Wreath products of cocommutative Hopf algebras
aa r X i v : . [ m a t h . R A ] J u l WREATH PRODUCTS OF COCOMMUTATIVE HOPFALGEBRAS
LAURENT BARTHOLDI, OLIVIER SIEGENTHALER, AND TODD TRIMBLE
Abstract.
We define wreath products of cocommutative Hopf alge-bras, and show that they enjoy a universal property of classifying cleftextensions, analogous to the Kaloujnine-Krasner theorem for groups.We show that the group ring of a wreath product of groups is thewreath product of their group rings, and that (with a natural definitionof wreath products of Lie algebras) the universal enveloping algebra of awreath product of Lie algebras is the wreath product of their envelopingalgebras.We recover the aforementioned result that group extensions may beclassified as certain subgroups of a wreath product, and that Lie algebraextensions may also be classified as certain subalgebras of a wreathproduct. Introduction
Let
A, Q be cocommutative Hopf algebras. We construct the wreath prod-uct A ≀ Q of A and Q , and show that it satisfies a universal property withrespect to containing all extensions of A by Q . The definition is very simple,in terms of measuring algebras , see § A ≀ Q := A Q Q. Our first main result is that the wreath product of Hopf algebras classifiestheir extensions:
Theorem A (Generalized Kaloujnine-Krasner theorem) . There is a bijec-tion between, on the one hand, cleft extensions E of A by Q , up to isomor-phism of extensions, and, on the other hand, Hopf subalgebras E of A ≀ Q with the property that E maps onto Q via the natural map A Q Q → Q and E ∩ A Q ∼ = A via the evaluation map A Q → A, f f @1 , up to conjugationin A ≀ Q . Extensions of groups — and of Hopf algebras — with abelian kernel areclassified by the cohomology group H ( Q, A ); see [17]. Kaloujnine and Kras-ner considered wreath products as a means to classify arbitrary extensions.There are two fundamental examples of cocommutative Hopf algebras:the group ring k G of a group G , with coproduct ∆( g ) = g ⊗ g for all g ∈ G ;and the universal enveloping algebra U ( g ) of a Lie algebra g , with coproduct∆( x ) = x ⊗ ⊗ x for all x ∈ g .Wreath products of groups were already considered since the beginningsof group theory [7, § II.I.41]. The wreath product A ≀ Q may be definedas the semidirect product A Q ⋊ Q ; its universal property of containing all Date : 15 July 2014. extensions of A by Q is known as the Kaloujnine-Krasner theorem. We showthat the group ring of A ≀ Q is the wreath product of the group rings of A and Q , recovering in this manner the Kaloujnine-Krasner theorem: Theorem B (Group rings) . If A = k A and Q = k Q be group rings, then A ≀ Q ∼ = k ( A ≀ Q ) qua Hopf algebras. Corollary C (Kaloujnine-Krasner) . There is a bijection between, on theone hand, group extensions E of A by Q , up to isomorphism of extensions,and, on the other hand, subgroups E of A ≀ Q with the property that E mapsonto Q via the natural map A Q ⋊ Q → Q and E ∩ A Q ∼ = A via the evaluationmap A Q → A , f f (1) , up to conjugation in A ≀ Q . Special cases of wreath products of Lie algebras were considered in variousplaces in the literature [2, 3, 8, 13, 15, 18, 19]. In case k is a field of positivecharacteristic, then by “Lie algebra” we always mean “restricted Lie al-gebra”, and by “universal enveloping algebra” we always mean “restricteduniversal enveloping algebra”.The wreath product a ≀ q may be defined as Vect ( U ( q ) , a ) ⋊ q . An ana-logue of the Kaloujnine-Krasner theorem was proven in [14]. We show thatthe universal enveloping algebra of a ≀ q is the wreath product of universalenveloping algebras of a and q , recovering in this manner the Kaloujnine-Krasner theorem: Theorem D (Lie algebras) . If A = U ( a ) and G = U ( q ) be universal en-veloping algebras, then A ≀ G ∼ = U ( a ≀ q ) qua Hopf algebras. Corollary E (Kaloujnine-Krasner for Lie algebras, see [14]) . There is abijection between, on the one hand, Lie algebra extensions e of a by q , up toisomorphism of extensions, and, on the other hand, subalgebras e of a ≀ q withthe property that e maps onto q via the natural map Vect ( U ( q ) , a ) ⋊ q → q and e ∩ Vect ( U ( q ) , a ) ⋊ q ∼ = a via the evaluation map Vect ( U ( q ) , a ) ⋊ q → a , f f (1) , up to conjugation in a ≀ q . Assumptions.
All algebras are assumed to be defined over the com-mutative ring k . All Hopf algebras are cocommutative, and all extensions ofHopf algebras are cleft. We assume that k is sufficiently well behaved thatthe Poincar´e-Birkhoff-Witt theorem holds for Lie algebras. If k has posi-tive characteristic, we consider restricted Lie algebras, and their restricteduniversal envelopes.As references for Hopf algebras, we based ourselves on [20] and [10]. Forextensions of Hopf algebras, we consulted [12].1.2. Thanks. (check with Todd) We are very grateful to Todd Trimble fornumerous enlightening explanations on the measuring coalgebra.2.
