AbstractR. Larson and M. Sweedler recently proved that for free finitely generated Hopf algebras H over a principal ideal domain R the following are equivalent: (a) H has an antipode and (b) H has a nonsingular left integral. In this paper I give a generalization of this result which needs only a minor restriction, which, for example, always holds if pic(R) = 0 for the base ring R. A finitely generated projective Hopf algebra H over R has an antipode if and only if H is a Frobenius algebra with a Frobenius homomorphism ψ such that Σ h(1) ψ(h(2)) = ψ(h) · 1 for all h ϵ H. We also show that the antipode is bijective and that the ideal of left integrals is a free rank 1, R-direct summand of H

Pareigis, Bodo

English

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Hopf Algebras,&quot;

Kategorien und Funktoren,&quot; Teubner,

When Hopf Algebras are Frobenius Algebras

Elsevier - Publisher Connector 

JOURNAL OF ALGEBRA 18, X3-596 (1971) 
When Hopf Algebras Are Frobenius Algebras 
BODO PAREIGIS 
Mathematisches Institut der UniversitZt Uiincheu 
8 A&incJaen 13, Schellingstrasse 2-8, Germany 
Communicated by A. Friihlich 
Received August 19, 1970 
R. Larson and M. Sweedler recently proved that for free finitely generated 
Hopf algebras H over a principal ideal domain R the following are equivalent: 
(a) H has an antipode and (b) H has a nonsingular left integral. In this paper 
I give a generalization of this result which needs only a minor restriction, 
which, for example, abvays holds if pit(R) = 0 for the base ring R. A finitely 
generated projective Hopf algebra H over R has an antipode if and only if H 
is a Frobenius algebra with a Frobenius homomorphism # such that 
C/z(r) #(&)) = $(h) . 1 for all h E H. We also show that the antipode is 
bijective and that the ideal of left integrals is a free rank 1, R-direct summand 
of H. 
I, In Ref. [4], Larson and Sweedler proved the equivalence for a 
finite-dimensional Hopf algebra over a principal ideal domain to have a 
(necessarily unique) antipode and to have a nonsingular left integral. It is 
easy to see that this result implies that a finite-dimensional Hopf algebra 
over a principal ideal domain is a Frobenius algebra, which generalizes the 
well-known fact that a group ring of a finite group is Frobenius as well as the 
result of Berkson [l], that the restricted universal enveloping algebra of a 
finite-dimensional restricted Lie algebra is Frobenius. This result has 
consequences with respect to a cohomology theory of Hopf algebras which 
will be exhibited in a subsequent paper. 
In this paper we want to generalize the main result of [4] to arbitrary 
commutative rings R and finitely generated projective Hopf algebras H. 
We need only a slight restriction on H or on R, viz., pit(R) = 0 to get the 
equivalence between the existence of an antipode and the fact that H 
is a Frobenius algebra with a Frobenius homomorphism 16, such that 
z(h) h(l)#(h(2)) = 4(h) . 1 for all h E H, where &) 1~) @ h(s) = d(h) is the 
Sweedler notation. We do not know whether the imposed restrictions on R 
or Hare necessary for the above result1 In this context we also prove that the 
r See footnote at the end. 
588 
WHEN HOPF ALGEBRAS ARE FROBENIUS ALGEBRAS 589 
antipode of a finitely generated projective Hopf algebra is bijective. This 
holds without any further restrictions. 
Since integrals are also of interest in this general situation, we shall prove 
