Homology theory valued in the category of bicommutative Hopf algebras
aa r X i v : . [ m a t h . A T ] M a y HOMOLOGY THEORY VALUED IN THE CATEGORY OF BICOMMUTATIVEHOPF ALGEBRAS
MINKYU KIMA bstract . The codomain category of a generalized homology theory is the category of mod-ules over a ring. For an abelian category A , an A -valued (generalized) homology theory isdefined by formally replacing the category of modules with the category A . It is known thatthe category of bicommutative (i.e. commutative and cocommutative) Hopf algebras over afield k is an abelian category. Denote the category by Hopf bicom k . In this paper, we give someways to construct Hopf bicom k -valued homology theories. As a main result, we give Hopf bicom k -valued homology theories whose coe ffi cients are neither group Hopf algebras nor functionHopf algebras. The examples contain not only ordinary homology theories but also extraor-dinary ones. C ontents Acknowledgement 21. Bicommutative Hopf algebras 21.1. Group Hopf algebras and function Hopf algebras 21.2. Nontrivial bisemisimple bicommutative Hopf algebras 42.
Hopf bicom k -valued homology theory 52.1. Construction by the group Hopf algebra functor 62.2. Construction of ordinary homology theory 62.3. Construction by exponential functors 7Appendix A. A bicommutative bisemisimple nontrivial Hopf algebra with dimensionprime 9References 10In a functorial point of view, the codomain of a generalized homology theory is the cate-gory of abelian groups, Ab , or more generally the category of modules over a ring R , Mod R .For an abelian category A , an A -valued (generalized) homology theory is defined by formallyreplacing the category of modules with the category A (see Definition 2.1). With this termi-nology, a generalized homology theory is an Ab -valued homology theory or Mod R -valuedhomology theory.The category of bicommutative Hopf algebras over a field k is an abelian category [6][7]. Denote the category by Hopf bicom k . The main purpose of this paper is to give a way toconstruct Hopf bicom k -valued homology theories which are nontrivial in the following sense.A group induces a Hopf algebra called a group Hopf algebra. Furthermore, a finite groupinduces a Hopf algebra called a function Hopf algebra. A Hopf bicom k -valued homology theoryis trivial if each coe ffi cient Hopf algebra is a group Hopf algebra or a function Hopf algebra. Theorem 0.1.
For a bicommutative Hopf algebra A over a field k, there exists an ordinaryhomology theory E ‚ such that E p pt q – A. Denote the ordinary homology theory by E ‚ p´q “ H ‚ p´ ; A q . If the Hopf algebra A is nota group Hopf algebra or a function Hopf algebra, the obtained ordinary homology theory is nontrivial . The existence of such a Hopf algebra A depends on the ground field k . Someexamples are given in subsection 1.2 and appendix.On the one hand, we give a way to construct a Hopf bicom k -valued homology theory which isnot only nontrivial and possibly extraordinary . Theorem 0.2.
