|
This article is cited in 7 scientific papers (total in 7 papers)
The PBW Filtration, Demazure Modules and Toroidal Current Algebras
Evgeny Feiginab a I. E. Tamm Department of Theoretical Physics, Lebedev Physics Institute, Leninski Prospect 53, Moscow, 119991, Russia
b Mathematical Institute, University of Cologne, Weyertal 86-90, D-50931, Cologne, Germany
Abstract:
Let $L$ be the basic (level one vacuum) representation of the affine Kac–Moody Lie algebra
$\widehat{\mathfrak g}$. The $m$-th space $F_m$ of the PBW filtration on $L$ is a linear span of vectors of the form $x_1\cdots x_lv_0$, where $l\le m$, $x_i\in\widehat{\mathfrak g}$ and $v_0$ is a highest weight vector of $L$. In this paper we give two descriptions of the associated graded space $L^{\mathrm{gr}}$ with respect to the PBW filtration. The “top-down” description deals with a structure of $L^{\mathrm{gr}}$ 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 $e_\theta(z)^2$, which corresponds to the longest root $\theta$. The “bottom-up” description deals with the structure of $L^{\mathrm{gr}}$ as a representation of the current algebra $\mathfrak g\otimes\mathbb C[t]$. We prove that each quotient $F_m/F_{m-1}$ can be filtered by graded deformations of the tensor products of $m$ copies of $\mathfrak g$.
Keywords:
affine Kac–Moody algebras; integrable representations; Demazure modules.
Received: July 4, 2008; in final form October 6, 2008; Published online October 14, 2008
Citation:
Evgeny Feigin, “The PBW Filtration, Demazure Modules and Toroidal Current Algebras”, SIGMA, 4 (2008), 070, 21 pp.
Linking options:
https://www.mathnet.ru/eng/sigma323 https://www.mathnet.ru/eng/sigma/v4/p70
|
Statistics & downloads: |
Abstract page: | 294 | Full-text PDF : | 68 | References: | 32 |
|