The deformed Virasoro algebra at roots of unity

Abstract

We discuss some aspects of the representation theory of the deformed Virasoro algebra $\virpq$. In particular, we give a proof of the formula for the Kac determinant and then determine the center of $\virpq$ for $q$ a primitive N-th root of unity. We derive explicit expressions for the generators of the center in the limit $t=q{p}^{-1}\to \infty$ and elucidate the connection to the Hall-Littlewood symmetric functions. Furthermore, we argue that for $q=\sqrtN{1}$ the algebra describes `Gentile statistics' of order $N-1$, i.e., a situation in which at most $N-1$ particles can occupy the same state.

51 pages, TeX (with amssym.def)

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Cited by in corpus (18)

• Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra
• 2d-4d Connection between q-Virasoro/W Block at Root of Unity Limit and Instanton Partition Function on ALE Space
• $q$-Virasoro/W Algebra at Root of Unity and Parafermions
• Matrix model from N = 2 orbifold partition function
• Five-dimensional SU(2) AGT conjecture and recursive formula of deformed Gaiotto state
• $q$-Virasoro modular double and 3d partition functions
• Notes on Enhancement of Flavor Symmetry and 5d Superconformal Index
• $G$-covariant $\varphi$-coordinated quasi modules for quantum vertex algebras
• Central extensions of classical and quantum q-Viraroso algebras
• Kac determinant and singular vector of the level N representation of Ding-Iohara-Miki algebra
• Norm of the Whittaker vector of the deformed Virasoro algebra
• Twisted reduction of quiver W-algebras
• On deformed W-algebras and quantum affine algebras
• On refined Chern-Simons and refined ABJ matrix models
• Associating quantum vertex algebras to certain deformed Heisenberg Lie algebras
• Whittaker vector of deformed Virasoro algebra and Macdonald symmetric functions
• Twisted and Non-Twisted Deformed Virasoro Algebra via Vertex Operators of $U_q({\widehat{\mathfrak{sl}}}_2)$
• Singular Vector of Ding-Iohara-Miki Algebra and Hall-Littlewood Limit of 5D AGT Conjecture