Examples of polynomial identities distinguishing the Galois objects over finite-dimensional Hopf algebras

We define polynomial H-identities for comodule algebras over a Hopf algebra H and establish general properties for the corresponding T-ideals. In the case H is a Taft algebra or the Hopf algebra E(n), we exhibit a finite set of polynomial H-identities which distinguish the Galois objects over H up to isomorphism.


Introduction
By the celebrated Amitsur-Levitzki theorem [4], the standard polynomial of degree 2n is a polynomial identity for the algebra M n (C) of n×n-matrices with complex entries, and M n (C) has no non-zero polynomial identity of degree less than 2n. It follows that the identities S 2n distinguish the finite-dimensional simple associative algebras over C up to isomorphism.
an algebraically closed field of characteristic zero is determined up to Ggraded isomorphism by its G-graded polynomial identities. Aljadeff and Haile [1] extended their result to non-abelian groups. Similar results exist for other classes of algebras.
Let now H be a Hopf algebra over a field k. Consider the class of Hcomodule algebras. This class contains the G-graded k-algebras; indeed, such an algebra is nothing but a comodule algebra over the group algebra kG equipped with its standard Hopf algebra structure. Similarly, a comodule algebra over the Hopf algebra of k-valued functions on a finite group G is the same as a G-algebra, i.e., an associative k-algebra equipped with a left G-action by algebra automorphisms.
In this context we may wonder whether the following assertion holds: if H is a Hopf algebra over an algebraically closed field of characteristic zero, then any finite-dimensional simple H-comodule algebra is determined up to H-comodule algebra isomorphism by its polynomial H-identities.
In this note we provide evidence in support of this assertion by means of examples. When H is the n 2 -dimensional Taft algebra H n 2 or the 2 n+1dimensional Hopf algebra E(n), we exhibit (finitely many) polynomial H-identities that distinguish the H-Galois objects over an algebraically closed field. Denoting the T -ideal of polynomial H-identities for a comodule algebra A by Id H (A), we deduce that Id H (A) = Id H (A ) implies that A and A are isomorphic Galois objects. Since each of our finite sets of identities determines the Galois object A up to isomorphism, it also determines the T -ideal Id H (A) completely; in a sense which we shall not make precise, these identities generate the T -ideal.
Before giving the explicit identities, we have to define the concept of a polynomial H-identity for a comodule algebra A over a Hopf algebra H; this is done in full generality in § 2. When A is obtained from H by twisting its product with the help of a two-cocycle, we produce in § 2.3 a universal map detecting all polynomial H-identities for A, i.e., a map whose kernel is exactly the T -ideal Id H (A).
In § 3 we deal with the Taft algebra H n 2 . After recalling the classification of its Galois objects, we show that the degree 2n polynomial is a polynomial H n 2 -identity and that it distinguishes the isomorphism classes of the Galois objects over an algebraically closed field. In § 3.3 we extend this to certain monomial Hopf algebras.
We prove a similar result for the Hopf algebra E(n) in § 4, exhibiting a finite set of polynomial E(n)-identities which distinguishes the Galois objects over E(n).

Polynomial identities for comodule algebras
This is a general section in which we define polynomial identities for comodule algebras and state general properties of the corresponding Tideals.
We fix a ground field k over which all our constructions will be defined. In particular, all linear maps are supposed to be k-linear and unadorned tensor product symbols ⊗ mean tensor products over k. Throughout the paper we assume that k is infinite.

Reminder on comodule algebras
We suppose the reader familiar with the language of Hopf algebra, as presented for instance in [15,19]. As is customary, we denote the coproduct of a Hopf algebra by ∆, its counit by ε, and its antipode by S. We also make use of a Heyneman-Sweedler-type notation for the image of an element x of a Hopf algebra H under its coproduct.
Recall that a (right) H-comodule algebra over a Hopf k-algebra H is an associative unital k-algebra A equipped with a right H-comodule structure whose (coassociative, counital) coaction is an algebra map. The subalgebra A H of coinvariants of an H-comodule algebra A is defined by A Galois object over H is an H-comodule algebra A such that A H = k 1 A and the map β : A ⊗ A → A ⊗ H given by a ⊗ a → (a ⊗ 1) δ(a ) (a, a ∈ A) is a linear isomorphism. For more on Galois objects, see [15,Chap. 8].
Let us now concentrate on a special class of Galois objects, which we call twisted comodule algebras. Recall that a two-cocycle α on H is a bilinear form α : H × H → k satisfying the cocycle condition for all x, y, z ∈ H. We assume that α is invertible (with respect to the convolution product) and normalized; the latter means that α(x, Let u H be a copy of the underlying vector space of H. Denote the identity map u from H to u H by x → u x (x ∈ H). We define the algebra α H as the vector space u H equipped with the product given by for all x, y ∈ H. This product is associative thanks to the cocycle condition; the two-cocycle α being normalized, u 1 is the unit of α H.
The algebra α H carries an H-comodule algebra structure with coaction It is easy to check that the subalgebra of coinvariants of α H coincides with k u 1 and that the map β : α H ⊗ α H → α H ⊗ H is bijective, turning α H into a Galois object over H. Conversely, when H is finite-dimensional, any Galois object over H is isomorphic to a comodule algebra of the form α H.

