$W_{1+\infty }$ and $\tilde{W}$ algebras, and Ward identities

Abstract

It was demonstrated recently that the $W_{1+\infty }$ algebra contains commutative subalgebras associated with all integer slope rays (including the vertical one). In this paper, we realize that every element of such a ray is associated with a generalized $\tilde{W}$ algebra. In particular, the simplest commutative subalgebra associated with the rational Calogero Hamiltonians is associated with the $\tilde{W}$ algebras studied earlier. We suggest a definition of the generalized $\tilde{W}$ algebra as differential operators in variables $p_k$ basing on the matrix realization of the $W_{1+\infty }$ algebra, and also suggest an unambiguous recursive definition, which, however, involves more elements of the $W_{1+\infty }$ algebra than is contained in its commutative subalgebras. The positive integer rays are associated with $\tilde{W}$ algebras that form sets of Ward identities for the WLZZ matrix models, while the vertical ray associated with the trigonometric Calogero-Sutherland model describes the hypergeometric $\tau$-functions corresponding to the completed cycles.

11 pages, 1 figure

References in corpus (7)

• Superintegrability for ($\beta$-deformed) partition function hierarchies with $W$-representations
• On KP-integrable skew Hurwitz $\tau$-functions and their $\beta$-deformations
• Interpolating Matrix Models for WLZZ series
• Many-body integrable systems implied by WLZZ models
• Non-Abelian W-representation for GKM
• Commutative subalgebras from Serre relations
• Spectral curves and $W$-representations of matrix models

Cited by in corpus (5)

• Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models
• Two $\beta$-ensemble realization of $\beta$-deformed WLZZ models
• On character expansion and Gaussian regularization of Itzykson-Zuber measure
• Generalized $\tilde{W}$ algebras
• Phases in WLZZ Matrix Models