hopf algebra dual to a polynomial algebra over a commutative ring

Mathematical Notes, vol. 71, no. 5, 2002, pp. 617–623.

Translated from Matematicheskie Zametki, vol. 71, no. 5, 2002, pp. 677–685.

Original Russian Text Copyright c©2002 by V. L. Kurakin.

Hopf Algebra Dual to a Polynomial Algebraover a Commutative Ring

V. L. Kurakin

Received October 2, 2001

Abstract—For a polynomial algebra A = R[X] or R[X, X−1] in several variables over acommutative ring R with a Hopf algebra structure (A, m, e,∆, ε, S) the existence of thedual Hopf algebra (A◦ ,∆◦ , ε◦ , m◦ , e◦ , S◦) is proved.

Key words: Hopf algebra, bialgebra, polynomial algebra, commutative ring, coalgerba, finitelygenerated module, linear recurring sequence.

Let R be a commutative ring with identity. We assume that the reader is familiar with thedefinitions of an algebra (A, m, e) , a coalgebra (A, ∆, ε) , a bialgebra (A, m, e, ∆, ε) , and a Hopfalgebra (A, m, e, ∆, ε, S) over R (see [1–4]) with multiplication m : A⊗A→ A , unit e : R→ A ,comultiplication ∆: A→ A⊗ A , counit ε : A→ R , and antipode S : A→ A . The multiplicationm(a ⊗ b) of elements of A will be denoted by ab , and the identity element e(1) of A by thesame symbol 1 as the identity of R . The term “map” in what follows stands for an R-linearmap, and ⊗ means the tensor product of modules or algebras over R . The map dual to a mapϕ ∈ HomR(A, B) is defined as ϕ∗ : B∗ → A∗ , ϕ∗(f) = fϕ , f ∈ B∗ , where A∗ = HomR(A, R) isthe module dual to A .

It is known [2] that if (A, ∆, ε) is a coalgebra, then a structure of algebra can be defined on A∗ .Consider the canonical map

σ : A∗ ⊗A∗ → (A⊗A)∗ , σ(f ⊗ g)(a ⊗ b) = f(a)g(b),

where f , g ∈ A∗ , a , b ∈ A . Thus σ(f ⊗ g) = mR(f ⊗ g) , where mR : R ⊗ R → R is themultiplication in R . Note that σ(f ⊗ g) is the value of σ at f ⊗ g , whereas mR(f ⊗ g) is thecomposition of maps. Let

∗ : A∗ ⊗A∗ σ→ (A⊗A)∗ ∆∗→ A∗

be the composition of the maps ∆∗ and σ . Then (A∗ , ∆∗ , ε∗) is an algebra called dual to acoalgebra (A, ∆, ε) .

The dual situation is more complicated. Let (A, m, e) be an algebra. Then in the general casethe dual map m∗ : A∗ → (A ⊗ A)∗ cannot in a natural way (with the help of σ) be turned intoa comultiplication on A∗ , since, first, the image m∗(A∗) is not always contained in σ(A∗ ⊗ A∗) ,and, second, the map σ can be noninjective or nonsurjective. For these reasons, the coalgebradual to an algebra A is determined on some submodule A◦ ⊆ A∗ .

If R is a field, then A◦ is defined as the set of maps f ∈ A∗ such that the kernel of f containsan ideal of a finite codimension (see [1]):

A◦ = {f ∈ A∗ : ∃I � A, I ⊆ Ker f , dimA/I <∞}.

0001-4346/2002/7156-0617$27.00 c©2002 Plenum Publishing Corporation 617

618 V. L. KURAKIN

For algebras over a commutative ring R , following [2], we define

A◦ = {f ∈ A∗ : ∃I � A, I ⊆ Ker f , A/I ∈ P (R)},

where P (R) is the category of finitely generated projective R-modules. Then, under the conditionthat R is Noetherian, the restriction of σ to A◦ ⊗ A◦ is an isomorphism A◦ ⊗ A◦ ∼= (A ⊗ A)◦ ,and m∗(A◦) ⊆ (A⊗A)◦ . Consider the restrictions of the dual maps m∗ and e∗ to A◦:

m◦ : A◦ m∗→ (A⊗A)◦ σ

−1→ A◦ ⊗A◦ , e◦ : A◦ e∗→ R∗ = R.

