paper

The alternating presentation of $U_{q} (g l_{2})$ from Freidel-Maillet algebras

arXiv:2011.01572 · doi:10.1016/j.nuclphysb.2021.115400

Abstract

An infinite dimensional algebra denoted $A ˉ_{q}$ that is isomorphic to a central extension of $U_{q}^{+}$ - the positive part of $U_{q} (s l_{2})$ - has been recently proposed by Paul Terwilliger. It provides an `alternating' PoincarÃ©-Birkhoff-Witt (PBW) basis besides the known Damiani's PBW basis built from positive root vectors. In this paper, a presentation of $A ˉ_{q}$ in terms of a Freidel-Maillet type algebra is obtained. Using this presentation: (a) finite dimensional tensor product representations for $A ˉ_{q}$ are constructed; (b) explicit isomorphisms from $A ˉ_{q}$ to certain Drinfeld type `alternating' subalgebras of $U_{q} (g l_{2})$ are obtained; (c) the image in $U_{q}^{+}$ of all the generators of $A ˉ_{q}$ in terms of Damiani's root vectors is obtained. A new tensor product decomposition for $U_{q} (s l_{2})$ in terms of Drinfeld type `alternating' subalgebras follows. The specialization $q \to 1$ of $A ˉ_{q}$ is also introduced and studied in details. In this case, a presentation is given as a non-standard Yang-Baxter algebra. This paper is dedicated to Paul Terwilliger for his 65th birthday.

39 pages; v2: Some notations improved; Typos corrected; References added

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

Abstract

The alternating presentation of $U_{q} (g l_{2})$ from Freidel-Maillet algebras