Polynomial H-identities
Let us now define the notion of a polynomial H-identity for an H-comodule algebra A. (Polynomial identities for module algebras over a Hopf algebra have been defined e.g. in [5,8] This algebra is isomorphic to the algebra of non-commutative polynomials in the indeterminates X xr i , where i = 1, 2, . . . and {x r } r is a linear basis of H. The algebra T is graded with all generators X x i homogeneous of degree 1.
There is a natural H-comodule algebra structure on T whose coaction δ : T → T ⊗ H is given by The coaction obviously preserves the grading.
When H is the trivial one-dimensional Hopf algebra k, then a H-comodule algebra A is nothing but an associative unital algebra. In this case a polynomial H-identity for A is a classical polynomial identity, i.e., a non-commutative polynomial P (X 1 , Example 2.3. When H = kG is a group algebra, a polynomial H-identity is the same as a G-graded polynomial identity, as defined for instance in [6].

Example 2.4.
Let H be an arbitrary Hopf algebra and A an H-comodule algebra. Assume that the subalgebra A H of coinvariants is central in A (such a condition is satisfied e.g. when A = α H is a twisted comodule algebra). For x, y ∈ H consider the following elements of T : and The proof follows the same lines as the proof of [3,Prop. 2.2]. Note that the assumption that k is infinite is needed to establish that the ideal I H (A) is graded. We summarize property (c) by saying that I H (A) is a T -ideal, a standard concept in the theory of polynomial identities (see [18]).
It is also clear that, if A → A is a map of H-comodule algebras, then In particular, if A and A are isomorphic H-comodule algebras, then In § § 3-4 we will consider certain finite-dimensional Hopf algebras H such that the equality I H (A) = I H (A ) for twisted comodule algebras A, A implies that A and A are isomorphic.
To this end, we next show how to detect polynomial H-identities for twisted comodule algebras.

Detecting polynomial identities
Let α H be a twisted comodule algebra for some normalized convolution invertible two-cocycle α, as defined in § 2.1. We claim that the polynomial H-identities for α H can be detected by a "universal" comodule algebra map which we now define.
For each i = 1, 2, . . ., consider a copy t H i of H, identifying x ∈ H linearly with the symbol t x i ∈ t H i , and define S to be the symmetric algebra on the The algebra S is isomorphic to the algebra of commutative polynomials in the indeterminates t xr i , where i = 1, 2, . . . and {x r } r is a linear basis of H. The map µ α : T → S ⊗ α H is given by The algebra S⊗ α H is generated by the symbols t x i u y (x, y ∈ H; i ≥ 1) as a k-algebra (we drop the tensor product sign ⊗ between the t-symbols and the u-symbols). It is an H-comodule algebra whose S(t H )-linear coaction extends the coaction of α H:  To prove Theorem 2.6 we need the following proposition.
Combining these observations yields a proof of the proposition. We can be even more precise: the algebra map χ : S → k uniquely associated to µ in the statement is determined on the generators t x i by Proof of Theorem 2.6. Let P ∈ T be in the kernel of µ α . Since by Proposition 2.7 any H-comodule algebra map µ : To prove the converse inclusion, start from a polynomial H-identity P and observe that for every algebra map χ : S → k, the composite map µ = (χ⊗id)•µ α from T to α H is a comodule algebra map. By definition of a polynomial H-identity, we thus have µ(P ) = 0. Now choose a basis {x r } r of H and expand µ α (P ) as µ α (P ) = r µ (r) Since the elements u xr are linearly independent, we have χ(µ (r) α (P )) = 0 for all r. This means that µ

Taft algebras
Let n be an integer ≥ 2 and k a field whose characteristic does not divide n. We assume that k contains a primitive n-th root of unity, which we denote by q.