The measuring coalgebra
Let
C, D be coalgebras over a field k . There is a coalgebra D C , whichfulfills the role of an internal ‘Hom( C, D )’, in the category of coalgebras. Itcomes equipped with an evaluation map D C ⊗ C → D , conveniently written D C ⊗ C ∋ f ⊗ c f @ c ∈ D . REATH PRODUCTS OF COCOMMUTATIVE HOPF ALGEBRAS 3
Sometimes D C is called the “measuring coalgebra” from C to D . It maybe described in two manners, one purely categorical and one more concrete.The category of coalgebras Coalg is equivalent to the category of left-exact functors
Lex ( fdRing , Set ) from finite-dimensional k -algebras to sets.The equivalence takes the coalgebra C to the left-exact functor R Coalg ( R ∗ , C ),with R ∗ denoting the k -dual of R , namely the coalgebra of linear maps R → k .Conversely, let F be a left-exact functor fdRing → Set , and consider theset F R ∈ fdRing { R ∗ } × F ( R ). It is a directed set, with a morphism ( R ∗ , f ) → ( S ∗ , g ) for each ring morphism φ : S → R satisfying F ( φ )( f ) = g . Thenassociate with F the colimit of the coalgebras R ∗ along this directed set.It is maybe psychologically reassuring to restrict oneself to “injective”markings f ∈ F ( R ). One may at leisure consider the set n ( R ∗ , f ) : R ∈ fdRing , f ∈ F ( R ) , and ∀ S ∈ fdRing , ∀ φ, ψ : S → R Ä φ ∗ f = ψ ∗ f if and only if φ = ψ ä o . It is also a directed set. At the heart of these constructions lies the factthat every coalgebra is the colimit of its finite-dimensional subcoalgebras,see [20, Theorem 2.2.1].The fact that these transformations define an equivalence of categoriesis the content of Gabriel-Ulmer duality [1]. This duality canonically rep-resents any left-exact functor as a filtered colimit of representable functorsHom( − , C i ) for some finite-dimensional coalgebras C i ; the coalgebra associ-ated with the functor is simply the filtered colimit of the C i .The natural property of an internal ‘Hom’ states D B ⊗ C = ( D C ) B ; so, inparticular, Coalg ( B ⊗ C, D ) =
Coalg ( B, D C ). Therefore, the measuringcoalgebra D C represents the functor R Coalg ( R ∗ ⊗ C, D ). Let us omitthe “ R ” from the descriptions of the functors, remembering that R isa placeholder for a ring that must be treated functorially. The coalgebrastructure is given by coproduct Coalg ( R ∗ ⊗ C, D ) → Coalg ( R ∗ ⊗ C, D ) × Coalg ( R ∗ ⊗ C, D ) f ∆( f ) := ( f, f ) , and counit Coalg ( R ∗ ⊗ C, D ) → Coalg ( R ∗ , k ) , f ε ( f ) := ε. The evaluation map is given by
Coalg ( R ∗ ⊗ C, D ) × C → Coalg ( R ∗ , D )( f, c ) f @ c := f ( − ⊗ c ) , or even more categorically by Coalg ( R ∗ ⊗ C, D ) × Coalg ( R ∗ , C ) → Coalg ( R ∗ , D )( f, g ) f @ g := ( R ∗ ∋ ξ X f ( ξ ⊗ g ( ξ ))) . The measuring coalgebra may also be constructed more directly, followingFox [6] and Sweedler [20, Theorem 7.0.4]. Let U denote the free coalgebra on Vect ( C, D ), and consider D C the maximal subcoalgebra of U that interlaces LAURENT BARTHOLDI, OLIVIER SIEGENTHALER, AND TODD TRIMBLE the counit and coproduct of C with that of D ; namely, there is an evaluationmap @ : U ⊗ C → D coming from U ’s universal property, and we considerthe sum of all coalgebras E ≤ U with ε ( u @ c ) = ε ( u ) ε ( c ) and ∆( u @ c ) =∆( u )@(∆( c )) for all u ∈ E, c ∈ C .This description can be made more concrete as follows. Firstly, U is nat-urally a subset of the set of power series over Vect ( C, D ); this follows fromthe description, by Sweedler, of the free (not yet cocommutative) coalgebraas U = T ( Vect ( C, D ) ∗ ) ◦ . Elements of U may be written u = X n ≥ X some φ ,...,φ n : C → D φ · · · φ n . This shows that U naturally sits inside grHom(Sym C, Sym D ): to such anexpression u , we associate the graded map( c ⊗ · · · ⊗ c m ) X n = m X those φ ,...,φ n : C → D φ ( c ) ⊗ · · · ⊗ φ n ( c n ) . We embedded U into far too big a space, but now we trim it down. We stillcall u the graded map Sym C → Sym D . The counit on grHom(Sym C, Sym D )is ε ( u ) = u (1); the coproduct ∆( u )( b ⊗ · · · ⊗ b m , c ⊗ · · · ⊗ c n ) is obtainedby computing u ( b ⊗ · · · ⊗ b m ⊗ c ⊗ · · · ⊗ c n ) and cutting at the ‘ ⊗ ’ betweenpositions m and m + 1. The evaluation is u @ c = u ( c ). The requirement thatthese maps satisfy ε ( u @ c ) = ε ( u ) ε ( c ) and ∆( u @ c ) = ∆( u )@(∆( c )) gives aconcrete model for D C .2.1. Aside: an illustration on group-like coalgebras.