Let L be a field. Consider a bicommutative Hopf algebra A over k with anL-action α . There exists an assignment of a Hopf bicom k -valued homology theory p A , α q F ‚ toa Vec fin L -valued homology theory F ‚ . If dim k A ą , then the homology theory p A , α q F ‚ isordinary if and only if so is F ‚ . In our subsequent paper, we show that a
Hopf bicom k -valued homology theory induces a (pos-sibly, empty) family of TQFT’s. This paper provides a class of examples to apply the results.Especially, this paper makes sure that our subsequent result is a strict generalization of abelianDijkgraaf-Witten [2] [3] and bicommutative Turaev-Viro [9] [1].For applications in our subsequent paper, we go further. We give a way to construct anontrivial Hopf bicom k -valued homology which assigns finite-dimensional bisemisimple Hopfalgebras to any spaces. See Definition 1.5 for bisemsimple Hopf algebras. The existence ofsuch homology theories depends on the ground field k . Our main examples contain k “ Q , R and finite fields. On the other hand, for the case k “ C (more generally, algebraically closedfield), all the Hopf bicom k -valued homology theories are trivial in the above sense.In section 1, we give examples of bicommutative Hopf algebras in a systematic way. Insubsection 1.1, we give Hopf algebras induced by a group. In particular, we compare the cate-gory of abelian groups and the category of finite-dimensional (co)semisimple bicommutativeHopf algebras. In subsection 1.2, we discuss a way to obtain bicommutative Hopf algebraswhich are not induced by groups as above. Moreover, we give some conditions that suchHopf algebras be bisemisimple for the purpose of our subsequent paper.In section 2, we give three ways to construct Hopf bicom k -valued homology theories. In sub-section 2.1, we use the group Hopf algebra functor so that the obtained homology theoriesare trivial in the above sense. In subsection 2.2, we construct an ordinary homology theorywith coe ffi cients in an arbitrary bicommutative Hopf algebra. In subsection 2.3, we use anexponential functor induced by a bicommutative Hopf algebra equipped with an action of afield. We show that a bicommutative Hopf algebra A over k with an L -action gives a wayto construct Hopf bicom k -valued homology theory from a Vec fin L -valued homology theory where L is another field. The examples in subsection 2.3 contain some extraordinary homologytheories which are nontrivial . A cknowledgement Takeuchi (University of Tokyo) informed the author of the existence of nontrivial Hopfalgebra structures on F ‘ F and F ‘ F ‘ F based on a commutative algebra analogueof the Gelfand duality. The author had a helpful discussion with Wakatsuki (University ofTokyo) about the nontriviality of Hopf algebra. The author appreciates them all.1. B icommutative H opf algebras Group Hopf algebras and function Hopf algebras.
Let k be a field. For a finite group G , we denote by kG the group Hopf algebra : the algebra and the coalgebra structures areinduced by the group structure and the diagonal map respectively. Dually, we denote by k G the function Hopf algebra. It is obvious that G is an abelian group if and only if the inducedgroup Hopf algebra and the function Hopf algebra are bicommutative. OMOLOGY THEORY VALUED IN THE CATEGORY OF BICOMMUTATIVE HOPF ALGEBRAS 3
The previous constructions are functorial. They give covariant and contravariant functorsfrom the category of finite groups to the category of finite-dimensional Hopf algebras respec-tively : k p´q : Grp fin Ñ Hopf cocom , fin k (1) k p´q : ` Grp fin ˘ op Ñ Hopf com , fin k (2) Proposition 1.1.
The functors (1) and (2) are fully faithful.Proof.
It su ffi ces to show the functor (1) is fully faithful. Since the functor k p´q is faithfulby definitions, we prove that it is full. Let G , H be finite groups. Let ξ : kG Ñ kH be amorphism in Hopf cocom , fin k , i.e. a Hopf homomorphism. For an element g P G Ă kG and h P H Ă kH , we define λ g , h by ξ p g q “ ÿ h P H λ g , h ¨ h . (3)Since the homomorphism ξ preserves comultiplications, for any g P G we obtain, ÿ h P H λ g , h ¨ h b h “ ÿ h , h P H λ g , h ¨ λ g , h ¨ p h b h q . (4)It implies λ g , h “ λ g , h and λ g , h ¨ λ g , h “ h ‰ h . Since the homomorphism ξ preservescounits, for any g P G we have 1 “ ř h P H λ g , h . Hence, for each g P G there exists h P H suchthat λ g , h ‰
0. For such h P H , the above condition λ g , h “ λ g , h implies λ g , h “
1. Moreover, λ g , h “ λ g , h ¨ λ g , h “ h ‰ h implies that for each g P G there exists a unique h P H suchthat λ g , h “ λ g , h “ h ‰ h . Denote such h by ̺ p g q . Then by definition, we have ξ p g q “ ̺ p g q P H Ă kH . Since ξ preserves algebra structures, one can verify that ̺ : G Ñ H preserves groups structures. Above all, ̺ : G Ñ H is a morphism in Grp fin such that k ̺ “ ξ .It completes the proof. (cid:3) Proposition 1.2.