Then (A◦ , m◦ , e◦) is a coalgebra called dual to the algebra (A, m, e) . Moreover, if (A, m, e, ∆, ε)is a bialgebra, then ∗(A◦ ⊗A◦) ⊆ A◦ , ε∗(R) ⊆ A◦ , and (A◦ , ∆◦ , ε◦ , m◦ , e◦) is a bialgebra calleddual to A . If, in addition, A is a Hopf algebra with antipode S , then S∗(A◦) ⊆ A◦ and A◦ is aHopf algebra with the antipode S◦ . Here ∆◦ , ε◦ , and S◦ denote the restrictions of the maps ∗ ,ε∗ , and S∗ to A◦ .

Note that in [2] in the definition of the coalgebra dual to an algebra the condition that Ris Noetherian is missed. Thus for an algebra over a non-Noetherian ring R , the canonical dualcoalgebra, bialgebra, and Hopf algebra are not defined.

In this paper we prove that if A is a polynomial algebra over a commutative ring R withidentity, then the above canonical constructions lead to a well-defined dual coalgebra, bialgebra,and Hopf algebra on A◦ . Here by a polynomial algebra (bialgebra, Hopf algebra) we mean thealgebra R[X] or R[X, X−1] , where X = (x1 , . . . , xk) , X−1 = (x−11 , . . . , x−1k ) , k ≥ 1, with theusual multiplication of polynomials and identity and with arbitrary comultiplication, counit andantipode. The results stated below contain proofs and developments of results announced in [5].

Hopf algebras dual to a polynomial algebra over a field were considered in [6–11]. Over commu-tative rings, these algebras (besides [5]), were considered in [12], where along with other results itwas proved that the Hopf algebra dual to the polynomial Hopf algebra R[x] in one variable overa commutative Noetherian ring R is well defined. Note that in A◦ another definition was used:instead of A/I ∈ P (R) , the condition that the R-module A/I is finitely generated was assumed.In the following proposition, we show that for polynomial algebras these definitions of A◦ areequivalent.

An ideal I of R[X] which contains a system of monic polynomials F1(x1), . . . , Fk(xk) in onevariable is called monic. An ideal I of R[X, X−1] will be called monic if it contains a system ofmonic polynomials F1(x1), . . . , Fk(xk) such that the lowest terms F1(0), . . . , Fk(0) are invertibleelements of R . These polynomials will be referred to as a system of elementary polynomials of amonic ideal.

Proposition 1. Let R be a commutative ring with identity, and let K be a left R-submodule inthe polynomial algebra A = R[X] or R[X, X−1] . Then the following conditions are equivalent :

(a) ∃I ⊆ K , I � A , A/I is a finitely generated free R-module;(b) ∃I ⊆ K , I � A , A/I is a finitely generated projective R-module;(c) ∃I ⊆ K , I � A , A/I is a finitely generated R-module;(d) ∃I ⊆ K , I � A , I is monic.

Proof. The implications (a) =⇒ (b) =⇒ (c) are obvious.(c) =⇒ (d). Let the R-module A/I be generated by the elements F 1 , . . . , Fn , where F =

F + I . Then x1F j(X) =∑i bijF i for some bij ∈ R , 1 ≤ i, j ≤ n . Consider the matrix

B = (bij) ∈ Rn,n . If c = (c1 , . . . , cn) ∈ Rn is the vector of coefficients in the representation of an

element G ∈ A/I in the generating system F 1 , . . . , Fn , then cB is the vector of coefficients in therepresentation of the element x1G . Let c be the vector of coefficients in the representation of theelement 1 . Then the sequence of vectors c, cB, cB2 , . . . , which are the vectors of coefficients in therepresentations of the elements 1, x1 , x