Let us consider,even though this is not logically necessary for the sequel, the special case C = k X and D = k Y finite-dimensional group-like coalgebras (∆( x ) = x ⊗ x for x ∈ X , etc.), and let us try to determine D C in that case, using itsdescription as a subspace of grHom(Sym C, Sym D ). Consider u ∈ D C .From the counit relation, we get u (1) ε ( c ) = ε ( u ( c )). Considering c = x ∈ X , we get u (1) = ε ( u ( x )) for all x ∈ X . Writing u ( x ) = P α y y , we get u (1) = P y ∈ Y α y . More generally, for any x , . . . , x n ∈ X and i ∈ { , . . . , n } ,we get u ( x ⊗ “ x i ⊗ x n ) = remove i th Y -letter from u ( x ⊗ · · · ⊗ x n ) . This means that u ( x ⊗ · · · ⊗ x n ) is determined by the value of u on anyelementary tensor that contains at least the letters x , . . . , x n .Consider then the coproduct. Writing again u ( x ) = P α y y , this means u ( x ⊗ x ) = P α y ( y ⊗ y ); and, more generally, u ( x ⊗ · · · ⊗ x i ⊗ x i ⊗ · · · ⊗ x n ) = double i th Y -letter in u ( x ⊗ · · · ⊗ x n ) . This means that u ( x ⊗ · · · ⊗ x n ) is determined by the value of u on theword obtained from x · · · x n by removing duplicates.Consider now an arbitrary f : X → Y . Associate with it the followinggraded map u f : Sym C → Sym D : u f ( x ⊗ · · · ⊗ x n ) = f ( x ) ⊗ · · · ⊗ f ( x n ) . Clearly, this is an element of (
C, D ) comm : its coproduct is ∆( u f ) = u f ⊗ u f and ε ( u f ) = 1, so it spans a 1-dimensional subcoalgebra. REATH PRODUCTS OF COCOMMUTATIVE HOPF ALGEBRAS 5
All in all, if X = { x , . . . , x n } , then u is determined by its value on x ⊗ · · · ⊗ x n . If we write u ( x ⊗ · · · ⊗ x n ) = P y =( y ,...,y n ) ∈ Y n α y y andidentify ( y . . . , y n ) ∈ Y n with f : X → Y given by f ( x i ) = y i , we haveexpressed u as P f : X → Y α y u f . This shows that, { u f : ( f : X → Y ) } is abasis of D C , and one has ( k Y ) k X = k ( Y X ).2.2. Hopf algebra structure.
Fox observed in [5] that when C and D areHopf algebras, the construction yields a natural Hopf algebra structure on D C . In fact, Fox’s formula does not use the Hopf algebra structure of C ,but only that of D .In the categorical language, the multiplication in D C is given by a map Coalg ( R ∗ ⊗ C, D ) × Coalg ( R ∗ ⊗ C, D ) → Coalg ( R ∗ ⊗ C, D )( f, g ) Ä ξ X f ( ξ ) g ( ξ ) ä , the unit is the map Coalg ( R ∗ , k ) → Coalg ( R ∗ ⊗ C, D ) , ε Ä ξ ⊗ c ε ( ξ ) ε ( c )1 ä , and the antipode is the map Coalg ( R ∗ ⊗ C, D ) → Coalg ( R ∗ ⊗ C, D ) , f S ( f ) := Ä ξ ⊗ c S ( f ( ξ ⊗ c )) ä . There is also a Hopf algebra action of C on D C , namely a coalgebramorphism C ⊗ D C → D C , given by Coalg ( R ∗ , C ) × Coalg ( R ∗ ⊗ C, D ) → Coalg ( R ∗ ⊗ C, D )( f, g ) Ä ξ ⊗ c X g ( ξ ⊗ cf ( ξ )) ä . It satisfies the properties given in (2)–(3).In the more concrete description, we have the convolution product
Vect ( C, D ) ⊗ Vect ( C, D ) → Vect ( C, D ) f ⊗ g f · g := m D ◦ ( f ⊗ g ) ◦ ∆ C , which induces by the universal property of U a map D C ⊗ D C → D C ; thesame arguments give a unit and antipode to D C , and make D C an algebra C -module. 3. Extensions of Hopf algebras
Let
A, Q be Hopf algebras. An extension of A by Q is a Hopf algebra E ,given with morphisms ι : A ֒ → E and π : E ։ Q , such that Hker( π ) = ι ( A ).Here(1) Hker( π ) = { e ∈ E | X e ⊗ π ( e ) = e ⊗ } is a normal Hopf subalgebra of E , and Q ∼ = E/ ( E Hker( π ) + ).Note that ι turns E into an A -module, and π turns E into a Q -comodule;explicitly, the A -module structure on E is A ⊗ E → E given by a ⊗ e ι ( a ) e ,and the Q -comodule structure on E is E → E ⊗ Q given by e e ⊗ π ( e ). LAURENT BARTHOLDI, OLIVIER SIEGENTHALER, AND TODD TRIMBLE An isomorphism between two extensions E, E ′ is a triple of isomorphisms α : A → A, φ : E → E ′ , ω : Q → Q with φι = ι ′ α and ωπ = π ′ φ : k / / A α (cid:15) (cid:15) ι / / E φ (cid:15) (cid:15) π / / Q ω (cid:15) (cid:15) / / kk / / A ι ′ / / E ′ π ′ / / Q / / k . The usual setting, in the literature, is to consider the extension of analgebra by a Hopf algebra. Here we assume both kernel and quotient areHopf algebras; the only difference amounts to, in appropriate places, replace“linear map” by “coalgebra map”.3.1.
