Area-Preserving Algebra Structure of Two-Dimensional Supergravity

The effective action for 2d-gravity in Weyl-invariant regularization is extended to supersymmetric case. The super area-preserving invariance and cocyclic properties under general supergravitational transformations of the last action is shown.Comment: latex file, no figures, 5 pages. Some references are adde

New solutions to the sℓq(2)s\ell_q(2)-invariant Yang-Baxter equations at roots of unity: cyclic representations

We find the all solutions to the $sl_q(2)$-invariant multi-parametric Yang-Baxter equations (YBE) at $q=i$ defined on the cyclic (semi-cyclic, nilpotent) representations of the algebra. We are deriving the solutions in form of the linear combinations over the $sl_q(2)$-invariant objects - projectors. The direct construction of the projector operators at roots of unity gives us an opportunity to consider all the possible cases, including also degenerated one, when the number of the projectors becomes larger, and various type of solutions are arising, and as well as the inhomogeneous case. We are giving a full classification of the YBE solutions for the considered representations. A specific character of the solutions is the existence of the arbitrary functions.Comment: 28 pages; extended version; new references are adde

Trace Anomalies and Cocycles of Weyl and Diffeomorphisms Groups

The general structure of trace anomaly, suggested recently by Deser and Shwimmer, is argued to be the consequence of the Wess-Zumino consistency condition. The response of partition function on a finite Weyl transformation, which is connected with the cocycles of the Weyl group in $d=2k$ dimensions is considered, and explicit answers for $d=4,6$ are obtained. Particularly, it is shown, that addition of the special combination of the local counterterms leads to the simple form of that cocycle, quadratic over Weyl field $\sigma$, i.e. the form, similar to the two-dimensional Lioville action. This form also establishes the connection of the cocycles with conformal-invariant operators of order $d$ and zero weight. Beside that, the general rule for transformation of that cocycles into the cocycles of diffeomorphisms group is presented.Comment: 14 page

Polynomial Realization of sℓq(2)s\ell_q(2) and Fusion Rules at Exceptional Values of qq

Representations of the $s\ell_q(2)$ algebra are constructed in the space of polynomials of real (complex) variable for $q^N=1$. The spin addition rule based on eigenvalues of Casimir operator is illustrated on few simplest cases and conjecture for general case is formulated

Integrable XYZ Model with Staggered Anisotropy Parameter

We apply to the XYZ model the technique of construction of integrable models with staggered parameters, presented recently for the XXZ case. The solution of modified Yang-Baxter equations is found and the corresponding integrable zig-zag ladder Hamiltonian is calculated. The result is coinciding with the XXZ case in the appropriate limit.Comment: 8 pages ; epic packag