Right On The Weak Hopf Algebra Twisted Weak Smash Product

In this paper, let H be a weak Hopf algebra with a weak antipode S and A be a weak H-bimodule algebra. Define a multiplication: (a ⊕ h)(b ⊕g) = a(h₁ — b — S(h₃)) ⊕h₂g, on tensor space A⊕ H, for all a, b ∈ A , h, g∈ H. If the following conditions hold: l₁ — a ⊕ l₂h = a⊕ h , a- S(l₂) ⊕ l₁h = a⊕ h, then the tensor space A ⊕H is called a right twisted weak smash product which is denoted by A * H. A * H is an algebra with the unit 1*1 and is a coalgebra, whose comutiplication is given by Δ,(a * h) = (a₁ * h₁) ⊕(a₂ * h₂), and whose counit is given by ε_*(a * h) = ε_A(a)εH(h₁ we mainly study a right twisted smash product A* H over weak Hopf algebras and investigate their properties.Refering to [13][14][15], let A be a weak bialgebra. Suppose A* H satisfies the following condition:In this situation, if (A, Sa), (H, Sh) are weak Hopf algebras, and satisfy the following conditions:then A * H is a weak Hopf algebra with a weak antipode S*(a * h) = (1 *In contrast to the study of Hopf algebras,it allows one to define the trace function for H on A. We get the conclusion that the map l : A - A^H given by l (a) = l- a is an A^H-bimodule map. Let H be a finite dimensional weak Hopf algebra and A an weak H-bimodule algebra.Assume that l : A — A^H is a surjective .Then there exists a non-zero idempotent e e A* H such that e(A * H)e = A^He ≌ A^H as algebra.The following conditions on a weak Hopf algebra H over K are equivalent:H is semisimple <==> There exists a normalized left integral e e H. We study the semisimple of A * H, that is, we find out a normalized left integral e*q ï¿¡ A*H. Let A * H be a weak Hopf algebra ,e and q be two left integrals of A and H .respectively. Then e * q is a left integral of A * 77 if and only if for all a e A , x ï¿¡ H a(x\ —s-e *- 5(2:2)) * 9 = nï¿¡(a)eff(z)e * g.In this case, if e and q are two normalized left integrals of A and H, respectively ,and satisfy e*(li(e J— 5(ii))*?) = ï¿¡/i(lie)ï¿¡//(iigr) ,thene*g is also a normalized left integral of A* H.On the other hand, the following conditions on a weak Hopf algebra H over K are also equivalent: H is semisimple <ï¿¡=> H is a separable K-algebra.It is easy to give a criterion for A * H to be separable over A.Let H be a weak Hopf algebra with . a bijective weak antipode S, and let A be a left weak H-module algebra. Assume there exists a normalized left(or right)integral in H and satisfy a J— S(h\) * h,2 = a *— Sih-i) * /ii.Then the right twisted smash product A * H is separable over A.