Every finite-dimensional function Hopf algebra is commutative and semisim-ple.Proof.
A finite-dimensional function Hopf algebra k G is commutative by definitions. Notethat the Hopf algebra k G has a normalized integral. By [5], it is semisimple. (cid:3) Equivalently, every finite-dimensional group Hopf algebra is cocommutative and cosemisim-ple. The functors (1), (2) factor as follows. k p´q : Grp fin Ñ Hopf cocom , fin , semisim k (5) k p´q : ` Grp fin ˘ op Ñ Hopf com , fin , cosemisim k (6)The converse of Proposition 1.2 is true if we require some conditions on the ground field k . We give a su ffi cient condition for the ground field k and a finite-dimensional commutativeHopf algebra over k to be isomorphic to a function Hopf algebra [4]. Proposition 1.3.
Suppose that the field k is algebraically closed. Then every finite-dimensionalsemisimple commutative Hopf algebra over k is isomorphic to a function Hopf algebra. Inparticular, we obtain a category equivalence ` Grp fin ˘ op » Hopf com , fin , semisim k .The analogous statement for group Hopf algebras holds : Grp fin » Hopf cocom , fin , cosemisim k .Proof. Let k be an algebraically closed field. Let A be a semisimple commutative algebraover k . We apply the Wedderburn’s theorem (here, we use the assumption of k and thesemisimplicity of A ) to A to obtain an algebra isomorphism A – À i M n i p k q . Since A is MINKYU KIM a commutative algebra, the natural numbers n i should be 1. Hence, we obtain an algebraisomorphism A – À i k .We compute group-like elements in the dual Hopf algebra A _ . By definitions, group-likeelements of A are given by algebra homomorphisms from A to the unit algebra k . Under theabove isomorphism A – À i k , such algebra homomorphisms from A to k corresponds to theprojections À i k Ñ k one-to-one. Hence, the order of the group-like elements of A _ is theorder of such projections, i.e. dim A . Above all, the Hopf algebra A coincides with the Hopfalgebra generated by its group-like elements. It completes the proof. (cid:3) Corollary 1.4.
If the field k is algebraically closed, then every finite-dimensional semisimplecommutative Hopf algebra over k is isomorphic to a function Hopf algebra. In particular, weobtain a category equivalence ` Ab fin ˘ op » Hopf bicom , fin , semisim k .The analogous statement for group Hopf algebras holds : Ab fin » Hopf bicom , fin , cosemisim k . Nontrivial bisemisimple bicommutative Hopf algebras.Definition 1.5.
A finite-dimensional Hopf algebra is bisemisimple if the underlying algebrais semisimple and the dual algebra of its underlying coalgebra is semisimple.
Definition 1.6.
A Hopf algebra is trivial if it is a group Hopf algebra or a function Hopfalgebra.The semisimplicity or the cosemisimplicity is a necessary condition for a finite-dimensionalbicommutative Hopf algebra to be trivial. Moreover, if the ground field is algebraicallyclosed, then it is a su ffi cient condition by Corollary 1.4.Denote by GE p A q the group consisting of group-like elements in the Hopf algebra A : GE p A q def . “ t ‰ v P A ; ∆ p v q “ v b v u . (7)Then we have a natural isomorphism GE p A b B q – GE p A q ˆ GE p B q . The group Hopfalgebra kGE p A q is naturally a sub Hopf algebra of A . It is obvious that the Hopf algebra A is a group Hopf algebra if and only if the sub Hopf algebra kGE p A q coincides with A .Especially, a finite-dimensional Hopf algebra A is trivial if and only if | GE p A q| ă dim k A and | GE p A _ q| ă dim k A . Definition 1.7.