Smash and wreath products.
An important special case of exten-sion, for which the operations can be written out explicitly, is the smashproduct . Let
H, Q be Hopf algebras, and assume that H is a Hopf Q -module;namely, there is a coalgebra morphism ⋆ : Q ⊗ H → H satisfying q ⋆ ε ( q )1 , q ⋆ ( hk ) = X ( q ⋆ h )( q ⋆ k ) , (2) 1 ⋆ h = h, q ⋆ ( r ⋆ h ) = qr ⋆ h. (3)The smash product H Q is, as a coalgebra, H ⊗ Q ; its elements are writtenas sums of elementary tensors h q , and ∆( h q ) = P h q ⊗ h q and ε ( h q ) = ε ( h ) ε ( q ) in Sweedler notation. The multiplication in H Q isdefined by ( h q )( k r ) = X h ( q ⋆ k ) q r, and the antipode is S ( h q ) = ( S ( q ) ⋆ S ( h )) S ( q ). The identity map θ : H ⊗ Q → H Q is an H -module, Q -comodule isomorphism. See [11] fordetails.The smash product is the Hopf algebra analogue to semidirect productsof groups and Lie algebras. We use it to define the wreath product: A ≀ Q = A Q Q. We write τ : A ≀ Q → Q the natural map h q ε ( h ) q , so that we have anexact sequence k / / A Q / / A ≀ Q τ / / Q / / k . If only condition (2) is satisfied, we say Q measures H . Assume nowthat there is given a convolution-invertible map σ ∈ Vect ( Q ⊗ Q, H ); itsconvolution inverse is a map δ : Q ⊗ Q → H such that m ◦ ( σ ⊗ δ ) ◦ (∆ ⊗ ∆) = η ( ε ⊗ ε ). The crossed product H σ Q is, as a coalgebra, H ⊗ Q ; itsmultiplication is given, in the same notation as above, by( h q )( k r ) = X h ( q ⋆ k ) σ ( q , r ) q r . As we shall see the crossed product is the Hopf algebra analogue to generalextensions of groups and Lie algebras.
REATH PRODUCTS OF COCOMMUTATIVE HOPF ALGEBRAS 7
Cleft extensions.
The next class of extensions we consider are the cleft extensions; these are the closest to group and Lie algebra extensions.We return to the general notation of an extension E of A by Q , k / / A ι / / E π / / Q / / k . The extension E is cleft if there exists a Q -comodule, coalgebra morphism γ : Q → E that is convolution-invertible, see [12, § γ iscalled a cleavage , and we often write it q e q . It is convolution-invertible ifit has a convolution inverse, namely if there exists a linear (not necessarily Q -comodule!) map κ : Q → E such that P κ ( q ) γ ( q ) = ǫ ( q )1.Recall that an extension E is Hopf-Galois if the natural map β : E ⊗ A E → E ⊗ Q , given by e ⊗ f P ef ⊗ π ( f ), is bijective. By [4] (seealso [12, Theorem 8.2.4]), the extension E is cleft if and only if it is Hopf-Galois and E ∼ = A ⊗ Q qua (left A -module, right Q -comodule).Let us write θ : A ⊗ Q → E such an isomorphism. We relate the twonotations as follows. Given a cleavage γ with inverse κ , we define an inversefor the canonical map β : E ⊗ A E → E ⊗ Q by e ⊗ q P eκ ( q ) ⊗ γ ( q ), andan A -module, Q -comodule isomorphism θ : A ⊗ Q → E by a ⊗ q aγ ( q ).On the other hand, given θ : A ⊗ Q → E , define a cleavage by q θ (1 ⊗ q ),and note that it is convolution-invertible. We refer to [16] for details onvarious other notions of Hopf algebra extensions. Theorem 3.1.
