The alternating presentation of Uq(gl2ˆ) from Freidel-Maillet algebras

• Main Author: Pascal Baseilhac
• Format: Article
• Language: English
• Published: Elsevier 2021-06-01
• Series: Nuclear Physics B
• Online Access: http://www.sciencedirect.com/science/article/pii/S0550321321000973

An infinite dimensional algebra denoted A¯q that is isomorphic to a central extension of Uq+ - the positive part of Uq(sl2ˆ) - has been recently proposed by Paul Terwilliger. It provides an ‘alternating’ Poincaré-Birkhoff-Witt (PBW) basis besides the known Damiani's PBW basis built from positive root vectors. In this paper, a presentation of A¯q in terms of a Freidel-Maillet type algebra is obtained. Using this presentation: (a) finite dimensional tensor product representations for A¯q are constructed; (b) explicit isomorphisms from A¯q to certain Drinfeld type ‘alternating’ subalgebras of Uq(gl2ˆ) are obtained; (c) the image in Uq+ of all the generators of A¯q in terms of Damiani's root vectors is obtained. A new tensor product decomposition for Uq(sl2ˆ) in terms of Drinfeld type ‘alternating’ subalgebras follows. The specialization q→1 of A¯q is also introduced and studied in details. In this case, a presentation is given as a non-standard Yang-Baxter algebra.