Categorical Hopf kernels and representations of semisimple Hopf algebras
aa r X i v : . [ m a t h . QA ] O c t CATEGORICAL HOPF KERNELS ANDREPRESENTATIONS OF SEMISIMPLE HOPFALGEBRAS
SEBASTIAN BURCIU
Abstract.
In the category of semisimple Hopf algebras the Hopfkernels introduced by Andruskiewitsch and Devoto in [1] coincidewith kernels of representation as introduced in [2]. Some new re-sults concerning the normality of kernels are also presented. Itis proven that the property for Hopf algebras to have all kernelsnormal Hopf subalgebras is a selfdual property. Introduction
Semisimple Hopf algebras were intensively studied in the last twentyyears. Many properties from finite groups are extended to the moregeneral setting of semisimple Hopf algebras.In order to work inside the category of Hopf algebras in [1] the au-thors introduced the notion of Hopf kernel of a morphism between Hopfalgebras. Using the exponent properties of semisimple Hopf algebras in[2] a notion of kernel of a representation of a semisimple Hopf algebrawas proposed. This notion extends the notion of kernel of finite grouprepresentations. In this paper we prove that the two notions of kernelscoincide. It will be proven in Proposition 2.8 that the Hopf kernel of amorphism between semisimple Hopf algebras is the kernel of a certainrepresentation. Theorem 2.10 shows that the converse of this fact isalso true, any kernel of a representation is also a Hopf kernel in thesense defined in [1].Although many properties of kernels were transferred from groupsto this more general setting of semisimple Hopf algebras there are stillsome unanswered questions in this direction. For example the normal-ity of kernels of representations was proven in [2] with an additionalassumption, that of centrality of the character in the dual Hopf alge-bra. It is not known yet if this additional assumption is necessary. Inthis paper we study Hopf algebras where all these kernels are normalHopf subalgebras. We say that a Hopf algebra has property ( N ) if any Date : November 20, 2018. kernel H χ is a normal Hopf subalgebra of H for any χ ∈ Irr( H ). Itwill be shown in Theorem 3.11 that property ( N ) is a self dual notion, H has property ( N ) if and only if H ∗ has property ( N ). We also givenew information on these kernels in terms of the central charactersdescribed in [9].We work over the base field C and all Hopf algebra notations arethose from [6]. We drop the sigma symbol in Sweedler’s notations forcomultiplication.2. The kernel of a representation as a Hopf kernel
Through all this paper H is a semisimple Hopf algebra over C . Itfollows that H is also cosemisimple and S = id H . The set of irreduciblecharacters of H is denoted by Irr( H ) and this is a base of the characteralgebra C ( H ) ⊂ H ∗ . Moreover C ( H ) is a semisimple algebra [10].To any irreducible character d ∈ Irr( H ∗ ) is associated a simple co-matrix coalgebra C = C < x ij > ≤ i,j ≤ q as in [4].If Λ is the idempotent integral of H it follows that dim C ( H )Λ is theregular character of H ∗ , that is:(2.1) dim C ( H )Λ = X d ∈ Irr( H ∗ ) ǫ ( d ) d. Let H be a semisimple Hopf algebra over C and M be an H -moduleaffording the character χ . Proposition 1.2 from [2] shows that | χ ( d ) | ≤ ǫ ( d ) χ (1). In fact the equivalence of the following assertions followsfrom the same Proposition. Proposition 2.2. ( see [2]
Remark 1.3) Let H be a semisimple Hopfalgebra over C and M be an H -module affording the character χ . (1) χ ( d ) = ǫ ( d ) χ (1) . (2) χ ( x ij ) = δ ij χ (1) for all i, j . (3) dm = ǫ ( d ) m for all m ∈ M . (4) x ij m = δ ij m for all i, j and m ∈ M . The kernel H M (or H χ ) is defined as follows (see [2]). Let ker H ( χ )be the set of all irreducible characters d ∈ Irr( H ∗ ) which satisfy theequivalent conditions above. It can be proven that this set is closedunder multiplication and “ ∗ ” and therefore it generates a Hopf subal-gebra H M (or H χ ) of H which is called the kernel of the representation M [7]. Remark 2.3. (1) For later use let us notice that ker H ( χ ) ⊂ ker H ( χ n ) for all n ≥
0. This is item 1 of Remark 1.5 from [2]. It follows
EMISIMPLE HOPF ALGEBRAS 3 that ∩ n ≥ ker H ( χ n ) = ker H ( χ ) which can also be written as ∩ n ≥ H M ⊗ n = H M .(2) We also need the following result proven in [8]. Suppose that d is a character of H ∗ and χ a character of H . Then χ ( d ∗ ) = χ ( d ).(3) Suppose that χ i ∈ Irr( H ) and χ = P si =1 m i χ i where m i > H ( χ ) = ∩ ri =1 ker H ( χ i ) Proposition 2.4.