Galois objects over a Taft algebra
The Taft algebra H n 2 has the following presentation as a k-algebra: H n 2 = k x, y | x n = 1 , yx = qxy , y n = 0 (see [20]). The set {x i y j } 0≤i,j≤n−1 is a basis of the vector space H n 2 , which therefore is of dimension n 2 .
The algebra H n 2 is a Hopf algebra with coproduct ∆, counit ε and antipode S defined by When n = 2, this is the four-dimensional Sweedler algebra. Given scalars a, c ∈ k such that a = 0, we consider the algebra A a,c with the following presentation: A a,c = k x, y | x n = a , yx = qxy , y n = c . This is a right H n 2 -comodule algebra with coaction given by the same formulas as (3.1).
By [ [12]), any Galois object over H n 2 is isomorphic to A a,c for some scalars a, c with a = 0. Moreover, A a,c is isomorphic to A a ,c as a comodule algebra if and only if there is v ∈ k × = k − {0} such that a = v n a and c = c. It follows that (a, c) → A a,c induces a bijection between k × /(k × ) n × k and the set of isomorphism classes of Galois objects over H n 2 . (Note that k × /(k × ) n is isomorphic to the cohomology group H 2 (G, k × ).) If k is algebraically closed, then k × = (k × ) n and any Galois object over H n 2 is isomorphic to A 1,c for a unique scalar c.

A polynomial identity distinguishing the Galois objects
Let A = A a,c be a Galois object as defined in § 3.1. Such a comodule algebra is a twisted comodule algebra α H n 2 for some normalized convolution invertible two-cocycle α. It can be checked that the map u : H n 2 → A a,c is such that u 1 = 1, u x = x and u y = y. This allows us to compute the corresponding universal comodule algebra map µ α : T → S ⊗ A a,c on certain elements of T .
For simplicity, we set E = X 1 1 , X = X x 1 , Y = X y 1 for the X-symbols, and t 1 = t 1 1 , t x = t x 1 , t y = t y 1 for the t-symbols. In view of (2.1) and (3.1), we have (3.4) (In the previous formulas we consider the commuting t-variables as extended scalars; this allows us to drop the unit u 1 of A a,c and the tensor symbols between the t-variables and the u-variables.) This polynomial is a generalization of the degree 4 identity obtained for the Sweedler algebra in [3,Cor. 10.4] (in the special case b = 0).
Proof. It suffices to check that µ α (P c ) = 0 using (3.4) and the defining relations of A a,c .
Since yx = qxy, we have Therefore, in view of x n = a, we obtain We also have µ α (E n ) = t n 1 and µ α (X n ) = t n x x n = at n x . To compute µ α (Y n ), we need the following well-known fact (see [14,Lemma 2.2]): if u and v satisfy the relation vu = quv for some primitive n-root of unity q, then (3.5) Since yx = qxy, we may apply (3.5) to u = t y x and v = t 1 y. We thus obtain µ α (Y n ) = t n y x n + t n 1 y n = at n y + ct n 1 . Combining the previous equalities, we obtain µ α (P c ) = 0.
The following result shows that the polynomial identity of Proposition 3.1 distinguishes the Galois objects of the Taft algebra.

Theorem 3.2. If k is algebraically closed, then
Id H n 2 (A a,c ) = Id H n 2 (A a ,c ) implies that A a,c and A a ,c are isomorphic comodule algebras.
Proof. By the last remark in § 3.1, we may assume a = a = 1. Consider the elements P c ∈ Id H n 2 (A 1,c ) and P c ∈ Id H n 2 (A 1,c ) given by Proposition 3.1. By the equality of T -ideals, both P c and P c are polynomial H n 2 -identities for A 1,c . Hence, so is the difference P c − P c . Therefore, µ α (P c − P c ) = 0. Now, The previous proof shows that the theorem holds if we only assume an inclusion Id H n 2 (A a ,c ) ⊂ Id H n 2 (A a,c ) of T -ideals. Note also that the single polynomial H n 2 -identity P c determines the full T -ideal Id H n 2 (A a,c ) over an algebraically closed field.

Remark 3.3.
If k is not algebraically closed, then the equality of T -ideals of Theorem 3.2 implies that A a ,c is a form of A a,c . Recall that an H n 2comodule algebra A is a form of A a,c if there is an algebraic extension k of k such that k ⊗ k A and k ⊗ k A a,c are isomorphic comodule algebras over the Hopf algebra k ⊗ k H n 2 .