that in a Hopf algebra Hwhich is Frobenius with a Frobenius homomorphism 
z/i such that Cch) h(,)z&,)) = 4(/z) . 1 f or all la E H the two-sided H ideal of 
left integrals is an R-free rank 1 R-direct summand of H. 
2. Let R be a commutative ring (associative with unit). All modules 
are assumed to be unitary R modules. All algebras are assumed to be associative 
R algebras with unit. 
A coalgebpa C is a module C together with homomorphisms d : C -+ C $3 C 
(the tensor product is taken over R), the diagonal, and E : C ---f R, the counit 
or augmentation, such that 
commute where we identify C @ R, C, and R @ C. We adopt the Sweedler 
notation a(c) = &, c(r) @ c(a) as explained in [7, p. IO]. 
A Hopf algebra H is an algebra H with structure maps p : H @ H -+ H 
and 7 : R + H, which is also a coalgebra with structure maps A : H -+ H @J H 
and E : H + R such that A and E are algebra homomorphisms. As in [7, 
Proposition 3.1. l] one shows that p and 7 are coalgebra homomorphisms. 
Let C be a coalgebra. A C right comodule is a module III together with a 
homomorphism x : M-h M Q C such that 
1 1834 and 
commute, where we identify M and M @) R. Here again we use the Sweedler 
notation x(m) = C(ln, m(,,) 0 m(,) . Observe that the m(,)‘s for i > 1 are 
elements in C, whereas the nq,)‘s are in M. 
Let H be a Hopf algebra. An H right Hopf module is an El right module AI 
which is also an H right comodule such that 
590 PAREIGIS 
Let H be a Hopf algebra. Then hom,(H, H) is an associative R algebra 
with unit ~6, if we define the multiplication by f*g : = ~(f @g)d, i.e., 
f*gO) = Col)f(kdg(bd 117, PO 709 exercise 11. The antipode S of H is 
(if it exists) the two-sided inverse of the identity lH of H in hom,(H, H). 
3. LEMMA 1. Let A, B, and C be R modules. If C is Jinitely 
generated and projective, then X : hom(./& B @ C) -+ hom(C* @ kl, B) with 
h(f)(c* @ a) := (1 @ c*)f (a) is an isomorphism. 
Proof. h defines a natural transformation 
hom(A, B @ -) -+ hom(- * 0 4 B), 
which is an isomorphism for C = R. By [6,4.11. Lemma 21 the above lemma 
holds. 
PROPOSITION 1. Let C be aJillitely generated projective R coalgebra nd M 
be an R module. Then x : M -+ M @ C dpfnes a C right comodule if and 
only if Ah) : C* @ M + M defines a C * left module, where X is deJLined as in 
Lemma 1. 
Proof. C* is an R algebra by c*d*(c) = &) c*(c(r)) d*(c(a)). Assume 
x : IkI -+ M @ C defines a C right comodule. Then 
and 
so AR is a C* left module. 
Now let M be a C* left module. The natural transformation 
p:MGJC@D-+hom(C*@D*,M) 
defined by ~(m @c @ d)(c* @ d*) = c*(c) d*(d)m is an isomorphism for 
finitely generated projective modules C and D, since it is an isomorphism for 
C=D=R [6,4.11.Lemma2].SoM~C~C~hom(C*~C*,M) 
for our finitely generated projective coalgebra C. 
WHEN HOPF ALGEBRAS ARE FROBENLDS ALGEBRQ 591 
Let .2^ = ((x @ 1)x - (1 @J d)x)(nz) E M @ C GJ C. Then p(xx)(cl* @ c2*) = 
cI*(cg*n2) - (cl*c2*)fn = 0 for all c 1*, cs* E C*, so x = 0, i.e., x is coasso- 
ciative. Furthermore, we have (1 @ <)x(e) = x:(,z) q,)~(q~~) = MX = m, 
since E is the unit element in C*. So NI is a C right comodule. 
PROPOSITION 2. Let H be a jkitely generated projective Hopf algebra with 
antipode S. Then H* is an H right Hopf module. 
Roof H* is an H* left module, so it is an H right comodule, We have 
for g*, iz* E H*, and lz E H and the comodule map x : El” -+ H* @ H with 
x(h*) = Cm h;, 0 h;, . 
