of extensions, 81 right exact sequence, 89 smash product, 63 split exact sequence, 61, 89 Filtration, 26
r(G),
17
Grouplike element, 24 Hopf algebra, 16 coconnected, 93 conilpotent,
22
164.
filtration, 26 24
G(H), Hg,
25
L(H),
94
split, 21
U
95
,
J , 26 K(x),
98
L , 121 Module,
9
(C.)H.A. H
as
module, H*,
48
21
module as a coalgebra, 43 e3' -module, n 2
7
structure on
Poincare-Birkhoff-Witt
X 0 X, 42 theorem, 140
Primitive element, 94
p
,
113
Restricted Universal Enveloping Algebra, 93, 95 Smash product, 63 m He Tr(G) -+ H , 66 Symmetric tensor, 97 Universal Enveloping Algebra,
93, 95
165.
Unit, 8,
V V,
,
13
115 119
Vector space, 7 X , 110
166.
Biography
Moss Eisenberg Sweedler was born in Brooklyn, New York, April 29, 1942.
He was graduated from the Massachusetts
Institute of Technology with a B.S. in June 1963. tinued study there as a graduate student.
He con-