21 , . . . , is annihilated by the characteristic polynomial χB(t)

MATHEMATICAL NOTES Vol. 71 No. 5 2002

HOPF ALGEBRA DUAL TO A POLYNOMIAL ALGEBRA OVER A COMMUTATIVE RING 619

of the matrix B . Therefore, χB(x1) = 0, i.e., χB(x1) ∈ I . Thus I contains a monic polynomialin x1 . In the same way, it can be proved that I contains a monic polynomial in xs , 1 ≤ s ≤ k . Itfollows in the case A = R[X] that I is monic. If A = R[X, X−1] , then let x−11 F j(X) =

∑i b′ijF i ,

b′ij ∈ R , 1 ≤ i, j ≤ n , and let B′ = (b′ij) . The equalities x−11 x1Fj(X) = Fj(X) , 1 ≤ j ≤ n , show

that BB′ = E . Hence B is an invertible matrix, and χB(0) is an invertible element of R . Thusin the case A = R[X, X−1] the ideal I is also monic.

(d) =⇒ (a). Let I ⊆ K be a monic ideal and F1(x1), . . . , Fk(xk) be a system of elementarypolynomials of I of degrees m1 , . . . , mk , respectively. Then any polynomial F (X) ∈ A can beuniquely divided with remainder by the polynomials F1 , . . . , Fk , i.e., represented in the form

F (X) = F1(x1)Q1(X) + · · ·+ Fk(xk)Qk(X) + r(X),

where Qj(X) ∈ A , r(X) ∈ R[X] , and the degree of r(X) in xj is less than mj , 1 ≤ j ≤ k . Itfollows that A/(F1 , . . . , Fk) is a free R-module of rank m1 . . .mk . �

We construct the bialgebra A◦ in the general case as far as it is possible. First we do notassume that A is a polynomial algebra. Our proofs follow the scheme worked out for algebras overfields [1]. The following proposition is well known and can be easily proved directly.

Proposition 2. For arbitrary R-modules A and B , one has (A⊕B)∗ ∼= A∗ ⊕B∗ .

Proposition 3. If A and B are finitely generated projective R-modules, then A∗⊗B∗ ∼= (A⊗B)∗ ,and

σ : A∗ ⊗B∗ → (A⊗B)∗ , σ(f ⊗ g)(a ⊗ b) = f(a)g(b),

where f ∈ A∗ , g ∈ B∗ , a ∈ A , b ∈ B , is an isomorphism of R-modules.

Proof. First assume that A and B are free, A = Rm and B = Rn . By Proposition 1, we haveA∗ = Hom(Rm , R) = Hom(R, R)m ∼= Rm . Then A∗ ⊗ B∗ ∼= Rm ⊗ Rn ∼= Rmn ∼= (A ⊗ B)∗ . Lete1 , . . . , em be a basis of A and f1 , . . . , fm be the dual basis of A∗ , i.e., fi(ej) = δij . In thesame way, let ε1 , . . . , εn be a basis of B and g1 , . . . , gn be the dual basis of B∗ . Straightforwardcalculations show that σ takes the basis fi ⊗ gj of A∗ ⊗B∗ to the basis of (A⊗B)∗ dual to thebasis ei ⊗ εj of A⊗B . Therefore, σ is an isomorphism.

Now let A and B be projective. Then A⊕A1 = Rm and B⊕B1 = Rn for some R-modules A1and B1 , and by the above, σ : (A⊕A1)

∗⊗ (B⊕B1)∗ → ((A⊕A1)⊗ (B⊕B1))

∗ is an isomorphism.Taking into account Proposition 2, we obtain

σ : (A∗ ⊕A∗1)⊗ (B∗ ⊕B∗1)→ ((A⊕A1)⊗ (B ⊕B1))∗

or

σ : (A∗ ⊗B∗)⊕ (A∗ ⊗B∗1 )⊕ (A∗1 ⊗B∗)⊕ (A∗1 ⊗B∗1)→ (A⊗B)∗ ⊕ (A⊗B1)

∗ ⊕ (A1 ⊗B)∗ ⊕ (A1 ⊗B1)∗.

