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Algebras and Representation Theory

, Volume 18, Issue 1, pp 103–135 | Cite as

Elliptic Algebra \(U_{q,p}(\widehat {\mathfrak {g}})\) and Quantum Z-algebras

  • Rasha M. Farghly
  • Hitoshi KonnoEmail author
  • Kazuyuki Oshima
Article

Abstract

A new definition of the elliptic algebra \(U_{q,p}(\widehat {\mathfrak {g}})\) associated with an untwisted affine Lie algebra \(\widehat {\mathfrak {g}}\) is given as a topological algebra over the ring of formal power series in p. We also introduce a quantum dynamical analogue of Lepowsky-Wilson’s Z-algebras. The Z-algebra governs the irreducibility of the infinite dimensional \(U_{q,p}({\widehat {\mathfrak {g}}})\)-modules. Some level-1 examples indicate a direct connection of the irreducible \(U_{q,p}(\widehat {\mathfrak {g}})\)-modules to those of the W-algebras associated with the coset \(\widehat {\mathfrak {g}}\oplus \widehat {\mathfrak {g}}\supset (\widehat {\mathfrak {g}})_{{diag}}\) with level (rg − 1, 1) (g:the dual Coxeter number), which includes Fateev-Lukyanov’s W B l -algebra.

Keywords

Quantum group Affine Lie algebra Virasoro algebra W-algebra Z-algebra 

Mathematics Subject Classification (2010)

17B37 20G42 81R10 81R50 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Rasha M. Farghly
    • 1
  • Hitoshi Konno
    • 2
    Email author
  • Kazuyuki Oshima
    • 3
  1. 1.Department of Mathematics, Graduate School of ScienceHiroshima UniversityHigashi-HiroshimaJapan
  2. 2.Department of MathematicsTokyo University of Marine Science and TechnologyKotoJapan
  3. 3.Department of Mathematics, Center for General EducationAichi Institute of TechnologyYakusa-choJapan

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