Let E be an extension of A by Q . The following are equiv-alent: (i) the extension is cleft; (ii) the extension is Hopf-Galois and there exists an A -module, Q -comoduleisomorphism E → A ⊗ Q ; (iii) the algebra Q measures A and there is a -cocycle σ : Q ⊗ Q → A ,such that E is of the form A σ Q .Proof. It suffices to carry previously known results from the (algebra-extension-by-Hopf algebra) setting to the (Hopf algebra-extension-by-Hopf algebra)setting. The equivalence (i) ⇔ (ii) is [12, Theorem 8.2.4]; the equivalence(i) ⇔ (iii) is [12, Theorem 7.2.2]. (cid:3) The Kaloujnine-Krasner theorem for cleft extensions
We are ready to prove that cleft extensions of A by Q are classified bycertain subalgebras of A ≀ Q . Recall the short exact sequence k / / A Q / / A ≀ Q τ / / Q / / k . Proof of Theorem A, ( ⇐ ). Consider a subalgebra E of A ≀ Q whichmaps onto Q via τ , and with E ∩ A Q ∼ = A via evaluation at 1 ∈ Q . We thenhave Hopf algebra maps π = τ | E : E ։ Q and ι : A ֒ → E , with Hker( π ) = E ∩ A Q = ι ( A ), so E is an extension of A by Q . Furthermore, the map θ − : E → A ⊗ Q given by E → A Q Q → A ⊗ Qe X f q → X ( f @1) ⊗ q LAURENT BARTHOLDI, OLIVIER SIEGENTHALER, AND TODD TRIMBLE is a Q -comodule isomorphism. Using it, define the Q -comodule map γ : q θ (1 ⊗ q ). To see that it is a cleavage, consider κ : Q → E by κ ( q ) = θ (1 ⊗ S ( q )), and note that it is a convolution inverse of γ . Therefore, E is a cleftextension.Assume now that two subalgebras E, E ′ of A ≀ Q are conjugate, say by anelement x ∈ A ≀ Q ; so we have E ′ = x E = P { x eS ( x ) : e ∈ E } . Define thenthe following maps: φ : E → E ′ , e x e := X x eS ( x ) , and α : A → A by α ( a ) = ( x ι ( a ))@1 and ω ( q ) = τ ( x ) q . It is easy to see that( α, φ, ω ) is an isomorphism of extensions.4.2. Proof of Theorem A, ( ⇒ ). Consider a cleft extension E of A by Q : k / / A ι / / E π / / Q / / k , with a cleavage γ : q e q .Define then the following map α : E → A ≀ Q , again expressing coalgebrasas functors fdRing → Set : α ( e ) = X β ( e ) π ( e ) , where β : E → A Q represents the natural transformation Coalg ( R ∗ , E ) → Coalg ( R ∗ ⊗ Q, A )given by ( f : R ∗ → E ) (cid:16) ξ ⊗ q X ‹ q f ( ξ ) S ( Â q π ( f ( ξ ) )) (cid:17) . First check that β ( e ) belongs to A Q for all e ∈ E , or equivalently that P ‹ q e S ( ‚ q π ( e )) belongs to A for all e := f ( ξ ) ∈ E and all q ∈ Q . Thisfollows immediately from (1).Then check that α is a homomorphism of Hopf algebras. For this, consider e, e ′ ∈ E , and compute α ( ee ′ ) = X β ( e e ′ ) π ( e e ′ ) , α ( e ) α ( e ′ ) = X β ( e )( π ( e ) ⋆β ( e ′ )) π ( e ) π ( e ′ );so it suffices to prove β ( ee ′ ) = P β ( e )( π ( e ) ⋆ β ( e ′ )). Now represent e bythe functor f : R ∗ → E and represent e ′ by the functor f ′ . We get β ( ee ′ ) = Ä ξ ⊗ q X ‹ q f ( ξ ) f ′ ( ξ ) S ( Â q π ( f ( ξ ) f ′ ( ξ ) )) ä , X β ( e )( π ( e ) ⋆ β ( e ′ )) = Ä ξ ⊗ q X ‹ q f ( ξ ) S ( Â q π ( f ( ξ ) )) Â q π ( f ( ξ ) ) f ′ ( ξ ) S ( Â q π ( f ( ξ ) ) π ( f ′ ( ξ ) )) ä , and both terms are equal.Next, check that α is injective. If e = ι ( a ) for some a ∈ A , then β ( e )@1 = a , so certainly α is injective on ι ( A ). On the other hand, E/ι ( A ) ∼ = Q underthe map π , so ker( α ) is contained in A .Finally, check that the two constructions above are inverses of each other:if E is simultaneously a subalgebra of A ≀ Q and an extension of A by Q ,then α ( E ) is conjugate to E . The proof of Theorem A is complete. REATH PRODUCTS OF COCOMMUTATIVE HOPF ALGEBRAS 9 Groups
We recall the universal property of wreath products of groups mentionedin the introduction:
Theorem 5.1 (Kaloujnine-Krasner, [9]) . Let E be an extension of A by Q : / / A / / E π / / Q / / . Then E is a subgroup of A ≀ Q .Conversely, if E is a subgroup of A ≀ Q which maps onto Q by the naturalmap ρ : A ≀ Q → Q , and such that ker ρ ∩ E is isomorphic to A via f f (1) ,then E is an extension of A by Q . Although the proof is classical, we cannot resist including it, since it isparticularly short, and is essentially the proof of Theorem A:
Sketch of proof.