Since the inclusions σ(A∗ ⊗ B∗) ⊆ (A ⊗ B)∗ , etc., are true, we see that the restriction of σ toA∗ ⊗B∗ is an isomorphism between A∗ ⊗B∗ and (A⊗B)∗ . �

Proposition 4. If I and J are ideals in the algebras A and B , respectively, then

A⊗B/(A⊗ J + I ⊗B) ∼= A/I ⊗B/J.


620 V. L. KURAKIN

Proof. Since the map (a, b)→ [a]I ⊗ [b]J is R-balanced, the map A⊗B → A/I ⊗B/J given bya⊗ b→ [a]I ⊗ [b]J is defined. The ideal K = A⊗ J + I ⊗B is contained in its kernel; hence

ϕ : (A⊗B)/K → A/I ⊗B/J , [a⊗ b]K → [a]I ⊗ [b]J ,

is well defined. On the other hand, the map ([a]I , [b]J )→ [a⊗ b]K is well defined and R-balanced.Therefore, there exists a homomorphism

ψ : A/I ⊗B/J → (A⊗B)/K, [a]I ⊗ [b]J → [a⊗ b]K .

Since ϕψ = 1 and ψϕ = 1, we see that ϕ is an isomorphism. �Proposition 5. If A and B are algebras, then σ(A◦ ⊗B◦) = (A⊗B)◦ .

Proof. Let f ∈ A◦ , g ∈ B◦ , and let I , J be ideals contained in Ker f and Ker g , respectively,such that A/I , B/J ∈ P (R) . Consider the ideal K = A⊗ J + I ⊗B of the algebra A⊗B . ThenK ⊆ Ker f ⊗ g , and by Proposition 4, (A⊗B)/K ∈ P (R) . Therefore, f ⊗ g ∈ (A⊗B)◦ .

Conversely, let h ∈ (A⊗B)◦ and K be an ideal contained in Kerh such that (A⊗B)/K ∈ P (R) .Let

I = {a ∈ A : a⊗ 1 ∈ K}, J = {b ∈ B : 1⊗ b ∈ K}.Then I and J are ideals in A and B , respectively, and A⊗J+I⊗B ⊆ K ⊆ Kerh . Therefore, themap h can be passed through A⊗B/(A⊗J+I⊗B) ∼= A/I⊗B/J , i.e., h(a⊗b) = h([a]I⊗[b]J ) forsome map h : A/I⊗B/J → R . Then h ∈ (A/I⊗B/J)∗ or by Proposition 3, h ∈ (A/I)∗⊗(B/J)∗ .Hence

h =∑i

fi ⊗ gi , where fi ∈ (A/I)∗ , gi ∈ (B/J)∗ .

Let fi : A → A/Ifi→ R and gi : B → B/J

gi→ R . Then fi ∈ A∗ , I ⊆ Ker fi , A/I ∈ P (R) ; hencefi ∈ A◦ , and, in the same way, gi ∈ B◦ . We have

h(a⊗ b) = h([a]I ⊗ [b]J ) =∑i

fi([a]I )gi([b]J ) =∑i

fi(a)gi(b),

i.e., h = σ(∑

fi ⊗ gi) ∈ σ(A◦ ⊗B◦) . �Proposition 6. If (A, m, e) is an algebra, then m∗(A◦) ⊆ (A⊗A)◦ . If (A, m, e, ∆, ε) is a bial-gebra, then ε∗(R∗) ⊆ A◦ . If A is a Hopf algebra with a surjective antipode S , then S∗(A◦) ⊆ A◦ .