Let A be a finite-dimensional bicommutative Hopf algebra. For the dual Hopfalgebra A _ , we define D p A q def . “ A b A _ by the tensor product of Hopf algebras. Proposition 1.8. (1)
If the Hopf algebra A is not a group Hopf algebra, then the Hopfalgebra D p A q is not trivial. (2) If the Hopf algebra A is bisemisimple, then the Hopf algebra D p A q is bisemisimple.Proof. We prove the first part. If A is not a group Hopf algebra, then | GE p A q| ă dim k A . Weobtain | GE p D p A qq| “ | GE p A q|¨| GE p A _ q| “ă dim k A ¨ dim k A _ “ dim k D p A q . Since we havean isomorphism of Hopf algebras D p A q _ – D p A q , we obtain | GE p D p A q _ q| ă dim k D p A q .Hence, D p A q is neither a group Hopf algebra nor a function Hopf algebra.We prove the second part. Suppose that A is bisemisimple. Recall that a finite-dimensionalHopf algebra is (co)semisimple if and only if it has a normalized (co)integral [5]. Thenthere exists a normalized integral and a normalized cointegral of A . A tensor product ofthem induces a normalized integral a normalized integral and a normalized cointegral of D p A q “ A b A _ . Hence, D p A q is bisemisimple. (cid:3) The following corollaries are immediate from Proposition 1.8.
Corollary 1.9.
Let k be a field. If there exists a finite-dimensional bicommutative (bisemisim-ple) Hopf algebra over k which is not a group Hopf algebra, then there exists a nontrivialfinite-dimensional bicommutative (bisemisimple) Hopf algebra.
OMOLOGY THEORY VALUED IN THE CATEGORY OF BICOMMUTATIVE HOPF ALGEBRAS 5
Corollary 1.10.
Let k be a field. Suppose that for an integer n, the number of n-th roots ofthe unit P k is lower than n. Then the Hopf algebra D p k Z { n q is not trivial. Furthermore, ifthe integer n is coprime to the characteristic of k, then it is bisemisimple. Example 1.11.
The real field R , the rational field Q and any finite field could be consideredas the field k having the integer n in Corollary 1.10. Remark 1.12.
The dimension of nontrivial Hopf algebras obtained by Proposition 1.8 is acomposite number since dim k D p A q “ p dim k A q . It depends on the field k whether there ex-ists a finite-dimensional bicommutative (bisemisimple) nontrivial Hopf algebra whose dimen-sion is prime. For the case k “ R , we have a negative answer. Let A be a finite-dimensionalbicommutative Hopf algebra over R with dim R A prime. If A is semisimple, then the Hopfalgebra A is a group Hopf algebra or a function Hopf algebra, i.e. trivial. On the other hand,for k “ F , we have a positive answer. A concrete example is given in the appendix. Hopf bicom k - valued homology theory Definition 2.1.
Let A be an abelian category. An A -valued homology theory E ‚ “ t E q , B q u q P Z is given by the following data :(1) For each integer q , the data E q consist of two assignments : The first one assigns anobject E q p K , K q of A to a pair of finite CW-spaces p K , K q . The second one assignsa morphism E q p f q : E q p K , K q Ñ E q p L , L q to a map f : p K , K q Ñ p L , L q where theobjects E q p K , K q , E q p L , L q are the corresponding objects by the first assignment.(2) For each pair of finite CW-spaces p K , K q , the data B q is a natural transformation B q : E q ` p K , K q Ñ E q p K q in A where E q p K q is a shorthand for E q p K , Hq .which are subject to following conditions :(1) The assignments satisfies a functoriality : E q p Id K , K q “ Id E q p K , K q , E q p g ˝ f q “ E q p g q ˝ E q p f q . Here, Id K , K is the identity on p K , K q , and the maps g , f are composable.(2) Let f , g be maps from p K , K q to p L , L q . A homotopy f » g induces E q p f q “ E q p g q . (3) A triple of finite CW-spaces p X , K , L q induces an isomorphism E q p K , K X L q Ñ E q p K Y L , L q .(4) A pair p K , K q induces a long exact sequence in A where i , j are inclusions : ¨ ¨ ¨ Ñ E q p K q E q p i q Ñ E q p K q E q p j q Ñ E q p K , K q B q ´ Ñ E q ´ p K q Ñ ¨ ¨ ¨ . The corresponding object E q p pt q to the pointed 0-sphere is called the q-th coe ffi cient of ho-mology theory. An A -valued homology theory is ordinary if any q -th coe ffi cient is isomor-phic to the zero object of A for q ‰
