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 SIGMA, 2008, Volume 4, 070, 21 pp. (Mi sigma323)

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

DOI: https://doi.org/10.3842/SIGMA.2008.070

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ArXiv: 0806.4851
MSC: 17B67
Received: July 4, 2008; in final form October 6, 2008; Published online October 14, 2008
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Citation: Evgeny Feigin, “The PBW Filtration, Demazure Modules and Toroidal Current Algebras”, SIGMA, 4 (2008), 070, 21 pp.

Citation in format AMSBIB
\Bibitem{Fei08} \by Evgeny Feigin \paper The PBW Filtration, Demazure Modules and Toroidal Current Algebras \jour SIGMA \yr 2008 \vol 4 \papernumber 070 \totalpages 21 \mathnet{http://mi.mathnet.ru/sigma323} \crossref{https://doi.org/10.3842/SIGMA.2008.070} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2470526} \zmath{https://zbmath.org/?q=an:05555842} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000267267800070} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84896060542} 

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This publication is cited in the following articles:
1. Feigin E., Fourier G., Littelmann P., “PBW filtration and bases for irreducible modules in type A(n)”, Transform Groups, 16:1 (2011), 71–89
2. Feigin E., Fourier G., Littelmann P., “PBW Filtration and Bases for Symplectic Lie Algebras”, Int Math Res Not, 2011, no. 24, 5760–5784
3. Feigin E., “Degenerate Flag Varieties and the Median Genocchi Numbers”, Math. Res. Lett., 18:6 (2011), 1163–1178
4. Feigin E., “G(a)(M) Degeneration of Flag Varieties”, Sel. Math.-New Ser., 18:3 (2012), 513–537
5. Bhimarthi Ravinder, “Demazure Modules, Chari–Venkatesh Modules and Fusion Products”, SIGMA, 10 (2014), 110, 10 pp.
6. Kus D., Littelmann P., “Fusion Products and Toroidal Algebras”, Pac. J. Math., 278:2 (2015), 427–445
7. Feigin E., Makedonskyi I., “Nonsymmetric Macdonald Polynomials and Pbw Filtration: Towards the Proof of the Cherednik-Orr Conjecture”, J. Comb. Theory Ser. A, 135 (2015), 60–84
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