Proof. Let f ∈ A◦ , and let I be an ideal contained in Ker f such that A/I ∈ P (R) . Consider theideal K = A⊗I+I⊗A of A⊗A . Then m∗(f)(K) = fm(K) = f(I) = 0, and (A⊗A)/K ∈ P (R)by Proposition 4. Therefore, m∗(f) ∈ (A⊗A)◦ .

Let A be a bialgebra. By definition, ε∗ : R∗ = R → A∗ , ε∗(1) = ε . Hence it is sufficientto prove that ε ∈ A◦ . Since A is a bialgebra, ε : A → R is an algebra homomorphism. Itfollows that Ker ε is an ideal in A , and also that ε(1) = 1 and ε is an epimorphism. Therefore,A/Ker ε ∼= R ∈ P (R) and ε ∈ A◦ .

Let A be a Hopf algebra, f ∈ A◦ , and let I be an ideal contained in Ker f such that A/I ∈P (R) . Since an antipode of a Hopf algebra is an algebra antiendomorphism, S−1(I) is an ideal

in A . Moreover, S−1(I) ⊆ KerS∗(f) . Since S is surjective, the map τ : AS→ A → A/I is also

surjective. Therefore, A/S−1(I) = A/Ker τ ∼= A/I ∈ P (R) . This proves that S∗(f) ∈ A◦ . �Subsequent properties are true for algebras A satisfying the following condition:

if I and J are ideals of A such that A/I, A/J ∈ P (R),

then there exists an ideal K such that K ⊆ I ∩ J and A/K ∈ P (R).(1)

Note that the polynomial algebras R[X] and R[X, X−1] satisfy this condition. Indeed, if I and Jare ideals of a polynomial algebra A such that A/I , A/J ∈ P (R) , then, by Proposition 1, thereexist a system of elementary polynomials F1(x1), . . . , Fk(xk) in I and a system of elementarypolynomials G1(x1), . . . , Gk(xk) in J . Then the ideal K = (F1G1 , . . . , FkGk) is contained inI ∩ J , and A/K ∈ P (R) .



Proposition 7. If an algebra A satisfies the condition (1), then A◦ is a submodule in A∗ . If thealgebras A and B satisfy condition (1), then the map σ : A◦⊗B◦ → (A⊗B)◦ is an isomorphismof modules.

Proof. Let f , g ∈ A◦ , and let I and J be ideals contained in Ker f and Ker g , respectively,such that A/I , A/J ∈ P (R) . Since Ker rf contains I for any r ∈ R , we have rf ∈ A◦ . SinceI ∩ J ⊆ Ker(f + g) , we obtain f + g ∈ A◦ by condition (1). Therefore, A◦ is a submodule in A∗ .

Prove that σ is injective. Let h =∑

fi⊗ gi ∈ A◦⊗B◦ , where fi ∈ A◦ , gi ∈ B◦ . Assume thatσ(h) = 0. Take the ideals Ii ⊆ Ker fi and Ji ⊆ Ker gi in A and B , respectively, so that A/Ii ,B/Ji ∈ P (R) . By condition (1), there exist ideals I ⊆ ∩iIi and J ⊆ ∩iJi such that we have A/I ,B/J ∈ P (R) . Since I ⊆ Ker fi and J ⊆ Ker gi , the maps fi ∈ (A/I)∗ , fi([a]I ) = fi(a) andgi ∈ (B/J)∗ , gi([b]J ) = gi(b) are well defined. Moreover,

∑i

fi([a]I )gi([b]J ) =∑i

fi(a)gi(b) = σ(h)(a ⊗ b) = 0.

