## Abstract

Let L be the basic (level one vacuum) representation of the affine Kac-Moody Lie algebra g. The m-th space Fm of the PBW filtration on L is a linear span of vectors of the form x_{1} ···x_{l}v_{0}, where l ≤ m, x_{i} ∈ g and v0 is a highest weight vector of L. In this paper we give two descriptions of the associated graded space L^{gr} with respect to the PBW filtration. The "top-down" description deals with a structure of L^{gr} as a representation of the abelianized algebra of generating operators. We prove that the ideal of relations is generated by the coefficients of the squared field eθ(z)^{2}, which corresponds to the longest root θ. The bottom-up description deals with the structure of L^{gr} as a representation of the current algebra g ⊗ C [t]. We prove that each quotient F_{m}/F_{m-1} can be filtered by graded deformations of the tensor products of m copies of g.

Original language | English |
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Article number | 070 |

Journal | Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) |

Volume | 4 |

DOIs | |

State | Published - 2008 |

Externally published | Yes |

## Keywords

- Affine Kac-moody algebras
- Demazure modules
- Integrable representations