0. An A -valued homology theory is extraordinary if it isnot ordinary. Example 2.2.
Consider A “ Ab the category of abelian groups. Then an Ab -valued homol-ogy theory is nothing but a generalized homology theory (defined on finite CW-spaces). Moregenerally, one can consider Mod R , the category of R-modules for a ring R. MINKYU KIM
Construction by the group Hopf algebra functor.
Recall the group Hopf algebrafunctor, k p´q : Ab Ñ Hopf bicom k ; G ÞÑ kG . (8)We give a way to construct a Hopf bicom k -valued homology theory starting from an Ab -valuedhomology theory (equivalently, a generalized homology theory). Definition 2.3.
Let F ‚ be an Ab -valued homology theory. Note that the group Hopf algebrafunctor (8) is an exact functor. By composing the functor (8), the homology theory F ‚ inducesa Hopf bicom k -valued homology theory denoted by kF ‚ .A Hopf bicom k -valued homology theory E ‚ is induced by the group Hopf algebra functor ifthere exists an Ab -valued homology theory F ‚ such that E ‚ D – kF ‚ . Proposition 2.4.
Suppose that the ground field k is algebraically closed. Then a
Hopf bicom k -valued homology theory E ‚ satisfying the following conditions is induced by the group Hopfalgebra functor : for any pair of finite CW-spaces p K , K q , (1) the Hopf algebra E q p K , K q is finite-dimensional. (2) the Hopf algebra E q p K , K q is cosemisimple, i.e. its dual Hopf algebra is semisimple.Proof. We sketch the proof. By Proposition 1.3, there is a natural isomorphism E q p K , K q – kF q p K , K q where F q p K , K q consists of group-like elements of the Hopf algebra E q p K , K q .Moreover the Hopf homomorphism E q p f q corresponding to a continuous map f lift to grouphomomorphisms, denoted by F q p f q , by Proposition 1.1. Analogously, so does the boundaryhomomorphism B q : E q ` p K , K q Ñ E q p K q . The obtained data F ‚ “ t F q , B q u give an Ab -valued homology theory by Proposition 1.1 again. (cid:3) Analogously, the function Hopf algebra functor ` Ab fin ˘ op Ñ Hopf bicom k ; G Ñ k G is alsoan exact functor so that an Ab fin -valued cohomology theory (equivalently, a generalized co-homology theory whose coe ffi cients are finite groups) induces a Hopf bicom k -valued homologytheory. We have a proposition analogous to Proposition 2.4.2.2. Construction of ordinary homology theory.
The ordinary homology theory with co-e ffi cients in an abelian group could be obtained from the chain complex related with theCW-complex structure. Analogously an ordinary Hopf bicom k -valued homology theory is con-structed. More generally, we give a way to construct an ordinary A -valued homology theoryfor any abelian category A .Let A be an object of the category A . For a CW-space K , choose a CW-complex structureof K . For a nonnegative integer q , we define an object C cellq p K ; A q of A by C cellq p K ; A q “ A ‘ n q ,i.e. the n q -times direct sum of A where n q is the number of q -cells. Denote by r e q : e q ´ s P Z the incidence number of a q -cell e q and a p q ´ q -cell e q ´ of K . Consider a morphism B q : C cellq p K ; A q Ñ C cellq ´ p K ; A q whose p e q ´ , e q q -component of the matrix representation is r e q : e q ´ s ¨ Id A . Then the data of C cell ‚ p K ; A q “ t C cellq p K ; A q , B q u q P Z form a chain complex in A since B q ˝ B q ` “ C cellq p K ; A q “ q ă p K , K q , a chain complex C cell ‚ p K , K ; A q is defined as thecokernel of the inclusion C cell ‚ p K ; A q Ñ C cell ‚ p K ; A q . Let H cellq p K , K ; A q be the q -th homol-ogy theory of the chain complex C cell ‚ p K , K ; A q . It is formally a classical result that theobject H cellq p K , K ; A q does not depend on the choice of the CW-complex structure on the pair p K , K q (up to an isomorphism). We define H q p K , K ; A q def . “ H cellq p K , K ; A q . The assignment p K , K q ÞÑ H q p K , K ; A q induces an A -valued homology theory. It is ordinary by definition.We apply the above construction to the category of bicommutative Hopf algebras A “ Hopf bicom k . Then we obtain an ordinary homology theory with coe ffi cients in a bicommutativeHopf algebra A . OMOLOGY THEORY VALUED IN THE CATEGORY OF BICOMMUTATIVE HOPF ALGEBRAS 7
Proposition 2.5.