Let H be a semisimple Hopf algebra over C and M be a representation of H with character χ . Consider the subalgebra of H given by S M = { h ∈ H | hm = ǫ ( h ) m for all m ∈ M } . Then H M is the largest Hopf subalgebra of H contained in S M .Proof. By the definition of the kernel it is clear that H M ⊂ S M . Item2 of Remark 2.3 implies that if C ⊂ S M then S ( C ) ⊂ S M . PreviousPorposition also shows that if a subcoalgebra C is included in S M then C is also included in H M . It is easy to see that if C and D are twosubcoalgebras included in S M then the product coalgebra CD is alsoincluded in H M . It follows that the largest Hopf subalgebra included in S M it is the sum of all subcoalgebras included in S M and therefore thisHopf subalgebra is also included in H M . Thus this Hopf subalgebracoincides with H M (cid:3) Corollary 2.5.
Let H be a semisimple Hopf algebra over C and M bea representation of H . Then H M is the largest Hopf subalgebra K of H such that K + ⊂ Ann H ( M ) .Proof. It is easy to see that for any Hopf subalgebra K of H one has K + ⊂ Ann H ( M ) if and only if K ⊂ H M (cid:3) Remark 2.6.
Let K be a normal Hopf subalgebra of H and L := H//K be thequotient Hopf subalgebra. From previous Corollary it follows thatIrr( L ) = { χ ∈ Irr( H ) | H χ ⊃ K } .2.1. Hopf kernel of a Hopf algebra map.
In this subsection itwill be shown that any Hopf kernel is a kernel of representation. Theconverse it will be proven in Theorem 2.10.Recall the Hopf kernel of a Hopf map defined in [1]. If f : A → B isa Hopf algebra map then the Hopf kernel of f is defined as follows:(2.7) HKer ( f ) = { a ∈ H | a ⊗ π ( a ) ⊗ a = a ⊗ π (1) ⊗ a } . SEBASTIAN BURCIU
It was proven in [1] that the Hopf kernel is a Hopf subalgebra of H . Proposition 2.8.
Let I be a Hopf ideal of H and π : H → H/I be thecanonical Hopf projection. Then S H/I = { h ∈ H | π ( h ) = ǫ ( h )1 } . Regarding
H/I as H -module it follows that HKer( π ) = H H/I .Proof.
It is straightforward to verify the formula for S H/I . It is alsoclear that HKer( π ) ⊂ S H/I . Since HKer( π ) is a Hopf subalgebra of H Proposition 2.4 implies that HKer( π ) ⊂ H H/I . On the other hand iteasy to check using the equivalencies from Proposition 2.2 that H H/I ⊂ HKer( π ). Therefore the equality HKer( π ) = H H/I holds. (cid:3)
Description of the categorical Hopf kernel as the kernel ofa representation.
Let M be an H -module with character χ . Consider I M = ∩ n =0 Ann H ( M ⊗ n ) . Then I M is a Hopf ideal and it is the largest Hopf ideal contained inthe annihilator Ann H ( M ) [3]. Therefore B = H/I M is a Hopf algebraand one has a canonical projection of Hopf algebras π : H → B. Proposition 2.9.
Let H be a semisimple Hopf algebra over C and M be an H -module. Using the above notations it follows that H M = H B where B is regarded as H -module via π .Proof. The representations of the Hopf algebra
H/I M are exactly therepresentations that are constituents in some power M ⊗ n of M . Thusone has H B = ∩ n ≥ H M ⊗ n by item 3 of Remark 2.3. Also item 1 of thesame Remark 2.3 implies that H M = H B . (cid:3) The next theorem gives the characterization of the kernel of a rep-resentation as a Hopf kernel.
Theorem 2.10.
