Deformed ${\cal W}_{N}$ Algebras from Elliptic sl ( N ) Algebras

Abstract

We extend to the sl(N ) case the results that we previously obtained on the construction of W q,p algebras from the elliptic algebra A q,p ( sl(2) c ). The elliptic algebra A q,p ( sl(N ) c ) at the critical level c = −N has an extended center containing trace-like operators t(z). Families of Poisson structures indexed by N (N − 1)/2 integers, defining q-deformations of the W N algebra, are constructed. The operators t(z) also close an exchange algebra when (−pIt becomes Abelian when in addition p = q N h where h is a non-zero integer. The Poisson structures obtained in these classical limits contain different q-deformed W N algebras depending on the parity of h, characterizing the exchange structures at p = q N h as new W q,p (sl(N )) algebras. RésuméNousétendons au cas sl(N ) les résultats que nous avons obtenus précédemment concernant la construction des algèbres W q,pà partir de l'algèbre elliptique A q,p ( sl (2)

Deformed Double Yangian Structures

“…This proof reproduces exactly the proof given in [20,21] for the identification of exchange properties in the elliptic case. The relation c = −N − Mr is the scaling limit of the relation p M/2 = q −c−N in [21].…”

“…Associated non-linear Poisson structures realising q-deformed classical W N algebras were derived for G = sl(N) [15,16] and quantised [16,17]. A similar, albeit more complex, structure of sub-exchange algebras, extended centres or Abelian algebras, and classical (Poisson bracket) limits thereof, was uncovered for A q,p ( sl(2) c ) [18,19] and A q,p ( sl(N) c ) [20,21]. They realise new quantum versions of the already known q-W N Poisson structure [15,16].…”

Towards a cladistics of double Yangians and elliptic algebras*

“…The central extension of this structure was proposed in [11] for sl (2), and later extended to A q,p ( sl(N) c ) in [12]. Its connection to q-deformed Virasoro and W N algebras [13,14,15] was established in [16,17].…”

On deformed $\mathcal{W}$-algebras and quantum affine algebras

“…In [22,23] it has been argued that this deformed Virasoro algebra plays the role of the dynamical symmetry algebra in the Andrews-Baxter-Forrester RSOS models. The higher rank generalizations, W q,t [sl N ], were introduced in [10,3,18,11,15,2]. The deformed W-algebras W q,t [g], for arbitrary simple finite dimensional Lie algebras g, were introduced recently by Frenkel and Reshetikhin [15] and further studied in, e.g., [21].…”