Monomial Hopf algebras
Taft algebras can be generalized as follows. Let G be a finite group and x a central element of G of order n. We also assume that there exists a homomorphism χ : G → k × such that χ n = 1 and χ(x) = q is the fixed primitive n-root of unity.
To these data we associate a Hopf algebra H as follows: as an algebra, H is the quotient of the free product kG * k[y] by the two-sided ideal generated by the relations y n = 0 and yg = χ(g) gy . (g ∈ G) The elements gy i , where g runs over the elements of G and i = 0, . . . , n−1, form a basis of H, whose dimension is equal to n|G|.
In the literature this Hopf algebra is called a monomial Hopf algebra of type I (see [10,Sect. 7], [11]).
When G = Z/n and x is a generator of G, then H is the Taft algebra H n 2 . Note that for an arbitrary finite group G the inclusion Z/n x ⊂ G induces a natural inclusion H n 2 ⊂ H of Hopf algebras.
Given a two-cocycle σ ∈ Z 2 (G, k × ) of the group G and a scalar c ∈ k, we define A σ,c as the algebra generated by the symbols u y and u g for all g ∈ G and the relations for all g, h ∈ G. The algebra A σ,c is an H-comodule algebra with coaction given by δ(u y ) = 1 ⊗ y + u y ⊗ x and δ(u g ) = u g ⊗ g . (g ∈ G) (3.8) Bichon proved that any Galois object over H is isomorphic to one of the form A σ,c . Moreover, A σ,c and A σ ,c are isomorphic comodule algebras if and only c = c and the two-cocycles σ and σ represent the same element of the cohomology group H 2 (G, k × ). In other words, the map (σ, c) → A σ,c induces a bijection between H 2 (G, k × ) × k and the set of isomorphism classes of Galois objects over H (see [9,Th. 2.1]).
Let now introduce the same X-symbols E = X 1 1 , X = X x 1 , Y = X y 1 as in § 3.2. Since H n 2 is a Hopf subalgebra of H, we can reproduce the same computation as in the proof of Proposition 3.1. It allows us to conclude that is a polynomial H-identity for the Galois object A σ,c .

Theorem 3.4. Suppose that k is algebraically closed. If
then A σ,c and A σ ,c are isomorphic comodule algebras.
Proof. Proceeding as in the proof of Theorem 3.2, we deduce c = c from the identity (3.9). It remains to check that σ and σ represent the same element of H 2 (G, k × ). Consider the following diagram: Here k σ G is the twisted group algebra generated by the symbols u g (g ∈ G) and Relations (3.6); it is the subalgebra of A σ,c generated by the elements u g , where g runs over all elements of G.
The vertical map ι T : is induced by the previous natural inclusion and the comodule algebra inclusion k σ G ⊂ A σ,c ; it sends a typical generator t g i u g of S(t kG ) ⊗ k σ G to the same expression viewed as an element of S(t H ) ⊗ A σ,c . The maps µ are the corresponding universal comodule maps; the horizontal sequences are exact in view of Theorem 2.6. The diagram is obviously commutative. Hence, the restriction ι of ι T to Id kG (k σ G) send the latter to Id H (A σ,c ) and is injective. Since ι S is injective, we have Consequently, the equality of the theorem implies the equality of T -ideals of graded identities. We now appeal to [2, Sect. 1], from which it follows that σ and σ are cohomologous two-cocycles.

Galois objects over E(n)
Fix an integer n ≥ 1. Assume that the field k is of characteristic = 2.
The algebra E(n) is generated by elements x, y 1 , . . . , y n subject to the relations x 2 = 1 , y 2 i = 0 , y i x + xy i = 0 , y i y j + y j y i = 0 for all i, j = 1, . . . , n. As a vector space, E(n) is of dimension 2 n+1 .
The algebra E(n) is a Hopf algebra with coproduct ∆, counit ε and antipode S determined for all i = 1, . . . , n by When n = 1, the Hopf algebra E(n) coincides with the Sweedler algebra. The Galois objects over E(n) can be described as follows. Let a ∈ k × , c = (c 1 , . . . , c n ) ∈ k n , and d = (d i,j ) i,j=1,...,n be a symmetric matrix with entries in k. To this collection of scalars we associate the algebra A(a, c, d) generated by the symbols u, u 1 , . . . , u n and the relations   A(1, c, d) for any a = 0, and the Galois objects A(1, c, d) and A(1, c , d ) are isomorphic if and only if c = c and d = d .

Two families of polynomial identities
Let us now compute the universal comodule algebra map µ α : T → S ⊗ A(a, c, d) corresponding to the comodule algebra A(a, c, d).
Proposition 4.1. The degree 4 polynomials are polynomial E(n)-identities for the Galois object A(a, c, d).
Proof. In view of (4.6) and of the defining relations of A(a, c, d), we obtain . From these equalities, it is easy to check that the above polynomials belong to the kernel of µ α , hence are polynomial E(n)-identities.

Theorem 4.2. Suppose that k is algebraically closed. If
Id E(n) (A(a, c, d)) = Id E(n) (A(a , c , d )) , then A(a, c, d) and A(a , c , d ) are isomorphic comodule algebras.
Proof. We proceed as in the proof of Theorem 3.2 by using the identities of Proposition 4.1. Note that there is such an identity for each scalar used to parametrize the Galois objects, and each such scalar appears as the coefficient of the monomial E 2 X 2 ; the latter cannot be an identity since its image under the universal comodule map, being equal to at 2 0 t 2 x , does not vanish.
We finally note that the set of n(n + 3)/2 polynomial E(n)-identities of Proposition 4.1 determines the T -ideal Id E(n) (A(a, c, d)).