Let H be a semisimple Hopf algebra and M be a rep-resentation of H . Let as above π : H → H/I M be the canonical projec-tion. Then H M = HKer( π ) . Proof.
By previous Proposition it is enough to show thatHKer( π ) = H B . It is easy to see that H B ⊂ HKer( π ). Indeed, if h ∈ H B then P h ⊗ π ( h ) ⊗ h = P h ⊗ ǫ ( h )1 ⊗ h = P h ⊗ ⊗ h since π ( h ) = ǫ ( h )1 for all h ∈ H B . EMISIMPLE HOPF ALGEBRAS 5
One can also see that HKer( π ) ⊂ H B . Indeed if h ∈ HKer( π ) then P h ⊗ π ( h ) ⊗ h = P h ⊗ ⊗ h . Applying ǫ ⊗ id ⊗ ǫ to this identityit follows that π ( h ) = ǫ ( h )1 and Lemma 2.8 implies HKer( π ) ⊂ H B .Since HKer( π ) is a Hopf subalgebra it follows from Proposition 2.4 thatHKer( π ) ⊂ H B . Thus HKer( π ) = H B . (cid:3) Hopf algebras with all kernels normal
In this section we describe some properties of Hopf subalgebras withall kernels H χ normal Hopf subalgebras.3.1. Central characters in the dual Hopf algebra.
Let H be fi-nite dimensional semisimple Hopf algebra over C . Define a centralsubalgebra of H by ˆZ( H ) := Z( H ) T C ( H ∗ ). It is the subalgebra of H ∗ -characters which are central in H . Let ˆZ( H ∗ ) := Z( H ∗ ) T C ( H ) bethe dual concept, the subalgebra of H -characters which are central in H ∗ .Let φ : H ∗ → H given by f f ⇁ Λ H where f ⇁ Λ H = f ( S (Λ H ))Λ H . Then φ is an isomorphism of vector spaces [6]. Remark 3.1.
It can be checked that φ ( ξ d ) = ǫ ( d ) | H | d ∗ and φ − ( ξ χ ) = χ (1) χ for all d ∈ Irr( H ∗ ) and χ ∈ Irr( H ) (see for example [6] ). Here ξ χ ∈ H is the central primitive idempotent of H associated to χ . Dually, ξ d ∈ H ∗ is the central primitive idempotent of H ∗ associated to d ∈ Irr( H ∗ ) . The following description of ˆZ( H ∗ ) and ˆZ( H ) was given in [9]. Since φ ( C ( H )) = Z( H ) and φ (Z( H ∗ )) = C ( H ∗ ) it follows that the restriction φ | ˆZ( H ∗ ) : ˆZ( H ∗ ) → ˆZ( H )is an isomorphism of vector spaces.Since ˆZ( H ∗ ) is a commutative semisimple algebra it has a vectorspace basis given by its primitive idempotents. Since ˆZ( H ∗ ) is a subal-gebra of Z( H ∗ ) each primitive idempotent of ˆZ( H ∗ ) is a sum of primitiveidempotents of Z( H ∗ ). But the primitive idempotents of Z( H ∗ ) are ofthe form ξ d where d ∈ Irr( H ∗ ). Thus, there is a partition {Y j } j ∈ J ofthe set of irreducible characters of H ∗ such that the elements ( e j ) j ∈ J given by e j = X d ∈Y j ξ d form a basis for ˆZ( H ∗ ). Since φ (ˆZ( H ∗ )) = ˆZ( H ) it follows that b e j := | H | φ ( e j ) is a basis for ˆZ( H ). Using the first formula from Remark 3.1one has SEBASTIAN BURCIU (3.2) b e j = X d ∈Y j ǫ ( d ) d ∗ . Proposition 3.3 of [2] shows that kernels of central characters arenormal Hopf subalgebras. Thus with the above notations ker H ∗ ( b e j ) isa normal Hopf subalgebra of H ∗ . Remark 3.3.
By duality, the set of irreducible characters of H canbe partitioned into a finite collection of subsets {X i } i ∈ I such that theelements ( f i ) i ∈ I given by (3.4) f i = X χ ∈X i χ (1) χ form a C -basis for ˆZ( H ∗ ) . Then the elements φ ( f i ) = P χ ∈X i ξ χ arethe central orthogonal primitive idempotents of ˆZ( H ) and therefore theyform a basis for this space. Clearly | I | = | J | . Write d ∼ d ′ if both appear in the same minimal central character b e j of H . Clearly ∼ is an equivalence relation with equivalence classes Y j .For an irreducible character d ∈ Irr( H ∗ ) let N ( d ) be the smallestHopf subalgebra of H containing d . This always exists since intersectionof normal Hopf subalgebras is always a normal Hopf subalgebra. Proposition 3.5.
