MASCHKE-TYPE THEOREM FOR PARTIAL SMASH PRODUCTS

In this paper, we mainly study the trace function for partial Hopf actions and give a Maschke-type theorem for partial smash products. Mathematics Subject Classification (2010): 16T05, 16S30


Introduction
In [10], Exel first considered partial group actions in the context of operator algebras, and studied C * -algebras generated by partial isometries on a Hilbert space.In [6], Caenepeel and Janssen introduced partial Hopf actions regarded as a generalization of partial group actions, who was motivated by an attempt to generalize the notion of partial Galois extensions of commutative rings (see [8]), and also introduced the concept of partial smash products, which is an unital subalgebra of the usual smash products.In [12], Lomp developed the theory of partial Hopf actions, and extended the well-known results of Hopf algebras concerning smash products, such as the Blattner-Montgomery and Cohen-Montgomery theorems in [13].Recently, the authors in [3,9] gave the Morita context between the invariant subalgebra and the partial smash product.
Let H be a finite-dimensional Hopf algebra over a field k and A a partial Hmodule algebra.Then, the partial smash product A#H is a ring extension of A, which is familiar as the partial skew group ring A * G for the partial group action.
In [11], the authors proved the Maschke-type theorem for the partial skew group rings.So, we naturally have the following question.Does the Maschke-type theorem for the partial smash product A#H hold?

LIANGYUN ZHANG AND RUIFANG NIU
In this note we give a positive answer to this question by using a new method which is not just a generalization of the proof of the classical result in [7].
We always work over a fixed field k.Unless otherwise specified, linearity, modules and ⊗ are all meant over k.And we freely use the Hopf algebras terminology introduced in [13].For a coalgebra C, we write its comultiplication ∆(c) = c 1 ⊗ c 2 , for any c ∈ C, in which we omit the summation symbols for convenience.
A partial action of the Hopf algebra H on the algebra A is a linear map α : We will also call A a partial H-module algebra.It is easy to see every action is also a partial action.
Given a Hopf algebra H and a partial H-module algebra A, one can form the partial smash product A#H which is the unital subalgebra of A ⊗ H defined as follows: put an algebra structure in A ⊗ H with the product The partial smash product is given by that is, the subalgebra A#H is spanned by the elements of the form {a(h 1 •1 A )⊗h 2 , for any a ∈ A, h ∈ H}.One can easily verify that the multiplication of partial smash product satisfies For a partial H-module algebra A and its enveloping action B given in [4], a special case which will be useful for further results is the case when θ(A) is an ideal of H-module algebra B, where the map θ : A → B is a monomorphism of algebras.
The authors in [4, Proposition 4] gave the sufficient and necessary condition, that is, for any h, g ∈ H, a ∈ A, for the element θ(1 A ) to be a central idempotent in B. In our note we always assume that A is an ideal of B, since the map θ : A → B is a monomorphism of algebras.So, 1 A becomes a central idempotent in B.
Throughout this note we suppose that H is always a finite dimensional Hopf algebra.

Central trace functions and invariants
Similar to the partial group action in [11], we can define the invariants for a partial H-module algebra A as follows: Note that A H is a subalgebra of A with identity 1 A .Define the trace map tA : where 0 = t ∈ l H (the space of left integrals in H).It is clear that tA is a right A H -linear map.But we hope that it is an A Hbimodule map.
According to the references [1,5], we know that lazy 1-cocycles are related with (co)homology and extensions.
A lazy 1-cocycle is a map ∈ Hom(H, A) which is convolution invertible and satisfies for any h ∈ H, where A is a left H-module algebra.In particular, the unit of For a partial H-module algebra A, if for any h ∈ H, the condition of lazy 1cocycles (forgetting about the condition of being convolution invertible) holds: then, it is easy to check that H In what follows, we call the partial H-module algebra A satisfying the equality (3) a strong partial H-module algebra. Remark.
(1) The invariant subalgebra A H as above in this case becomes (2) If H is cocommutative as coalgebra, then A is a strong partial H-module algebra automatically.
In particular, for the partial group action, we know that it is a strong partial H-module algebra obviously.
(3) Let B be an H-module algebra.Then B is a trivial strong partial H-module algebra.
Before the next lemma we recall the definition of trace map for H-module algebras: let H be a finite-dimensional Hopf algebra acting on an algebra B with action " " and choose 0 = t ∈ In what follows, we discuss the surjectivity of trace map for a partial H-module algebra A, and throughout the rest of this section we always assume that for a partial H-module algebra A, (1) tA : Let (B, θ) be an enveloping action of a partial H-module algebra A. Then (2) tA (a) = tB (a)1 A , for any a ∈ A; (3) tB (B) = tB (A). Proof.
(3) We only show that tB (B) ⊆ tB (A), the opposite is obvious.Assume that there exists an element x ∈ B such that tB (x) = b ∈ tB (B), where the element x is of the form Σ i h i a i , for a finite number of elements (1) tA is onto A H if and only if there exists an element a ∈ A such that tA (a) = 1 A .
(2) Assume that (B, θ) is an enveloping action of a partial H-module algebra Proof.(1) Let there exist an element a ∈ A such that tA (a) = 1 A .Then, for any c ∈ A H , c = c1 A = c tA (a) = tA (ca), that is, tA is onto A H . Conversely, it is straightforward.
( implies tA (a) = tB (a)1 A = 1 A .According to the above conclusion, we know that tA is onto A H .