Therefore, σ(h) = 0, where σ : (A/I)∗ ⊗ (B/J)∗ → (A/I ⊗B/J)∗ is defined in a natural way (asin Proposition 3) and h =

∑fi ⊗ gi ∈ (A/I)∗ ⊗ (B/J)∗ . By Proposition 3, σ is an isomorphism;

hence h = 0.Since A→ A/I and B → B/J are epimorphisms, the dual maps (A/I)∗ → A∗ , (B/J)∗ → B∗

are monomorphisms. We consider these monomorphisms as imbeddings. Then (A/I)∗ ⊆ A◦ , sincefor any map f ∈ A∗ such that f ∈ (A/I)∗ we have I ⊆ Ker f and A/I ∈ P (R) . In the same way,(B/J)∗ ⊆ B◦ . We have proved above that h = 0 in the module (A/I)∗ ⊗ (B/J)∗ . Then, by thedefinition of the tensor product, h = 0 in the module A◦ ⊗B◦ . It follows that σ is injective. ByProposition 5, σ is an isomorphism. �Theorem 1. Let (A, m, e) be an algebra satisfying condition (1). Then (A◦ , m◦ , e◦) is a coal-gebra with well-defined operations.

Proof. By Propositions 6 and 7, the map m◦ (defined at the beginning of the article) is welldefined. The coassociativity of m◦ and the counitarity of e◦ follow from the associativity of mand unitarity of e in the same way, as for algebras over a field [1]. �Theorem 2. If A = R[X] or R[X, X−1] is a polynomial bialgebra, then (A◦ , ∆◦ , ε◦ , m◦ , e◦) isa bialgebra. If in addition A is a Hopf algebra with antipode S , then A◦ is a Hopf algebra withthe antipode S◦ .

Proof. Since polynomial algebra is commutative, S2 is the identity (the proof is the same as foralgebras over a field, see [1, Proposition 4.0.1(6)]). Therefore, S is bijective. By Proposition 6, themaps ε◦ and S◦ are well defined on A◦ . The associativity of ∆◦ , the unitarity of ε◦ , the antipodeaxiom for S◦ , and also the fact that m◦ and e◦ are homomorphisms of the algebra (A◦ , ∆◦ , ε◦) ,are proved in the same way as for algebras over a field [1]. Taking into account Theorem 1, itremains to prove that the multiplication ∆◦ is well defined, i.e., that ∗(A◦ ⊗A◦) ⊆ A◦ .

Let f , g ∈ A◦ , and let I and J be ideals of A contained respectively in Ker f and Ker gsuch that A/I , A/J ∈ P (R) . By Proposition 1, we can assume that A/I and A/J are finitelygenerated free R-modules. Let π1 : A → A/I and π2 : A → A/J be canonical projections andπ = π1 ⊗ π2 . Since I ⊆ Ker f , the map f can be passed through π1 , i.e., f = fπ1 for the mapf : A/I → R defined by f([a]I) = f(a) , a ∈ A . In the same way, g = gπ2 , where g : A/J → R ,g([a]J ) = g(a) . According to the definition of ∗ = ∆∗σ , we have

f ∗ g = ∆∗σ(f ⊗ g) = mR(f ⊗ g)∆ = mR(f ⊗ g)π∆ = mR(f ⊗ g)∆,

where ∆ = π∆: A→ A/I ⊗A/J . Let e1 , . . . , et be a basis of the free R-module B = A/I ⊗A/J

and M be the matrix of the map b → ∆(xs)b , b ∈ B , in this basis. Here xs , 1 ≤ s ≤ k ,


622 V. L. KURAKIN

are the elements (variables) of the polynomial algebra A . Then the characteristic polynomialχM (t) of the matrix M annihilates the vector 1 ∈ B with respect to the map b→ ∆(xs)b . Thismeans that χM (∆(xs)) = 0. Since A is a bialgebra, ∆ and ∆ are algebra homomorphisms.Therefore, ∆(χM (xs)) = 0, i.e., χM (xs) ∈ Ker∆. It follows that χM (xs) ∈ Ker(f ∗ g) . In thecase A = R[X, X−1] , the map b→ ∆(x−1s )b is the inverse of the map considered above; hence Mis an invertible matrix and χM (0) is an invertible element of R . This proves that Ker(f ∗ g)contains a monic ideal. By Proposition 1, we obtain that f ∗ g ∈ A◦ . �

Now we give some applications of the obtained results. A sequence over R is an arbitraryfunction u : N0 → R . A sequence u = (u(0), u(1), . . . ) is called a linear recurring sequence (LRS)of order m if there exist elements c0 , . . . , cm−1 ∈ R such that

u(i + m) = cm−1u(i + m− 1) + · · ·+ c1u(i + 1) + c0u(i), i ≥ 0.

