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We give solutions of the q-deformed equations of quantum conformal Weyl gravity in terms of q-deformed plane waves., Comment: 13 pages, LATEX with revtex4; Proceedings of the XII International Conference on Symmetry Methods in Physics, (Yerevan, 2006) eds. G.S. Pogosyan et al; V2: added Summary and Outlook

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

In the classification of solutions of the Yang--Baxter equation, there are solutions that are not deformations of the trivial solution (essentially the identity). We consider the algebras defined by these solutions, and the corresponding dual algebras. We then study the representations of the latter. We are also interested in the Baxterisation of these $R$-matrices and in the corresponding quantum planes., Comment: 11 pages, LaTeX2e, requires ws-procs9x6.cls; using packages: amsfonts, amsmath, amssymb; talk by D.A. at 5-th International Workshop "Lie Theory and Its Applications in Physics", Varna, Bulgaria, (June 2003), to appear in the Proceedings (World Scientific, Singapore, 2004)

We give new solutions of the quantum conformal deformations of the full Maxwell equations in terms of deformations of the plane wave. We study the compatibility of these solutions with the conservation of the current. We also start the study of quantum linear conformal (Weyl) gravity by writing the corresponding q-deformed equations., Comment: 18 pages, Latex2e; v2: corrected misprints; v3: small text improvement; v4: misprints in f-la (3.14) corrected

We study the geometric aspects of two exotic bialgebras S03 and S14 introduced in math.QA/0206053. These bialgebras are obtained by the Faddeev-Reshetikhin-Takhtajan RTT prescription with non-triangular R-matrices which are denoted $R_{03}$ and $R_{14}$ in the classification of Hietarinta, and they are not deformations of either GL(2) or GL(1/1). We give the spectral decomposition which involves two, resp., three, projectors. These projectors are then used to provide the Baxterisation procedure with one, resp., two, parameters. Further, the projectors are used to construct the noncommutative planes together with the corresponding differentials following the Wess-Zumino prescription. In all these constructions there appear non-standard features which are noted. Such features show the importance of systematic study of all bialgebras of four generators., Comment: 14 pages, LaTeX2e, using packages: cite, amsfonts, amsmath, subeqn; v2: small improvements, coincides with journal version

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 3.4

In the classification of Hietarinta, three triangular $4\times 4$ $R$-matrices lead, via the FRT formalism, to matrix bialgebras which are not deformations of the trivial one. In this paper, we find the bialgebras which are in duality with these three exotic matrix bialgebras. We note that the $L-T$ duality of FRT is not sufficient for the construction of the bialgebras in duality. We find also the quantum planes corresponding to these bialgebras both by the Wess-Zumino R-matrix method and by Manin's method., Comment: 25 pages, LaTeX2e, using packages: cite, amsfonts, amsmath, subeqn

Combined $(q,h)$-deformations proposed by Kupershmidt and Ballesteros-Herranz-Parashar are studied. In each case a transformation is shown to lead to an equivalent, standard $q$-deformation. We briefly indicate that appropriate singular limits of the same type of transformations can however lead from standard biparametric $(p,q)$-deformations to non-hybrid but biparametric nonstandard $(g,h)$ ones. Finally a case of hybrid $(q,h)$-deformation is recalled, related to the superalgebra $GL(1|1)$., Comment: 17 pages, LaTeX2e, using packages: amsfonts, amsmath; V2: added preprint number, corrected misprints; J. Math. Phys., to appear