Maschke-type theorem for partial smash products
In this section, we assume that A is a strong partial H-module algebra, and give the Maschke-type theorem for partial smash product by using a kind of new method.
Lemma 3.1.In partial smash product A#H: Proof.For any a ∈ A, h ∈ H, we have Lemma 3.2.Let V be a left A#H-module, W a submodule of V and tA (1 A ) be invertible in A. Assume that λ : V → W is a projection as A-modules.Then, there is also a projection from V to W as A#H-modules.
Proof.Assume that λ : V → W be the projection as A-modules.Define the map We show that λ is a projection as A#H-module.First we check that λ is A#Hlinear.Since S is bijective, we can choose a#S(h) ∈ A#H: = (a(S(h 3 ) = (a(S(h 2 ) Since x is a right integral in H, we have

Now we use above equation to compute:
( From the above computation, we conclude that It remains to check that λ is a projection.If w ∈ W , then we have According to Lemma 3.2, we get the following main result.
Theorem 3.3.Under the same assumptions as above.If A is semisimple Artinian, then A#H is semisimple Artinian.
Remark.Since H is not a subalgebra of the partial smash product A#H, from the proof of Lemma 3.2, we can see that we use a new method which is not just a generalization of the proof of the classical result in [7] to prove the Maschke-type theorem.
Note that an H-module algebra B is a trivial strong partial H-module algebra, H is semisimple iff ε(t) = 0, where 0 = t ∈ l H , and tB (1 The above corollary is a generalization of Corollary 3.3 in [11].
In what follows, we consider the separability of A#H under the condition that tA

It is easy to prove u ∈ C(A). Moreover, for any
In the following, we will show that w is a separability idempotent for A#H.
Let µ : A#H ⊗ A A#H → A#H denote the multiplication map.Then As in Lemma 3.2, we choose S(x) = t, where 0 = t ∈ Question.In [3], the authors defined the partial invariants A H = {a ∈ A | h • a = (h • 1 A )a = a(h • 1 A ), for any h ∈ H}, and gave the Morita context between the invariant subalgebra A H and the partial smash product A#H.In our note, we introduce the condition (3) of lazy 1-cocycles related with cohomology and extensions in order to prove the Maschke-type theorem.We hope that this condition in the future can be improved.

lH.
Then the map tB : B → B H given by tB (b) = t b is a B H -bimodule map.We call tB a (left) trace function for H on B. From [2] we know that if B is an H-module algebra, the surjectivity of tB onto B H is equivalent to the existence of an element b ∈ B with tB (b) = 1 B .
If there is an element b ∈ B with tB (b) = 1 B , then, by Lemma 2.1, there exists an element a ∈ A such that tB (a) = 1 B .So, the fact that h•a = 1 A (h a) = (h a)1 A

Corollary 3 . 5 .
this case the semisimplity of H is equivalent to the invertibility of tB (1 B ) in B. What's more, the partial smash product A#H become a partial skew group ring A α G in case of replacing H by kG.Therefore, we have the following results.Corollary 3.4.Let H be a finite-dimensional semisimple Hopf algebra, and B an H-module algebra.If B is semisimple, then B#H is semisimple.The above corollary is a generalization of Theorem 6 in[7].Let α be a partial action of a finite group G on a unital algebra R.If R is semisimple and tR (1 R ) is invertible in R, then the partial skew group ring R α G is semisimple.