Weak Quasi Hopf Algebras Of Integral Theory And Quantum Trace

Weak quasi-bialgebra (quasi-Ilopf algebra) is an associative algebra with unit 1 a non-unital algebra map i. (comultiplication), an algebra map e (counit). In this paper, we study mainly from four respects and get some special characterizations.In chapter one, we begin by defining weak quasi-bialgebras, next we give a new construction of modular categories and make it become tensor category. In particular, modular categories of two equivalent weak quasi-bialgebras are tensor equivalence. Our main results are:Proposition 1.1.4(a)A weak quasi-bia]gebra A is quasitriangular if and only if the correspondingcategory of the left A-Mod is a tensor category.(b)Moreover, the tensor category A-Mod is symmetric if and only if relations (1.5-1.7)are satisfied together with the additional relation R21 -Proposition 1.2.8 Let A and A be weak quasi-bialgebras. Let a: A -* A'be amorphism between the underlying algebras. Suppose that the induced functor A' - Mod -> A - Mod extends to a tensor functor (^*, ^d, v,). Show that thereexists an invertible element Fin A'?A' such that 'p(t¹(u?v)) = F¹(Z^l(u?v)).Prove that necessarily:= e, (a? a)1^a(a)F = FtV(a(a))^F12(L ? td)(F) = F^{id? .&}(F)a(⁾Lemma 1.2.10 Under the hypothesis of proposition 1.3.9, the tensor functor (id, id, ²F) is a quasitriangular tensor equivalence from A-Mod to A-Mod.The second chapter is discussed to the antipode in the weak quasi-Hopf algebra.which is not an anticoalgebra morphism. We also proved that in any quasitriangular weak quasi-Hopf algebra the square of the antipode is an inner automorphisin.Theorem 2.1.5 Suppose given a k-algebra B. ahomomorphism h: A -+ B an antihomnophism g: A -+ B, and elements p, a E B such that g(a1)ph(a2) = 4a)p, ^h(a1)o'g(a2) = E(a)u(2.9)41^h(X')ag(X2)ph(X3) - 1(2.10)Z g(z1)pf(x2)a^g(x3) 1(2.11)In addition, suppose given p1, CT1 C B and an antihomophism g' A - B also satisfying (2.9-2.11). Then there exists exactly one invertible element f C B such that:p' = fp, CT1 = cr1, g'(a) = fg(a)f1, V a cAFurthermore, f = ^g'(x')p'f(a?)ag(x3), f1 = ^g(a1)pf(x2)a'g'(^r3).Theorem 2.2.3: Let A be a quasitriangular weak quasi-Hopf algebra, we haveProposition 2.2.4 Under the previous hypothesis, u is invertible in A. for all a A we have: s2(a) = uau1. where: u = J&²I3(^ra))s(R2)aRlzl.Proposition 2.2.6 Let u, P^, .1? be defined as above, a satisfying (2.1-2.2)then s(cK)u = ^>(R2)aR'.In chapter three, we generalize the fundamental structure Theorem on Hopf (bi)-modules to weak quasi-Hopf algebras. After defining integral and cointegral, we study the symmetry and semisimplicity of finite dimension weak quasi-Ilopf algebra.Theorem 3.1.8 Let M be a weak quasi-HopfA-bimodule. Consider N as a left A-module with A-action t as in (3.4), and N ? A as a weak quasi-Hopf A-bimodule as in Lemma 3.1.2. ThenI : N ? A D n a ---k n a C Mprovides an isamcrphism of weak quasi-Hopf A-bimodules with inverse given by f'(m) = E(mo) ? mmProposition. 3.3.4 Let A be a f.d. weak quasi-Hopf algebra and Let ) c L be a nonzero left cointegral on A. Then F,¹(4i) = (td?⁾(b(r)), Q = (8?id)(^r)where r R C A is the unique right integral satisfying...