Suppose that d, d ′ ∈ Irr( H ∗ ) with d ∼ d ′ . Then N ( d ) = N ( d ′ ) .Proof. Since N ( d ) is central it follows from [5] that the idempotentintegral Λ of N ( d ) is central in H . Since N ( d ) is a semisimple Hopfalgebra Equation 2.1 implies that Λ is a scalar multiple of the sum P e ∈ Irr( N ( d ) ∗ ) ǫ ( e ) e . But Irr( N ( d ) ∗ ) ⊂ Irr( H ∗ ) and decomposition 3.2of central characters of H ∗ shows that d ′ ∈ N ( d ). Therefore N ( d ′ ) ⊂ N ( d ). Symmetry implies that N ( d ′ ) = N ( d ). (cid:3) Remark 3.6. (see Remark 2.3 of [2] .)1) Suppose that K is a normal Hopf subalgebra of H and let L = H//K be the quotient Hopf algebra of H via π : H → L . Then π ∗ : L ∗ → H ∗ is an injective Hopf algebra map. It follows that π ∗ ( L ∗ ) isa normal Hopf subalgebra of H ∗ . Moreover ( H ∗ //L ∗ ) ∗ ∼ = K as Hopfalgebras.2) There is a bijection between normal Hopf subalgebras of H and H ∗ . To any normal Hopf subalgebra K of H one associates ( H//K ) ∗ EMISIMPLE HOPF ALGEBRAS 7 as normal Hopf subalgebra of H ∗ . Conversely to any L , a normal Hopfsubalgebra in H ∗ , one associates ( H ∗ //L ) ∗ as normal Hopf subalgebraof H . The fact that these maps are inverse one to the other followsfrom the previous item of this Remark. Hopf algebras with all kernels normal.Definition 3.7.
We say that a semisimple Hopf algebra H has prop-erty ( N ) if and only if H χ is a normal Hopf subalgebra of H for allirreducible characters χ ∈ Irr( H ) . Dually H ∗ has property ( N ) if and only if H ∗ d is normal for anyirreducible character d ∈ Irr( H ∗ ).For a semisimple Hopf algebra H denote by t H ∈ H ∗ the idempotentintegral of H ∗ It follows as in Equation 2.1 that dim C ( H ) t H is theregular character of H . The following theorem from [2] will be used inthe sequel. Theorem 3.8.
Let H be a finite dimensional semisimple Hopf algebra.Any normal Hopf subalgebra K of H is the kernel of a character whichis central in H ∗ . More precisely, with the above notations one has: K = H | L | π ∗ ( t L ) where L = H//K and t L is the idempotent integral of L . From the proof of this theorem also follows that the regular characterof L is dim C ( L ) t L = ǫ K ↑ HK . Theorem 3.9.
Let H be a semisimple Hopf algebra. If H ∗ has property ( N ) then ker d = ker d ′ for all d ∼ d ′ .Proof. Suppose H ∗ d is a normal Hopf subalgebra of H . Then by Remark3.6 there is some normal Hopf subalgebra K of H such that H ∗ d =( H//K ) ∗ . From the definition of the kernel and Remark 2 one has thatker H ∗ ( d ) = { χ | ker χ ⊃ K } . We claim that d ∈ K . Indeed as above the regular character of H//K is ǫ K ↑ HK . Note that d ∈ ker H ( χ ) if and only if χ ∈ ker H ∗ ( d ).This implies that d ∈ H ǫ K ↑ HK . But Corollary 2.5 of [2] implies that H ǫ K ↑ HK = K .Since N ( d ) is the minimal normal Hopf subalgebra containing d onehas that N ( d ) ⊂ K . On the other hand since Λ K is central in H decomposition 3.2 of central characters of H ∗ implies d ′ ∈ K . Thereforeker d ′ ⊃ ker d . By symmetry one obtains ker d ′ ⊂ ker d and thus theequality ker d ′ = ker d . (cid:3) SEBASTIAN BURCIU
Corollary 3.10.