Let q e q : Q → E be a (set-theoretic) section of π . Wedefine φ : E → A ≀ Q by φ ( e ) = (cid:16) q e qe ( fl qπ ( e )) − , π ( e ) (cid:17) . It is clear that φ is injective, and an easy check shows that φ is a homomor-phism. Conversely, if E is a subgroup of A ≀ Q as in the statement of thetheorem, then π = τ | E defines the extension. (cid:3) Proof of Theorem B.
The wreath product of groups A , Q is thesemidirect product A Q ⋊ Q ; and the group ring of a semidirect product isa smash product of the group rings. It is therefore sufficient to prove thatthe group ring of A Q is the measuring coalgebra ( k A ) k Q . In fact, the groupstructures are defined naturally from the sets A , Q to Q Q , so Theorem Bfollows from the Proposition 5.2.
Let
X, Y be sets, and let k X, k Y be their group-like coal-gebras, with ∆( x ) = x ⊗ x and ε ( x ) = 1 for all x ∈ X ; and similarly for Y .Then the coalgebras ( k Y ) k X and k ( Y X ) are isomorphic. Todd Trimble generously contributed the following proof:
Proof.
The coalgebra k Y represents the functor R Coalg ( R ∗ , k Y ), againabbreviated Coalg ( R ∗ , k Y ). Assume for a moment that Y is finite. Then Coalg ( R ∗ , k Y ) = Alg ( k Y , R ), the set of algebra morphisms from the prod-uct of Y copies of k to R . Such an algebra morphism k Y → R picks out Y many mutually orthogonal idempotents in R which sum to 1. Therefore, k Y represents the functor that takes R to the set of functions e : Y → R suchthat { e ( y ) } y ∈ Y are mutually orthogonal idempotents summing to 1.For Y infinite, the coalgebra k Y is the union, or filtered colimit, of k Y i with Y i ranging over finite subsets of Y . Consequently, k Y represents thefunctor which takes R to the set of functions e : Y → A with finite support,and again where the e ( y ) are mutually orthogonal idempotents summing to1. Let us call such functions “distributions”, although “quantum probabilitydistribution” might be more accurate. Now ( k Y ) k X represents the functor Coalg ( R ∗ ⊗ k X, k Y ) = Y x ∈ X Coalg ( R ∗ , k Y ) , which takes R to X -tuples of Y -indexed distributions in R . In this language,there is a natural map between X -tuples of Y -indexed distributions and Y X -indexed distributions, essentially given by currying: Coalg ( R ∗ , k Y X ) → Y x ∈ X Coalg ( R ∗ , k Y )( e : Y X → R ) Ñ x e x : Y → R, e x ( y ) := X φ : Y → X,x y e ( φ ) é φ Y x ∈ X e x ( φ ( x )) ! ← [ ( x e x )defines a natural bijection between the functors associated with ( k Y ) k X and k ( Y X ).(The sum and product in the bijection above range over infinite argu-ments, but they are in fact finite sums and products, because the finite-dimensional algebra R has only finitely many distinct idempotents.) (cid:3) Proof of Corollary C. By G ( A ) we denote the group-like elementsof a Hopf algebra A , defined as G ( A ) = { x ∈ A : ∆( x ) = x ⊗ x and ε ( x ) = 1 } . Lemma 5.3.
Let A be a Hopf algebra. Then G ( A ) is linearly independentin A . The following are equivalent: (1) A is a group algebra; (2) A ∼ = k G ( A ) ; (3) G ( A ) is a linear basis of A .Proof. Let x , . . . , x n be linearly independent in G ( A ), and consider x = P i c i x i ∈ G ( A ). Then X i c i x i ⊗ x i = ∆( x ) = x ⊗ x = X i,j c i c j x i ⊗ x j . Therefore c i c j = 0 for all i = j , and c i = c i for all i , so x ∈ { x , . . . , x n } .The equivalence follows immediately. (cid:3) Corollary 5.4.
Let
A, Q be the group rings of groups A , Q respectively.Then there is a bijection between cleft extensions of A by Q and group ex-tensions of A by Q , which relates each extension of A by Q to its groupring.Proof. Consider first an extension1 / / A ι / / E π / / Q / / , and set E = k E . Then the natural maps k ι : A → E and k π : E → Q turn E into an extension of A by Q , which is cleft because k π is split qua coalgebramap. REATH PRODUCTS OF COCOMMUTATIVE HOPF ALGEBRAS 11
Conversely, consider a cleft extension(4) k / / A ι / / E π / / Q / / k , and set E = G ( E ). Then the restriction ι : A → E is injective because ι isinjective, and the restriction π : E → Q is surjective because π is split quacoalgebra map. We certainly have π ◦ ι = 1, because (4) is exact. Finally,consider e ∈ ker( π ) ∩ E ; then e ∈ Hker( π ) ∩ E = ι ( A ), so1 / / A ι / / E π / / Q / / (cid:3) Corollary C now follows from Theorems A and B, and Corollary 5.4.6.