Let G be an abelian group and H ‚ p´ ; G q the ordinary homology theorywith coe ffi cients in the abelian group G. Let kH ‚ p´ ; G q be a Hopf bicom k -valued homologytheory induced by the group Hopf algebra functor (see subsection 2.1). Then we have anisomorphism of homology theories,H ‚ p´ ; kG q – kH ‚ p´ ; G q . (9) Remark 2.6.
By subsection 1.2, there are nontrivial Hopf algebras for an appropriate fieldk. For such a field k, there is a
Hopf bicom k -valued ordinary homology theory which could notbe obtained from subsection 2.1. Construction by exponential functors.Definition 2.7.
Let A be a bicommutative Hopf algebra. We define a unital ring End p A q asfollows. Its underlying set is given by the set of Hopf endomorphisms on A . Its abelian groupstructure is defined by the convolution ˚ of Hopf endomorphisms : for Hopf endomorphisms f , g on A , we define the convolution f ˚ g , which is also a Hopf endomorphism, by f ˚ g def . “ ∇ A ˝ p f b g q ˝ ∆ A . (10)The multiplication on End p A q is the composition of endomorphisms. The identity on A is theunit of the ring. Definition 2.8.
For a ring R , an R-action on a bicommutative Hopf algebra A is a ring homo-morphism α from R to End p A q . Proposition 2.9.
Denote by P R the category of finitely-generated projective modules over Rfor a ring R. The category of bicommutative Hopf algebras with an R-action is equivalentwith the category of symmetric monoidal functors from p P R , ‘q to p Vec k , bq .Proof. It follows from [8]. (cid:3)
Remark 2.10.
The symmetric monoidal functor in Proposition 2.9 is called an exponentialfunctor.
Definition 2.11.
Let L be another field. Let A be a bicommutative Hopf algebra over k with an L -action α . Denote by J : p Vec fin L , ‘q Ñ p Vec k , bq the induced symmetric monoidal functorin Proposition 2.9. We denote by p A , α q p´q : Vec fin L Ñ Hopf bicom k the additive functor inducedby J . In fact, a bicommutative Hopf monoid in p Vec fin L , ‘q is nothing but an object of Vec fin L since the direct sum ‘ is a biproduct in the category Vec fin L . Hence, the symmetric monoidalfunctor J assigns a bicommutative Hopf monoid in p Vec k , bq to each object of Vec fin L in afunctorial way. Proposition 2.12. (1)
The additive functor p A , α q p´q is an exact functor. (2) Let F ‚ be a Vec fin L -valued homology theory. The functor p A , α q p´q induces a Hopf bicom k -valued homology theory p A , α q F ‚ by composing the functor p A , α q p´q .Proof. The second part is immediate from the first part. We sketch the proof of the first part.Note that any short exact sequence in A “ Vec fin L splits. In general, for an arbitrary abeliancategory A where any short exact sequence splits, an additive functor F : p A , ‘q Ñ p B , ‘q is an exact functor for any abelian category B . (cid:3) Proposition 2.13.