Let H be a semisimple Hopf algebra. Then H ∗ hasproperty ( N ) if and only if ker d = ker d ′ for all d ∼ d ′ .Proof. We have already shown that if H ∗ has ( N ) then ker d = ker d ′ for all d ∼ d ′ . The converse follows by item 3 of Remark 2.3 sinceker H ∗ ( d ) = ∩ d ′ ∈Y j ker H ∗ ( d ′ ) = ker H ∗ ( b e j ) which is normal by Proposi-tion 3.3 of [2]. (cid:3) Now we can prove the main result of this subsection.
Theorem 3.11.
Let H be a semisimple Hopf algebra. Then H ∗ hasproperty ( N ) if and only if H has property ( N ) .Proof. We show that if H ∗ has ( N ) then H has also ( N ). Write χ ∼ χ ′ if both characters appear in a minimal decomposition of a centralcharacters f i of H from Remark 3.3. Suppose χ ∼ χ ′ and let d ∈ ker H ( χ ). Since H ∗ has ( N ) as before we have that H ∗ d = ( H//K ) ∗ forsome K , a normal Hopf subalgebra of H . Then Theorem 3.8 impliesthat χ is a constituent of ǫ K ↑ HK . Since by the same Theorem ǫ K ↑ HK iscentral in H ∗ it follows from decomposition 3.4 of central characters of H that χ ′ is also a constituent of ǫ K ↑ HK . Therefore also d ∈ ker χ ′ . Thisshows ker χ ⊂ ker χ ′ . By symmetry one has the equality ker χ = ker χ ′ .Previous Corollary shows that H has also property ( N ). (cid:3) Proposition 3.12.
Suppose that H is a semisimple Hopf algebra withproperty ( N ) and let d ∈ Irr( H ∗ ) . In this situation H ∗ d = ( H//N ( d )) ∗ .Proof. It is enough to show that ker H ∗ ( d ) = Irr( H//N ( d )). ClearlyIrr( H//N ( d )) ⊂ ker H ∗ ( d ) by Remark 2. Suppose now that χ ∈ ker H ∗ ( d ).Since H has ( N ) it follows that H χ is a normal Hopf subalgebra of H .Since d ∈ ker H ( χ ) definition of N ( d ) shows that N ( d ) ⊂ H χ . Thus χ ∈ Irr(
H//N ( d )). (cid:3) Acknowledgments.
This work was supported by the strategic grantPOSDRU/89/1.5/S/58852, Project ”Postdoctoral programme for train-ing scientific researchers” cofinanced by the European Social Foundwithin the Sectorial Operational Program Human Resources Develop-ment 2007 - 2013.
References
1. N. Andruskiewitsch and J. Devoto, Extensions of Hopf algebras, Algebra iAnaliz (1995), 22–69.2. S. Burciu, Normal Hopf subalgebras of semisimple Hopf Algebras, Proc. Amer.Mat. Soc. (2009), 3969–3979.3. Y. Kashina, Y. Sommerh¨auser, and Y. Zhu, Higher Frobenius-Schur indicators,Mem. Am. Math. Soc.,Am. Math. Soc., Providence (2006), no. 855. EMISIMPLE HOPF ALGEBRAS 9
4. R. G. Larson, Characters of Hopf algebras, J. Algebra (1971), 352–368.5. A. Masuoka, Semisimple Hopf algebras of dimension 2 p , Comm. Algebra (1995), no. 5, 1931–1940.6. S. Montgomery, Hopf algebras and their actions on rings, vol. 82, 2nd revisedprinting, Reg. Conf. Ser. Math, Am. Math. Soc, Providence, 1997.7. S. Natale, Semisolvability of semisimple Hopf algebras of low dimension, no.186, Mem. Am. Math. Soc., Am. Math. Soc., Providence, RI, 2007.8. W. D. Nichols and M. B. Richmond, The Grothendieck group of a Hopf algebra,I, Comm. Algebra (1998), no. 4, 1081–1095.9. S. Zhu, On finite dimensional semisimple Hopf algebras, Comm. Algebra (1993), no. 11, 3871–3885.10. Y. Zhu, Hopf algebras of prime dimension, Int. Math. Res. Not. (1994), 53–59. Inst. of Math. “Simion Stoilow” of the Romanian Academy P.O. Box1-764, RO-014700, Bucharest, Romania
E-mail address ::