The alternating presentation of Uq ((gl2)over-cap) from Freidel-Maillet algebras

An infinite dimensional algebra denoted (A) over bar (q) that is isomorphic to a central extension of U-q(+)- the positive part of U-q((sl(2)) over cap) - has been recently proposed by Paul Terwilliger. It provides an 'alternating' Poincare-Birkhoff-Witt (PBW) basis besides the known Damiani's PBW basis built from positive root vectors. In this paper, a presentation of (A) over bar (q) in terms of a Freidel-Maillet type algebra is obtained. Using this presentation: (a) finite dimensional tensor product representations for (A) over bar (q) are constructed; (b) explicit isomorphisms from (A) over bar (q) to certain Drinfeld type 'alternating' subalgebras of U-q((gl(2)) over cap) are obtained; (c) the image in U-q(+) of all the generators of (A) over bar (q) in terms of Damiani's root vectors is obtained. A new tensor product decomposition for U-q ((gl(2)) over cap) in terms of Drinfeld type 'alternating' subalgebras follows. The specialization q -> of (A) over bar (q) is also introduced and studied in details. In this case, a presentation is given as a non-standard Yang-Baxter algebra. (C) 2021 The Author(s). Published by Elsevier B.V.