g”h” = 1 h&g*@&) 
(h’b 
and 
g”h*‘@) = c g*(hd h*(b) = C h;,(h)g”(h&). (h) (h*) 
H* is also an H right module by h* . h = S(h) 0 h*, where (h 0 h*)(a) = 
h*(ah) for all a E H. For g*, h* E H*, and a, b E H and x : H* -+ H* @I H s 
hom(H*, H*), we have 
x(h* . a)(g*)(b) = (g*(h* . a))(b) 
= ($‘*, (P& . acl,)g*(h~,aizj))(b>. 
592 PAREIGIS 
This implies X(/Z* . a) = C(a)(hr) A& . a(,) @ h;kl,a(,) which proves the 
proposition. 
Let M and N be H right Hopf modules over a Hopf algebra H with antipode 
S. We define an R module P(M) = (m E M j x(m) = m @ 11. Letf : M + N 
be a module and comodule homomorphism. We define P(f) as restriction off 
to P(M). Then x(P(f)(m)) = x(f(m)) = (f @ 1)x(m) =f(m) @ 1 E P(N). 
Obviously P is a functor from the category of H right Hopf modules to the 
category of R modules. 
LEMMA 2. Let M be a Hopf module over a Hopf algebra H with antipode S. 
Then M g P(M) @ H as r&ht Hopf modules. Furthermore, P(M) is an R 
direct summand of M. 
Proof. The natural injection P(M) + M has a retraction iU3 m w 
C(m) %)shl)) E VW for 
Now a: : P(M) @ H--f M and p : M -+ P(M) @ H, defined by a(m @ h) = mh 
and P(m) = J&j ~m~,~S(wz~,~) @ tz(~) , are inverse R homomorphisms of each 
other. 01 being an H module homomorphism, 6 is an H module homo- 
morphism. Furthermore, #(m) = (p @ 1)x(m) implies that fl and, conse- 
quently, also ~11 is a comodule homomorphism. 
PROPOSITION 3. Let H be a JiTzitely genuated projective Hopf algebra zuith 
antipode S. Then P(H*) is a jinitely genuated projective rank 1 R module. 
Proof. For each prime ideal p in R the isomorphism H* E P(H*) @ H 
implies Hv* g P(H*), BRp HP . Now HP* g HP as free finite-dimensional 
R, modules. So dim(P(H*)& = 1 for all prime ideals p in R. Thus, P(H*) 
has rank 1. Furthermore, it is finitely generated projective as a direct summand 
of H”. 
4. An R algebra H is a Frobenius a&ebra if H is a finitely generated 
projective R module and if there is an isomorphism Cp : HH E HH*, where 
we consider H* as an H left module via h 0 h*(a) = h*(alz) for all a, h E H, 
h* E H* [2]. CD is called a Frobenizzs isomorphisnz. @( 1) ==: # is a free generator 
of H* as an H left module called Frobenius homomorphism. By [3, p. 220, (4)], 
# is also a free generator of H* as an H right module, where h* 0 h(a) : = h*(ha) 
for all h* E H*, and h, a E H. This is a consequence of the proof that the 
WHEN HOPF ALGEBRAS ARE FROBENIUS ALGEBR.AS 593 
conditions for a Frobenius algebra are independent of the choice of the sides, 
i.e., &l z &* implies HtI g liH* (if H is finitely generated and projective). 
# is unique up to multiplication with an invertible element of II [5, Satz 11. 
Tf a Frobenius algebra H has an augmentation E, then the element N with 
N o 4 = E is called a left fzorrn of H. A left norm N is also unique up to 
multiplication with an invertible element of H from the right side. We have 
((hN) 0 z&z) = $h(ahN) = (N 0 t&h) = E(Uh) = c(u)e(h) = (N 0 $j(u)e(h) = 
(E(Iz)N 0 $)(a) for all a, h E H. This implies 
hN = e(h)N for all h E N. 
An element a E H of an augmented algebra H with Jza = ma for all h E His 
called a left integral of the augmented algebra H. So a left norm is in particular 
a left integral. 
PROPOSKTION 4. Let H be u jkitely gemmted projective Hopf algebra zoith 
afztipode 5’. Then S is bijective. 
Proof. S is injective: Let f3 : H* z P(H*) @ H be the isomorphism of N 
