Quantum doubles from a class of noncocommutative weak Hopf algebras

The concept of biperfect (noncocommutative) weak Hopf algebras is introduced and their properties are discussed. A new type of quasi-bicrossed products is constructed by means of weak Hopf skew-pairs of the weak Hopf algebras which are generalizations of the Hopf pairs introduced by Takeuchi. As a special case, the quantum double of a finite dimensional biperfect (noncocommutative) weak Hopf algebra is built. Examples of quantum doubles from a Clifford monoid as well as a noncommutative and noncocommutative weak Hopf algebra are given, generalizing quantum doubles from a group and a noncommutative and noncocommutative Hopf algebra, respectively. Moreover, some characterizations of quantum doubles of finite dimensional biperfect weak Hopf algebras are obtained. (C) 2004 American Institute of Physics.

Some twisted results

The Drinfeld twist for the opposite quasi-Hopf algebra, H-COP, is determined and is shown to be related to the (second) Drinfeld twist on a quasi-Hopf algebra. The twisted form of the Drinfeld twist is investigated. In the quasi-triangular case, it is shown that the Drinfeld u-operator arises from the equivalence of H-COP to the quasi-Hopf algebra induced by twisting H with the R-matrix. The Altschuler-Coste u-operator arises in a similar way and is shown to be closely related to the Drinfeld u-operator. The quasi-cocycle condition is introduced and is shown to play a central role in the uniqueness of twisted structures on quasi-Hopf algebras. A generalization of the dynamical quantum Yang-Baxter equation, called the quasi-dynamical quantum Yang-Baxter equation, is introduced.

Braided m-Lie algebras

Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of End(F)M, where M is a Yetter-Drinfeld module over B with dimB < infinity. In particular, generalized classical braided m-Lie algebras sl(q,f)(GM(G)(A),F) and osp(q,l)(GM(G)(A),M,F) of generalized matrix algebra GMG(A) are constructed and their connection with special generalized matrix Lie superalgebra sl(s,f)(GM(Z2)(A(s)),F) and orthosymplectic generalized matrix Lie super algebra osp(s,l) (GM(Z2)(A(s)),M-s,F) are established. The relationship between representations of braided m-Lie algebras and their associated algebras are established.

Central elements of the elliptic Zn monodromy matrix algebra at roots of unity

The central elements of the algebra of monodromy matrices associated with the Z(n) R-matrix are studied. When the crossing parameter w takes a special rational value w = n/N, where N and n are positive coprime integers, the center is substantially larger than that in the generic case for which the quantum determinant provides the center. In the trigonometric limit, the situation corresponds to the quantum group at roots of unity. This is a higher rank generalization of the recent results by Belavin and Jimbo. (c) 2004 Elsevier B.V. All rights reserved.