Lie algebras
Let a and q be Lie algebras. Their wreath product is a ≀ q = Vect ( U ( q ) , a ) ⋊ q , where the semidirect product is defined by the action ( q ⋆ f )( u ) = f ( uq ) = − f ( qu ) on f : U ( q ) → a . If elements be represented as pairs f ⊕ q , then theLie bracket can be given explicitly by the formula(5) [ f ⊕ q, g ⊕ r ] = Ä u X [ f ( u ) , g ( u )] + f ( ur ) − g ( uq ) ä ⊕ [ q, r ] , where we write ∆( u ) = P u ⊗ u in the classical Sweedler notation.As in the case of groups, we have a “Kaloujnine-Krasner”-type embeddingresult for Lie algebras: Theorem 6.1.
Let e be an extension of a by q : / / a / / e π / / q / / . Then e is a subalgebra of a ≀ q .Conversely, if a is a subalgebra of a ≀ q which maps onto q by the naturalmap ρ : a ≀ q → q , and such that ker ρ ∩ e is isomorphic to a via f f (1) ,then e is an extension of a by q . Proof.
We include the proof for directness, though in the end we willalso deduce it from Theorem A. We start by choosing a linear section q e q : q → e of π : e → q . Lemma 6.2.
The map q e q extends to a map u e u : U q → e which is acoalgebra morphism.Proof. Take an ordered basis V = { v < v < . . . } of q ; then, by thePoincar´e-Birkhoff-Witt theorem, a basis of U q may be chosen as { w w · · · w n : w i ∈ V, w ≤ w ≤ · · · ≤ w n } . Set „ w · · · w n = › w · · · › w n . (cid:3) We may now define φ : e → a ≀ q by e φ = Ä u X f u S ( fl u e π − f u e ) ä ⊕ e π =: ( α, e π ) , where S is the antipode. Clearly φ is injective. Lemma 6.3. α ( u ) ∈ a for all u ∈ U q . Proof.
Clearly α ( u ) ∈ U e . We readily compute α ( u ) π = X u S ( u e π − u e π ) = 0 , so α ( u ) ∈ U a . We also compute ∆ α ( u ), using freely the facts that U q iscocommutative, and that ∆ commutes with S and q e q :∆ α ( u ) = ∆ X f u S ( fl u e π ) − ∆ X f u eS ( f u )= X g u S ( ‡ u e π ) ⊗ g u S ( g u ) + X g u S ( g u ) ⊗ g u S ( ‡ u e π ) − X g u S ( g u e ) ⊗ g u S ( g u ) − X g u S ( g u ) ⊗ g u S ( g u e )= α ( u ) ⊗ ⊗ α ( u ) , since P g u S ( g u ) and P g u S ( g u ) vanish except when u ∗ = u ∗ = 1,in which case they are equal to 1. It follows that α ( u ) ∈ e ∩ U a = a asrequired. (cid:3) To check that φ is a Lie homomorphism, we will need the Lemma 6.4.
For all q ∈ q and u ∈ U q we have X f u S ( g u q ) f u = − f uq. Proof.
Set v = f uq . We then have v = µ ( ηε ⊗ v = µ ( µ ⊗ ⊗ S ⊗ ⊗ v = X v S ( v ) v = X g u qS ( f u ) f u + X f u S ( g u q ) f u + X f u S ( f u ) g u q = v + X f u S ( g u q ) f u + v. (cid:3) Let us now write [ e φ , f φ ] = ( α, [ e π , f π ]); we have α ( u ) = X hg u S ć u e π − g u e ä , g u S (cid:16) ‡ u f π − g u f (cid:17)i − X · ( uf π ) S (cid:16) „ ( uf π ) e π − · ( uf π ) e (cid:17) + X · ( ue π ) S (cid:16) „ ( ue π ) f π − · ( ue π ) f (cid:17) = X hg u S ( ‡ u e π ) , g u S ( ‡ u f π ) i| {z } A − X hg u S ( g u e ) , g u S ( ‡ u f π ) i| {z } B − X î g u S ( ‡ u e π ) , g u S ( g u f ) ó | {z } C + X î g u S ( g u e ) , g u S ( g u f ) ó − X fl u f π S ( fl u e π ) | {z } A − X f u S ( · u f π e π ) + X f u S ( fl u f π e ) | {z } B + X fl u f π S ( f u e ) | {z } B + X fl u e π S ( fl u f π ) | {z } A + X f u S ( · u e π f π ) − X f u S ( fl u e π f ) | {z } C − X fl u e π S ( f u f ) | {z } C ;the terms A , B , C cancel by Lemma 6.4, leaving[ e φ , f φ ] = X f u S (cid:16) ‰ u [ e π , f π ] (cid:17) − X f u S ( f u [ e, f ]) ⊕ [ e π , f π ] = [ e, f ] φ . REATH PRODUCTS OF COCOMMUTATIVE HOPF ALGEBRAS 13
Proof of Theorem D.