right modules defined by Proposition 2 and Lemma 2. Let S(h) = 0 and 
p @ a E P(H*) @H, then 
6-Q @ ah) = 6F(p @ a) . h = S(h) 0 F(p @ a) = 0. 
So p @ alz = 0 for all p @ a E P(H*) @ Ii and the R endomorphism of 
P(H*) @ H, defined by the multiplication with lz, is the zero endomorphism. 
Let m be a maximal ideal of R. If we localize with respect to m, we get an 
R, Hopf algebra H, with antipode S, . Furthermore, (N*), s (H,,)*, 
where the second * means dualization with respect to R, I Also, P(M), g 
P(&&) for an H Hopf module M since P(M) is the kernel of x - lnl @ 17 
and localization is an exact functor. So (P(H*) (2~ H), s P((H,)*) 6JRn, H,. 
As in Proposition 3, P((H,,)*) is a free R, module on one generator so 
P((Hm)*) @jRm Hm z H,, . The multiplication of Hm with h on the right 
is a zero morphism for all maximal ideals m of R. So multiplication of H with 
R on the right must be zero which implies h = 0. So S is injective. 
S is surjective: Let 0 + H 5 H -+ Q + 0 be an R exact sequence. Then 
for all maximal ideals m C R, we have 0 + Hm -+ Hm -+ Q, + 0 is R,, exact. 
So Hm/mHm f Hm/~ttHm + Qm/mQm ---f 0 is R,,JmR, exact, where T is the 
antipode of the Rm/mRm Hopf algebra Hm/ntHm . Since this Hopf algebra is 
finitely generated, T is injective so that T is bijective and Qm/ntQm = 0. 
Since Q and Q, are finitely generated, Q, = 0 for all maximal ideals nt C R. 
So Q = 0 and S is surjective. 
THEOREM 1. Let H be a Jinitely generated projective Hopf algebra zciith 
594 PAREIGIS 
antipode S. Let P(H*) E R as R modules. Then H is a Frobenius algebra with 
a Frobenius homomorphism # such that 
c h&(h(,)) = #(h)l 
(h) 
for all h E H. 
Proof. By Proposition 2 and Lemma 2 there exists an isomorphism 
H* z P(H*) @H of H right modules where (h* * h)(a) = h*(aS(h)) or 
h” . Jz = S(h) 0 h * is the definition of the module structure on H*. Since 
P(H*) g R, let 0 : H* z H be an isomorphism of H right modules. Define 
CD = &?Y-1 : H gg H”, where we use Proposition 4. Then @(ha) = 
B-%!-l(ha) = &l(S-l(a)S-l(h)) = &l(S-l(a)) * S-l(h) = @(a) * S-l(h) = 
h 0 @(a), so H is a Frobenius algebra. Before we prove the formula on the 
Frobenius homomorphism we prove the following: 
LEMMA 3. Let H be a jinitely generated projective Hopf algPbra with 
antipode S with P(H*) g R. Let CD : H s H* be the Frobenius isomorphism 
constructed above a?zd let # = @(I) be a Frobemk homomorphism. Then 
# E P(H*) and # is a left integral in H*. 
Proof. @(l) = + implies se(#) = 1 and also O(#) = 1. 6 is a comodule 
homomorphism so C O(&,)) @ #(r) = (0 @ 1)x(Q) = d(B(#)) = d(1) = 
l@l=B(#)@l.B@lb’g em an isomorphism this implies x(1,5) = # @ 1 
so # E P(H*). Now 
(by UN 
for all h E II and h* E H*. This implies 
c h&(h(g)) = #(h)l 
(h) 
for all h E H. 
This also means (h*+)(h) = h*(l)+(h), which proves that zj is a left integral 
in H”. 
COROLLARY 1. Let R be a commutative ring with pit(R) = 0. Then each 
$nitely generated projective Hopf algebra with antipode is a Frobenius algebra. 
WHEN HOPF ALGEBRAS ARE FROBENIUS ALGEBRAS 595 
5, LEMMA 4. Let P be a finitely generated projective R module and 
let f : P -+ P be an epimorphism. Then f is an isomorphism. 
Proo$ The sequences 
are all split exact. But p is an isomorphism by reasorrs of dimension so 
A,/?nA, = 0. A, is finitely generated so that A, = 0 for all maximal 
ideals m C R, so that A = 0 andf is an isomorphism. 
THEOREM 2. Let H be a Hopf algebra and a Frobenius algebra with a 