Suppose that dim k A ą . For a Vec fin L -valued homology theory F ‚ , the Hopf bicom k -valued homology theory p A , α q F ‚ is ordinary if and only if F ‚ is ordinary.Proof. We sketch the proof. Note that we have a natural isomorphism p A , α q L ‘¨¨¨‘ L – A b¨ ¨ ¨ b A . Hence, F q p pt q is the zero vector space if and only if p A , α q F q p pt q is a one-dimensionalHopf algebra since dim k A ą (cid:3) MINKYU KIM
We give a typical example of a bicommutative Hopf algebra with an L -action for L “ F p for a prime number p . We compute the ring End p A q for some A . Lemma 2.14.
Let G be a finite abelian group. For the group Hopf algebra A “ kG, the ringEnd p A q is isomorphic to the ring consisting of group endomorphisms on G. We denote thelatter one by End p G q .The analogous statement is true. For the function Hopf algebra A “ k G , the ring End p A q is isomorphic to the opposite ring of End p G q .Proof. It is immediate from Proposition 1.1, (cid:3)
Proposition 2.15.
Suppose that a finite-dimensional bicommutative Hopf algebra A is cosemisim-ple (resp. semisimple). Then there exists a finite abelian group G such that the order | G | co-incides with the dimension dim k A and the unital ring End p A q is a subring of End p G q (resp.the opposite ring of End p G q ).Proof. We consider the case that A is cosemisimple. Let ¯ k be the algebraic closure of thefield k . Denote by ¯ k b k A the Hopf algebra over ¯ k obtained by coe ffi cient extension. Then theextension of coe ffi cients induces an injection End p A q Ñ End p ¯ k b k A q . The field extensionis also cosemisimple since it has a normalized cointegral which is obtained by the coe ffi cientextension of that of A . By Proposition 1.3, the Hopf algebra ¯ k b k A is a function Hopf algebra¯ k G for a finite abelian group G . We have an isomorphism of rings End p ¯ k b k A q – End p G q byLemma 2.14. It proves the claim. (cid:3) Corollary 2.16.
Suppose that a finite-dimensional bicommutative Hopf algebra A is cosemisim-ple or semisimple. If the dimension of A is a prime number p, then A has a canonical non-trivial F p -action.Proof. The ring
End p A q is a finite field with order p by Proposition 2.15. (cid:3) Example 2.17.
Consider k “ R . Let A “ D p R p Z { q qq for an odd prime q. We have F q ˆ F q – End p R p Z { q qq ˆ End p R Z { q q Ă End p R p Z { q q b R Z { q q “ End p A q . The field L “ F q diagonallyacts on A via this inclusion. By A “ D p R p Z { q qq , any Vec fin F q -valued homology theory inducesa Hopf bicom R -valued homology theory. The Hopf algebra A is neither a group Hopf algebranor a function Hopf algebra by Corollary 1.10. Thus the induced Hopf bicom R -valued homologytheory has coe ffi cients which are neither group Hopf algebra nor a function Hopf algebra. Example 2.18.
Consider an arbitrary bicommutative Hopf algebra A in Corollary 2.16. Thenwe obtain an assignment of a
Hopf bicom k -valued homology theory E ‚ “ A F ‚ to a Vec fin F p -valuedhomology theory F ‚ . By Proposition 2.13, if F ‚ is extraordinary, then the induced homologytheory E ‚ is extraordinary. Especially, if A is a nontrivial Hopf algebra, then E ‚ is an extra-ordinary homology theory whose coe ffi cients are neither a group Hopf algebra functor nor afunction Hopf algebra.For example, consider k “ F . Let A “ D , D in the appendix. Since the dimension ofA is p “ , any Vec fin L -valued homology theory induces a Hopf bicom k -valued homology theorywhere L “ F and k “ F . Remark 2.19.