The wreath product of Lie algebras a , q is thesemidirect product Vect ( U ( q ) , a ) ⋊ q ; and the universal enveloping algebra ofa semidirect product is a smash product of the universal enveloping algebras.It is therefore sufficient to prove that the universal enveloping algebra of Vect ( U ( q ) , a ) is the measuring coalgebra ( U a ) U q . In fact, the Lie algebrastructures are defined naturally from the vector spaces a , q to Vect ( U ( q ) , a ),and the coalgebra structure on U ( g ) is that of Sym g , so Theorem D followsfrom the Proposition 6.5.
Let
X, Y be vector spaces, and let
Sym X, Sym Y be theirsymmetric algebras, with ∆( x ) = x ⊗ ⊗ x and ε ( x ) = 0 for all x ∈ X ;and similarly for Y .Then the coalgebras (Sym Y ) Sym X and Sym(
Vect (Sym
X, Y )) are iso-morphic. Todd Trimble generously contributed the following proof:
Proof.
The coalgebra Sym Y represents the functor R Coalg ( R ∗ , Sym Y ),again abbreviated Coalg ( R ∗ , Sym Y ). As a first step, take Y to be 1-dimensional. Then Sym Y = k [ y ] with deconcatenation ∆( y n ) = P i + j = n y i ⊗ y j . It is the filtered colimit of the finite-dimensional subcoalgebras spannedby { , y, . . . , y n − } . The dual of this coalgebra is the algebra k [ y ] / ( y n ).Therefore, the functor represented by Sym Y is the colimit of the func-tors Alg ( k [ y ] / ( y n ) , R ); such a functor chooses a nilpotent element in R .Therefore, Sym k represents the functor J , computing the nil-radical of R ;equivalently, R Coalg ( R ∗ , Sym k ) = Vect ( J ( R ) ∗ , k ) . Consider then finite-dimensional Y ; say Y = k { y , . . . , y n } . Then Sym Y = N ni =1 Sym( k y i ) represents R Coalg ( R ∗ , Sym Y ) = ( J ( R )) Y = Vect ( J ( R ) ∗ , Y ) , since tensor products of coalgebras correspond to Cartesian products. Fi-nally, for arbitrary Y , we write Y as a filtered colimit of finite-dimensionalspaces Y i . Since Sym( − ) and Coalg ( R ∗ , − ) both preserve filtered colimits,we get the same statement in general.Now (Sym Y ) Sym X represents the functor R Coalg ( R ∗ , (Sym Y ) Sym X ) = Coalg ( R ∗ ⊗ Sym X, Sym Y )= Vect ( J ( R ) ∗ ⊗ Sym
X, Y ) =
Vect ( J ( R ) ∗ , Vect (Sym
X, Y ))=
Coalg ( J ( R ) ∗ , Sym(
Vect (Sym
X, Y )))so (Sym Y ) Sym X and Sym( Vect (Sym
X, Y )) represent the same functor andthus are isomorphic. (cid:3)
Proof of Corollary E. By P ( A ) we denote the primitive elementsof a Hopf algebra A , defined as P ( A ) = { x ∈ A − : ∆( x ) = x ⊗ ⊗ x } . Lemma 6.6.
Let A be a Hopf algebra, and let x , . . . , x n be linearly inde-pendent in P ( A ) . Then { x i · · · x i s : 1 ≤ i ≤ · · · ≤ i s ≤ n } is linearlyindependent. The following are equivalent: (1) A is a universal enveloping algebra; (2) A ∼ = U P ( A ) ; (3) P ( A ) generates A .Proof. Let x = P c i x k ( i, · · · x k ( i,s i ) = 0 be a linear dependence among theordered monomials { x i · · · x i s } . Assume that this linear dependence is suchthat s = max { s i } is minimal among all linear dependencies. Then ∆( x ) = 0;this expression has two summands 1 ⊗ x and x ⊗
1, and all other summandsare of the form P v ⊗ v ′ for ordered monomials v, v ′ of length (cid:8) s . Theyare therefore linearly independent, and must all vanish. We deduce s ≤ { x i } are linearly independent. The equivalencefollows immediately. (cid:3) Corollary 6.7.
Let
A, Q be the universal enveloping algebras of Lie algebras a , q respectively. Then there is a bijection between cleft extensions of A by Q and Lie algebra extensions of a by q , which relates each extension of a by q to its universal enveloping algebra.Proof. Consider first an extension0 / / a ι / / e π / / q / / , and set E = U ( e ). Then the natural maps U ( ι ) : A → E and U ( π ) : E → Q turn E into an extension of A by Q , which is cleft because U ( π ) is split quacoalgebra map, by Lemma 6.2.Conversely, consider a cleft extension(6) k / / A ι / / E π / / Q / / k , and set e = P ( E ). Then the restriction ι : a → e is injective because ι isinjective, and the restriction π : e → q is surjective because π is split quacoalgebra map. We certainly have π ◦ ι = 0, because (6) is exact. Finally,consider e ∈ ker( π ) ∩ e ; then e ∈ Hker( π ) ∩ e = ι ( a ), so0 / / a ι / / e π / / q / / (cid:3) Corollary E now follows from Theorems A and D, and Corollary 6.7.
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