arXiv subjects
Publications and source records attributed to Sergey Sinel'shchikov.
A complete list of Uq(sl2)-module algebra structures on the quantum plane is produced and the (uncountable family of) isomorphism classes of these structures are described. The composition series of representations in question are computed. The classical limits of the Uq(sl2)-module algebra structures are discussed.
The main result of this work is to present the complete list of Uq(sl2)-symmetries of quantum plane. For that, the structure of quantum plane automorphisms is used. Our idea in classifying the above symmetries is in introducing some special symbolic matrices to label the series of actions in question. The matrices depict which generators of Uq(sl2) produce nonzero images by acting on the generators of quantum plane. This data determines the weights of an action unambiguously. It turns out that there are uncountably many isomorphism classes of the symmetries. Another result is in describing composition series of the corresponding representations. The classical limit of the above symmetries is given.
Quantum bialgebras derivable from Uq(sl2) which contain idempotents and von Neumann regular Cartan-like generators are introduced and investigated. Various types of antipodes (invertible and von Neumann regular) on these bialgebras are constructed, which leads to a Hopf algebra structure and a von Neumann-Hopf algebra structure, respectively. For them, explicit forms of some particular R-matrices (also, invertible and von Neumann regular) are presented, and the latter respects the Pierce decomposition.