The polynomial F (x) = xm−cm−1xm−1−· · ·−c1x−c0 ∈ R[x] is called a characteristic polynomialof u .

It is known that the Hopf algebra dual to the polynomial algebra P [x] over a field P is thealgebra of linear recurring sequences [6]. For the polynomial algebra P [X] in several variables overa field, see [7, 11]. This property is also true for a polynomial algebra R[x] over a commutativering R (see [5]; for Noetherian R , see [12]). Indeed, a sequence u can be identified with the mapu ∈ R[x]∗ , u(xi) = u(i) , i ≥ 0. Then u is an LRS if and only if Keru contains a monic ideal(F (x)) , i.e., u ∈ R[x]◦ .

Coordinatewise multiplication and convolution of sequences u and v are defined by

(uv)(i) = u(i)v(i), (u ∗ v)(i) =

i∑j=0

(ij

)u(j)v(i − j), i ≥ 0.

By [13], if u and v are LRS of orders m and n with characteristic polynomials F (x) and G(x) ,respectively, then uv is an LRS of order mn , and the characteristic polynomial of the matrixS(F )⊗ S(G) is a characteristic polynomial of the LRS uv . Here S(F )m×m is the accompanyingmatrix of the polynomial F (x) and ⊗ is the tensor product of matrices. We prove a similar resultfor the convolution of sequences in the following generalized sense.

Let (R[x], m, e, ∆, ε) be a polynomial bialgebra. Then the multiplication ∗ = ∆◦ in thedual bialgebra (R[x]◦ , ∆◦ , ε◦ , m◦ , e◦) will be called a (generalized) convolution of linear recurringsequences.

If the comultiplication in R[x] is defined by ∆x = x ⊗ x or ∆x = x ⊗ 1 + 1 ⊗ x , thenthe generalized convolution ∗ coincides, respectively, with the coordinatewise multiplication andconvolution defined above. In general case, let

∆x =∑

cijxi ⊗ xj , cij ∈ R.

Theorem 3. If u and v are linear recurring sequences over R of orders m and n with char-acteristic polynomials F (x) and G(x) , respectively, then their generalized convolution u ∗ v isan LRS of order mn , and the characteristic polynomial of the matrix M =

∑cijS(F )i ⊗ S(G)j

is a characteristic polynomial of the LRS u ∗ v .

Proof. The fact that u∗v is an LRS of order mn was practically shown in the proof of Theorem 2.Consider this proof in more detail.

By assumption, I = (F ) ⊆ Keru and J = (G) ⊆ Ker v . Let e1 = ([1]F , [x]F , . . . , [xm−1]F ) bethe basis of the R-module A/I and e2 = ([1]G , [x]G , . . . , [xn−1]G) be the basis of the R-moduleA/J . Then

e1S(F )i = ([xi]F , [xi+1]F , . . . , [xi+m−1]F ) = [xi]F e1 , e2S(G)j = [xj ]Ge2.



Let identify the vectors e1 and e2 with the vectors ([1]F ⊗ [1]G , [x]F ⊗ [1]G , . . . , [xm−1]F ⊗ [1]G)and ([1]F ⊗ [1]G , [1]F ⊗ [x]G , . . . , [1]F ⊗ [xn−1]G) , respectively. Then e = e1 ⊗ e2 (the tensorproduct of the row-matrices) is a basis of the R-module B = A/I ⊗A/J . For b ∈ B denote by bethe vector of coefficients of b in the basis e , so that b = ebe . Then

e(Mbe) = (eM)be = (e1 ⊗ e2)

(∑cij S(F )i ⊗ S(G)j

)be.

