Realisations of GLp,q(2)GL_{p,q}(2) quantum group and its coloured extension through a novel Hopf algebra with five generators

A novel Hopf algebra $( {\tilde G}_{r,s} )$, depending on two deformation parameters and five generators, has been constructed. This ${\tilde G}_{r,s}$ Hopf algebra might be considered as some quantisation of classical $GL(2) \otimes GL(1)$ group, which contains the standard $GL_q(2)$ quantum group (with $q=r^{-1}$) as a Hopf subalgebra. However, we interestingly observe that the two parameter deformed $GL_{p,q}(2)$ quantum group can also be realised through the generators of this ${\tilde G}_{r,s}$ algebra, provided the sets of deformation parameters $p,~q$ and $r,~s$ are related to each other in a particular fashion. Subsequently we construct the invariant noncommutative planes associated with ${\tilde G}_{r,s}$ algebra and show how the two well known Manin planes corresponding to $GL_{p,q}(2)$ quantum group can easily be reproduced through such construction. Finally we consider the `coloured' extension of $GL_{p,q}(2)$ quantum group as well as corresponding Manin planes and explore their intimate connection with the `coloured' extension of ${\tilde G}_{r,s}$ Hopf structure.Comment: 24 page

Algebraic aspect and construction of Lax operators in quantum integrable systems

An algebraic construction more general and intimately connected with that of Faddeev$^1$, along with its application for generating different classes of quantum integrable models are summarised to complement the recent results of ref. 1 ( L.D. Faddeev, {\it Int. J. Mod. Phys. } {\bf A10}, 1845 (1995) ).Comment: 8 pages, plain TEX, no figure

Nilpotent Completely Positive Maps

We study the structure of nilpotent completely positive maps in terms of Choi-Kraus coefficients. We prove several inequalities, including certain majorization type inequalities for dimensions of kernels of powers of nilpotent completely positive maps.Comment: 10 page

Construction of some special subsequences within a Farey sequence

Recently it has been found that some special subsequences within a Farey sequence play a crucial role in determining the ranges of coupling constant for which quantum soliton states can exist for an integrable derivative nonlinear Schrodinger model. In this article, we find a novel mapping which connects two such subsequences belonging to Farey sequences of different orders. By using this mapping, we construct an algorithm to generate all of these special subsequences within a Farey sequence. We also derive the continued fraction expansions for all the elements belonging to a subsequence and observe a close connection amongst the corresponding expansion coefficients.Comment: latex, 8 page