Frobenius homomorphism Z/J such that &,) h&(h(r)) = #(h)l fGr all h E H. 
Then H has an antipode S. 
Proof. We define S : H -+ H by S(h) = x(N) N&(IIN~~~), where N is a 
left norm, i.e., No 4 = E. Then 
C bVd = c hw~wJ(hc~~~cd (h) (h)(N) 
= #(hN)l 
= c(h)1 
= Ah) 
for all h E H by definition of N. So 1 * S = ~6. 
Now hom,(H, H) is an associative R algebra with multiplication j*g = 
~(f @ g)4. Hom,(H, H) is also finitely generated and projective since H is. 
The map 
hom,(H, H) sf~f* SE hom,(H, H) 
is an R epimorphism for (f*l)*S=f*(l*S)=f~77~=f. By the 
preceeding lemma, - * S is an isomorphism with inverse map - * 1. So 
S * 1 = 7~ * S si: 1 = TE, i.e., S is an antipode. 
THEOREM 3. Let H be a Hopf atgebra and a Frobenius algebra. Then the 
two-sided ideal of left integrals h E H (with ah = c(a)h for all a E H) is a free 
R direct summand of H of rank 1 with basis {N), a left norm qf El. 
Proof. Let h be a left integral, then #(ah) = #(c(a)h) = #(h)E(a) = 
#(h)#(aN) = #(a#(h)N) for all a E H, so that 12 0 $J = #(h)N o # or h = #(h)N. 
596 PAREIGIS 
Furthermore, E is an epimorphism since ET = 1, . So E* : R --f H* is a 
monomorphism. Also p : R + H* z H is a monomorphism. But ~*(r)(h) = 
G”(l)@) = rr(h) = C/+N), so that p(r) = rN. Since p is injective, 
R 3 Y t-’ rN E H is injective. Thus, RN is free of rank 1. 
Finally, H 3 12 tt $(h)N E RN is a retraction for the inclusion RN C H, 
in which case RN splits off as a direct summand. 
By Theorem 1, 9 is a left integral and H 3 k F+ h 0 # E H* as well as 
H 3 h H Z/J 0 h E H* are isomorphisms, so that + is a nonsingular left integral 
in H*. Now let pit(R) = 0. If H is a finitely generated projective Hopf 
algebra with an antipode, then so is H*. Furthermore, H** g H as Hopf 
algebras with antipode and also H has a nonsingular left integral. This 
implies the main result of (see remark on page 588) [4] for the case that R 
is a principal ideal domain, for then pit(R) = 0 holds. 
Note added iz proof. P. Gabriel gave an example of a finitely generated projective 
Hopf algebra with antipode which is not a Frobenius algebra, showing that pit(R) = 0 
is a necessary condition for Corollary 1. We have however the following 
Theorem. Let H be a finitely generated projective Hopf algebra over a commutative 
ring R. H has an antipode if and only if H is a quasi Frobenius algebra and H* has 
a finite set of generators #r ,..., $R as an H left module (h 0 h*(a) = k*(ah)) such 
that for all K = l,..., it 
1 h&,(h(z)) = &r,(h)1 
(9 
for all k E H. 
Here a quasi Frobenius algebra is taken in the sense of B. Mtiller, Quasi-Frobenius- 
Erweiterungen, M&h. Zeitschr. 85, 345-368 (1964). This theorem guarantees that 
the cohomology theory of Hopf algebras can be developed without the restrictive 
condition pit(R) = 0. 
REFERENCES 
1. A. BERKSON, Proc. Amr. Muth. Sot. 15 (1964), 14, 15. 
2. P. KASCH, Sitzungsber. Heidelberg. Akad. Wiss. (1960/61), 89-109. 
3. F. KASCH, Math. Z. 77 (1961), 219-227. 
4. R. G. LARSON AND M. E. SWEEDLER, Amer. J. Math. 91 (1969), 75-94. 
5. B. PAEXEIGIS, Ann. Math. 153 (1964), 1-13. 
6. B. PAREIGIS, “Kategorien und Funktoren,” Teubner, Stuttgart, 1969. 
7. hf. E. SWEEDLER, “Hopf Algebras,” W. A. Benjamin Inc., New York, 1969. 


When Hopf algebras are Frobenius algebras 

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When Hopf Algebras are Frobenius Algebras

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