Using the property (A⊗B)(C ⊗D) = AC ⊗BD of the tensor product of matrices, we obtain

e(Mbe) =∑

cij([xi]F ⊗ [xj ]G)(e1 ⊗ e2)be = ∆x · b,

where ∆ = π∆ is the map introduced in the proof of Theorem 2. Thus M is the matrix of themap realizing the multiplication by ∆x ∈ B . By the proof of Theorem 2, χM (x) ∈ Ker(u ∗ v) .Therefore, χM (x) is a characteristic polynomial of the LRS u ∗ v , in particular, u ∗ v is an LRSof order mn . �

The proof shows that along with the characteristic polynomial of the matrix M any monicpolynomial annihilating M , for example, any minimal polynomial of M is also a characteristicpolynomial of the LRS u ∗ v . As a corollary, we also see that characteristic (and minimal) polyno-mials of the matrices S(F )⊗S(G) and S(F )⊗En+Em⊗S(G) are characteristic polynomials ofthe LRS uv and u ∗ v , respectively, where, in this particular case, u ∗ v is the usual convolutionof sequences.

Properties of Hopf algebras dual to polynomial algebras prompted by their representation asalgebras of linear recurring sequences will be considered in more detail in [14].

REFERENCES

1. M. F. Sweedler, Hopf Algebras, Benjamin, New York, 1969.2. V. A. Artamonov, “Structure of Hopf algebras,” in: Algebra, Geometry, Topology [in Russian], vol. 29,

Itogi Nauki i Techniki, VINITI, Moscow, 1991, pp. 3–63.3. S. MacLane, Homology, Springer, Berlin, 1963.4. Yu. A. Bakhturin, Basic Structures of Modern Algebra [in Russian], Nauka, Moscow, 1990.5. V. L. Kurakin, “Hopf algebras of linear recurring sequence over commutative rings,” in: Proceedingsof 2 Math. Conf. of Moscow State Social Univ. (Nakhabino, January 26–February 2, 1994 ) [in Russian],Moscow, 1994, pp. 67–69.

6. B. Peterson and E. Y. Taft, “The Hopf algebra of linear recursive sequences,” Aequat. Math., 20 (1980),1–17.

7. L. Cerlienco and F. Piras, “On the continuous dual of a polynomial bialgebra,” Comm. Algebra, 19(1991), no. 10, 2707–2727.

8. V. L. Kurakin, “Convolution of linear recurring sequences,” Uspekhi Mat. Nauk [Russian Math. Sur-veys], 48 (1993), no. 4, 235–236.

9. V. L. Kurakin, “Structure of the Hopf algebras of linear recurring sequences,” Uspekhi Mat. Nauk[Russian Math. Surveys], 48 (1993), no. 5, 177–178.

10. W. Chin and J. Goldman, “Bialgebras of linearly recursive sequences,” Comm. Algebra, 21 (1993),no. 11, 3935–3952.

11. V. L. Kurakin, A. S. Kuzmin, A. V. Mikhalev, and A. A. Nechaev, “Linear recurring sequences overrings and modules,” J. Math. Sci., 76 (1995), no. 6, 2793–2915.

12. J. Y. Abuhlail, J. Gomez-Torrecillas, and R. Wisbauer, “Dual coalgebras of algebras over commutativerings,” J. Pure Appl. Algebra, 153 (2000), 107–120.

13. P. Lu, G. Song, and J. Zhou, “Tensor product with application to linear recurring sequences,” J. Math.Res. Exposition, 12 (1992), no. 4, 551–558.

14. V. L. Kurakin, “Hopf algebras of linear recurring sequences over rings and modules,” to appear.

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hopf algebra dual to a polynomial algebra over a commutative ring

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