The alternating presentation of $U_q(\widehat{g{l}_2})$ from Freidel-Maillet algebras

Abstract

An infinite dimensional algebra denoted ${\bar{\mathcal{A}}}_q$ that is isomorphic to a central extension of $U_q^{+}$ - the positive part of $U_q(\widehat{s{l}_2})$ - 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 ${\bar{\mathcal{A}}}_q$ in terms of a Freidel-Maillet type algebra is obtained. Using this presentation: (a) finite dimensional tensor product representations for ${\bar{\mathcal{A}}}_q$ are constructed; (b) explicit isomorphisms from ${\bar{\mathcal{A}}}_q$ to certain Drinfeld type `alternating' subalgebras of $U_q(\widehat{g{l}_2})$ are obtained; (c) the image in $U_q^{+}$ of all the generators of ${\bar{\mathcal{A}}}_q$ in terms of Damiani's root vectors is obtained. A new tensor product decomposition for $U_q(\widehat{s{l}_2})$ in terms of Drinfeld type `alternating' subalgebras follows. The specialization $q\to 1$ of ${\bar{\mathcal{A}}}_q$ 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

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Cited by in corpus (9)

• The alternating central extension of the $q$-Onsager algebra
• A compact presentation for the alternating central extension of the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• The $q$-Onsager algebra and its alternating central extension
• On the second realization for the positive part of $U_q(\widehat{s{l}_2})$ of equitable type
• Using Catalan words and a $q$-shuffle algebra to describe the Beck PBW basis for the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• Tridiagonal pairs, alternating elements, and distance-regular graphs
• A generating function associated with the alternating elements in the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• Freidel-Maillet type presentations of $U_q(s{l}_2)$
• Using a $q$-shuffle algebra to describe the basic module $V({\Lambda }_0)$ for the quantized enveloping algebra $U_q({\widehat{\mathfrak{sl}}}_2)$