Exotic Bialgebra S03: Representations, Baxterisation and Applications

The exotic bialgebra S03, defined by a solution of the Yang-Baxter equation, which is not a deformation of the trivial, is considered. Its FRT dual algebra $s03_F$ is studied. The Baxterisation of the dual algebra is given in two different parametrisations. The finite-dimensional representations of $s03_F$ are considered. Diagonalisations of the braid matrices are used to yield remarkable insights concerning representations of the L-algebra and to formulate the fusion of finite-dimensional representations. Possible applications are considered, in particular, an exotic eight-vertex model and an integrable spin-chain model.Comment: 24 pages, Latex; V2: revised subsection 4.1, added 9 references, to appear in Annales Henri Poincare in the volume dedicated to D. Arnaudo

On a "New" Deformation of GL(2)

We refute a recent claim in the literature of a "new" quantum deformation of GL(2).Comment: 4 pages, LATE

Duality and Representations for New Exotic Bialgebras

We find the exotic matrix bialgebras which correspond to the two non-triangular nonsingular 4x4 R-matrices in the classification of Hietarinta, namely, R_{S0,3} and R_{S1,4}. We find two new exotic bialgebras S03 and S14 which are not deformations of the of the classical algebras of functions on GL(2) or GL(1|1). With this we finalize the classification of the matrix bialgebras which unital associative algebras generated by four elements. We also find the corresponding dual bialgebras of these new exotic bialgebras and study their representation theory in detail. We also discuss in detail a special case of R_{S1,4} in which the corresponding algebra turns out to be a special case of the two-parameter quantum group deformation GL_{p,q}(2).Comment: 33 pages, LaTeX2e, using packages: cite,amsfonts,amsmath,subeqn; reference updated; v3: corrections in subsection 3.

Fifty years of spellchecking

A short history of spellchecking from the late 1950s to the present day, describing its development through dictionary lookup, affix stripping, correction, confusion sets, and edit distance to the use of gigantic databases

Duality for the Jordanian Matrix Quantum Group GLg,h(2)GL_{g,h}(2)

We find the Hopf algebra $U_{g,h}$ dual to the Jordanian matrix quantum group $GL_{g,h}(2)$. As an algebra it depends only on the sum of the two parameters and is split in two subalgebras: $U'_{g,h}$ (with three generators) and $U(Z)$ (with one generator). The subalgebra $U(Z)$ is a central Hopf subalgebra of $U_{g,h}$. The subalgebra $U'_{g,h}$ is not a Hopf subalgebra and its coalgebra structure depends on both parameters. We discuss also two one-parameter special cases: $g =h$ and $g=-h$. The subalgebra $U'_{h,h}$ is a Hopf algebra and coincides with the algebra introduced by Ohn as the dual of $SL_h(2)$. The subalgebra $U'_{-h,h}$ is isomorphic to $U(sl(2))$ as an algebra but has a nontrivial coalgebra structure and again is not a Hopf subalgebra of $U_{-h,h}$.Comment: plain TeX with harvmac, 16 pages, added Appendix implementing the ACC nonlinear ma