For special case of p A , α q , the induced homology theory is naturally isomor-phic to those given by previous subsections. Let A “ k p Z { p q with the canonical F p -action α p .Then we have isomorphisms of Hopf bicom k -valued homology theories : p k p Z { p q , α p q F ‚ – kF ‚ , (11) p k p Z { p q , α p q H ‚ p´ ; Z { p q – H ‚ p´ ; k p Z { p qq (12) OMOLOGY THEORY VALUED IN THE CATEGORY OF BICOMMUTATIVE HOPF ALGEBRAS 9
Here, F ‚ is an arbitrary Vec fin F p -valued homology theory. H ‚ p´ ; Z { p q and H ‚ p´ ; k p Z { p qq denote the singular homology theories with coe ffi cient in the abelian group Z { p and the Hopfalgebra k p Z { p q respectively. A. A bicommutative bisemisimple nontrivial H opf algebra with dimension prime We give bicommutative Hopf algebras over k “ F , D “ D , D , whose underlying algebrais the direct sum algebra F ‘ F ‘ F . Here, F “ F r ω s{p ω ` q is a field with order 9. Itsatisfies the following properties : ‚ The Hopf algebra D is not trivial. ‚ The Hopf algebra D is semisimple. ‚ The Hopf algebra D is cosemisimple.For simplicity, we introduce following notations.(1) a “ p , , q P F ‘ F ‘ F (2) b “ p , , q , b “ p , ω, q P F ‘ F ‘ F (3) c “ p , , q , c “ p , , ω q P F ‘ F ‘ F We give a bicommutative Hopf algebra structure D on the algebra F ‘ F ‘ F over F . Viathe Z { Z -action exchanging b , b and c , c respectively, the structure D induces anotherbicommutative Hopf algebra structure D on the algebra F ‘ F ‘ F over F . By definitionof D , the Hopf algebras D , D are isomorphic to each other, but no the same. Moreover, thealgebra F ‘ F ‘ F over F has only two bicommutative Hopf algebra structures, which are D , D .The structure D is given as follows. comultiplication. (1) ∆ p a q “ a b a ` b b b ` b b b ` c b c ` c b c .(2) ∆ p b q “ p a b b ` b b a q`p b b c ` c b b q`p b b c ` c b b q` c b c ` c b c .(3) ∆ p b q “ p a b b ` b b a q`p b b c ` c b b q`p b b c ` c b b q`p c b c ` c b c q .(4) ∆ p c q “ p a b c ` c b a q` b b b ` b b b `p b b c ` c b b q`p b b c ` c b b q .(5) ∆ p c q “ p a b c ` c b a q`p b b b ` b b b q`p b b c ` c b b q`p b b c ` c b b q . counit. (1) ǫ p a q “ ǫ p b q “ ǫ p b q “ ǫ p c q “ ǫ p c q “ antipode. (1) S p a q “ a , S p b q “ b , S p b q “ ´ b , S p c q “ c , S p c q “ ´ c .A finite-dimensional Hopf algebra is semisimple if and only if it has a normalized integral[5]. An element σ P A is an integral of a Hopf algebra A if it satisfies σ ¨ v “ ǫ p v q ¨ σ “ v ¨ σ for any v P A . An integral is normalized if ǫ p σ q “ P k . If a normalized exists, then it isunique and we denote by σ A P A . Dually, the notions of cointegral and normalized cointegralare defined.The Hopf algebra D is semisimple and cosemisimple. It is deduced from the existence ofboth of a normalized integral σ D and a normalized cointegral σ D : σ D “ a (13) σ D “ ´ δ a ` δ b ` δ c . (14)Here, δ is the delta functional related with the basis a , b , b , c , c P D . We directly verified it by Python.
The Hopf algebra D is not trivial. In fact, by direct calculations, the group-like elementsare given by GE p D q “ t η u (15) GE p D _ q “ t ǫ u (16)From the group-like elements, we see that the Hopf algebras D , D _ are not group Hopfalgebras. In other words, the Hopf algebra D is not trivial.R eferences [1] John Barrett and Bruce Westbury. Invariants of piecewise-linear 3-manifolds. Transactions of the AmericanMathematical Society , 348(10):3997–4022, 1996.[2] Robbert Dijkgraaf and Edward Witten. Topological gauge theories and group cohomology.
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