Two-sided (two-cosided) Hopf modules and Doi-Hopf modules for quasi-Hopf algebras

Let $H$ be a finite dimensional quasi-Hopf algebra over a field $k$ and ${\mathfrak A}$ a right $H$-comodule algebra in the sense of Hausser and Nill. We first show that on the $k$-vector space {\mathfrak A}\ot H^* we can define an algebra structure, denoted by {\mathfrak A}\ovsm H^*, in the monoidal category of left $H$-modules (i.e. {\mathfrak A}\ovsm H^* is an $H$-module algebra. Then we will prove that the category of two-sided $({\mathfrak A}, H)$-bimodules \hba is isomorphic to the category of relative ({\mathfrak A}\ovsm H^*, H^*)-Hopf modules, as introduced in by Hausser and Nill. In the particular case where ${\mathfrak A}=H$, we will obtain a result announced by Nill. We will also introduce the categories of Doi-Hopf modules and two-sided two-cosided Hopf modules and we will show that they are in certain situations isomorphic to module categories.Comment: 31 page

Integrals for (dual) quasi-Hopf algebras. Applications

A classical result in the theory of Hopf algebras concerns the uniqueness and existence of integrals: for an arbitrary Hopf algebra, the integral space has dimension $\leq 1$, and for a finite dimensional Hopf algebra, this dimension is exaclty one. We generalize these results to quasi-Hopf algebras and dual quasi-Hopf algebras. In particular, it will follow that the bijectivity of the antipode follows from the other axioms of a finite dimensional quasi-Hopf algebra. We give a new version of the Fundamental Theorem for quasi-Hopf algebras. We show that a dual quasi-Hopf algebra is co-Frobenius if and only if it has a non-zero integral. In this case, the space of left or right integrals has dimension one.Comment: 25 pages; new version with minor correction

Parity violation, anyon scattering and the mean field approximation

Some general features of the scattering of boson-based anyons with an added non-statistical interaction are discussed. Periodicity requirements of the phase shifts are derived, and used to illustrate the danger inherent in separating these phase shifts into the well-known pure Aharanov-Bohm phase shifts, and an additional set which arise due to the interaction. It is proven that the added phase shifts, although due to the non-statistical interaction, necessarily change as the statistical parameter is varied, keeping the interaction fixed. A hard-disk interaction provides a concrete illustration of these general ideas. In the latter part of the paper, scattering with an additional hard-disk interaction is studied in detail, with an eye towards providing a criterion for the validity of the mean-field approximation for anyons, which is the first step in virtually any treatment of this system. We find, consistent with previous work, that the approximation is justified if the statistical interaction is weak, and that it must be more weak for boson-based than for fermion-based anyons.Comment: 17 pages plus 3 encoded/compressed post-script figures, UdeM-LPN-TH-94-18

Effective potential for nonrelativistic non-Abelian Chern-Simons matter system in constant background fields

We present the effective potential for nonrelativistic matter coupled to non-Abelian Chern-Simons gauge fields. We perform the calculation using a functional method in constant background fields to satisfy Gauss's law and to simplify the computation. Both the quantum gauge and matter fields are integrated over. The gauge fixing is achieved with an $R_\xi$-gauge in the $\xi\to 0$ limit. Divergences appearing in the matter sector are regulated via operator regularization. We find no corrections to the Chern-Simons coupling constant and the system experiences conformal symmetry breaking to one-loop order except at the known value of self-duality. These results agree with previous analysis of the non-Abelian Aharonov-Bohm scattering.Comment: 17 pages in Tex (phyzzx), UdeM-LPN-TH-94-200, CRM-219