×

zbMATH — the first resource for mathematics

The PBW filtration. (English) Zbl 1229.17026
Summary: In this paper we study the PBW filtration on irreducible integrable highest weight representations of affine Kac-Moody algebras \(\widehat{\mathfrak{g}}\). The \(n\)th space of this filtration is spanned by the vectors \(x_1\dots x_sv\), where \(x_i\in\widehat{\mathfrak{g}}\), \(s\leq n\), and \(v\) is a highest weight vector. For the vacuum module we give a conjectural description of the corresponding adjoint graded space in terms of generators and relations. For \(\mathfrak{g}\) of the type \(A_1\) we prove our conjecture and derive the fermionic formula for the graded character.

MSC:
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Eddy Ardonne, Rinat Kedem, and Michael Stone, Fermionic characters and arbitrary highest-weight integrable \Hat \?\?\?\(_{+}\)\(_{1}\)-modules, Comm. Math. Phys. 264 (2006), no. 2, 427 – 464. · Zbl 1233.17017 · doi:10.1007/s00220-005-1486-3 · doi.org
[2] Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, vol. 88, American Mathematical Society, Providence, RI, 2001. · Zbl 0981.17022
[3] C. Calinescu, Principal subspaces of higher-level deformed \( \widehat{\mathfrak{sl}_2}\)-modules, math.0611534.
[4] C. Calinescu, J. Lepowsky, and A. Milas, Vertex-algebraic structure of the principal subspaces of certain \?\(_{1}\)\?\textonesuperior \?-modules. I. Level one case, Internat. J. Math. 19 (2008), no. 1, 71 – 92. · Zbl 1184.17012 · doi:10.1142/S0129167X08004571 · doi.org
[5] Chongying Dong, Vertex algebras associated with even lattices, J. Algebra 161 (1993), no. 1, 245 – 265. · Zbl 0807.17022 · doi:10.1006/jabr.1993.1217 · doi.org
[6] B. Feigin and E. Feigin, Two-dimensional current algebras and affine fusion product, J. Algebra 313 (2007), no. 1, 176 – 198. · Zbl 1132.17013 · doi:10.1016/j.jalgebra.2006.11.039 · doi.org
[7] B. Feigin, E. Feigin, M. Jimbo, T. Miwa, and Y. Takeyama, A \?_1,3-filtration of the Virasoro minimal series \?(\?,\?’) with 1<\?’/\?<2, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 213 – 257. · Zbl 1162.17025 · doi:10.2977/prims/1210167327 · doi.org
[8] B. Feigin, M. Jimbo, R. Kedem, S. Loktev, and T. Miwa, Spaces of coinvariants and fusion product. I. From equivalence theorem to Kostka polynomials, Duke Math. J. 125 (2004), no. 3, 549 – 588. · Zbl 1129.17304 · doi:10.1215/S0012-7094-04-12533-3 · doi.org
[9] B. Feigin, R. Kedem, S. Loktev, T. Miwa, and E. Mukhin, Combinatorics of the \Hat \?\?\(_{2}\) spaces of coinvariants, Transform. Groups 6 (2001), no. 1, 25 – 52. · Zbl 1004.17001 · doi:10.1007/BF01236061 · doi.org
[10] B. Feigin and S. Loktev, On generalized Kostka polynomials and the quantum Verlinde rule, Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl. Ser. 2, vol. 194, Amer. Math. Soc., Providence, RI, 1999, pp. 61 – 79. · Zbl 0974.17008 · doi:10.1090/trans2/194/04 · doi.org
[11] A. V. Stoyanovskiĭ and B. L. Feĭgin, Functional models of the representations of current algebras, and semi-infinite Schubert cells, Funktsional. Anal. i Prilozhen. 28 (1994), no. 1, 68 – 90, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 28 (1994), no. 1, 55 – 72. · Zbl 0905.17030 · doi:10.1007/BF01079010 · doi.org
[12] I. B. Frenkel and V. G. Kac, Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980/81), no. 1, 23 – 66. · Zbl 0493.17010 · doi:10.1007/BF01391662 · doi.org
[13] Galin Georgiev, Combinatorial constructions of modules for infinite-dimensional Lie algebras. I. Principal subspace, J. Pure Appl. Algebra 112 (1996), no. 3, 247 – 286. · Zbl 0871.17018 · doi:10.1016/0022-4049(95)00143-3 · doi.org
[14] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. · Zbl 0716.17022
[15] Victor Kac, Vertex algebras for beginners, University Lecture Series, vol. 10, American Mathematical Society, Providence, RI, 1997. · Zbl 0861.17017
[16] James Lepowsky and Mirko Primc, Structure of the standard modules for the affine Lie algebra \?\?\textonesuperior \?\(_{1}\), Contemporary Mathematics, vol. 46, American Mathematical Society, Providence, RI, 1985. · Zbl 0569.17007
[17] A. Meurman and M. Primc, Annihilating fields of standard modules of \( sl(2,\mathbb{C})\) and combinatorial identities, Mem. Amer. Math. Soc. 652 (1999). · Zbl 0918.17018
[18] M. Primc, Vertex operator construction of standard modules for \?\?\textonesuperior \?_\?, Pacific J. Math. 162 (1994), no. 1, 143 – 187. · Zbl 0787.17024
[19] Anne Schilling and S. Ole Warnaar, Supernomial coefficients, polynomial identities and \?-series, Ramanujan J. 2 (1998), no. 4, 459 – 494. · Zbl 0921.05007 · doi:10.1023/A:1009780810189 · doi.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.