The PBW filtration
HTML articles powered by AMS MathViewer
- by Evgeny Feigin
- Represent. Theory 13 (2009), 165-181
- DOI: https://doi.org/10.1090/S1088-4165-09-00349-5
- Published electronically: May 1, 2009
- PDF | Request permission
Abstract:
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 $x_1\dots x_s v$, where $x_i\in \widehat {\mathfrak {g}}$, $s\le n$, 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.References
- Eddy Ardonne, Rinat Kedem, and Michael Stone, Fermionic characters and arbitrary highest-weight integrable $\widehat {\mathfrak {sl}}_{r+1}$-modules, Comm. Math. Phys. 264 (2006), no. 2, 427–464. MR 2215613, DOI 10.1007/s00220-005-1486-3
- Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, vol. 88, American Mathematical Society, Providence, RI, 2001. MR 1849359, DOI 10.1090/surv/088
- C. Calinescu, Principal subspaces of higher-level deformed $\widehat {\mathfrak {sl}_2}$-modules, math.0611534.
- C. Calinescu, J. Lepowsky, and A. Milas, Vertex-algebraic structure of the principal subspaces of certain $A_1^{(1)}$-modules. I. Level one case, Internat. J. Math. 19 (2008), no. 1, 71–92. MR 2380473, DOI 10.1142/S0129167X08004571
- Chongying Dong, Vertex algebras associated with even lattices, J. Algebra 161 (1993), no. 1, 245–265. MR 1245855, DOI 10.1006/jabr.1993.1217
- B. Feigin and E. Feigin, Two-dimensional current algebras and affine fusion product, J. Algebra 313 (2007), no. 1, 176–198. MR 2326142, DOI 10.1016/j.jalgebra.2006.11.039
- B. Feigin, E. Feigin, M. Jimbo, T. Miwa, and Y. Takeyama, A $\phi _{1,3}$-filtration of the Virasoro minimal series $M(p,p’)$ with $1<p’/p<2$, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 213–257. MR 2426348, DOI 10.2977/prims/1210167327
- 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. MR 2166753, DOI 10.1215/S0012-7094-04-12533-3
- B. Feigin, R. Kedem, S. Loktev, T. Miwa, and E. Mukhin, Combinatorics of the $\widehat {\mathfrak {sl}}_2$ spaces of coinvariants, Transform. Groups 6 (2001), no. 1, 25–52. MR 1825167, DOI 10.1007/BF01236061
- 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. MR 1729359, DOI 10.1090/trans2/194/04
- 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. MR 1275728, DOI 10.1007/BF01079010
- 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. MR 595581, DOI 10.1007/BF01391662
- Galin Georgiev, Combinatorial constructions of modules for infinite-dimensional Lie algebras. I. Principal subspace, J. Pure Appl. Algebra 112 (1996), no. 3, 247–286. MR 1410178, DOI 10.1016/0022-4049(95)00143-3
- Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
- Victor Kac, Vertex algebras for beginners, University Lecture Series, vol. 10, American Mathematical Society, Providence, RI, 1997. MR 1417941, DOI 10.1090/ulect/010
- James Lepowsky and Mirko Primc, Structure of the standard modules for the affine Lie algebra $A^{(1)}_1$, Contemporary Mathematics, vol. 46, American Mathematical Society, Providence, RI, 1985. MR 814303, DOI 10.1090/conm/046
- A. Meurman and M. Primc, Annihilating fields of standard modules of $sl(2,\mathbb {C})$ and combinatorial identities, Mem. Amer. Math. Soc. 652 (1999).
- M. Primc, Vertex operator construction of standard modules for $A^{(1)}_n$, Pacific J. Math. 162 (1994), no. 1, 143–187. MR 1247147, DOI 10.2140/pjm.1994.162.143
- Anne Schilling and S. Ole Warnaar, Supernomial coefficients, polynomial identities and $q$-series, Ramanujan J. 2 (1998), no. 4, 459–494. MR 1665322, DOI 10.1023/A:1009780810189
Bibliographic Information
- Evgeny Feigin
- Affiliation: Tamm Theory Division, Lebedev Physics Institute, Russian Academy of Sciences, Russia, 119991, Moscow, Leninski prospect, 53 – and – Independent University of Moscow, Russia, Moscow, 119002, Bol’shoi Vlas’evskii, 11
- Email: evgfeig@gmail.com
- Received by editor(s): November 15, 2007
- Received by editor(s) in revised form: February 4, 2009
- Published electronically: May 1, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory 13 (2009), 165-181
- MSC (2000): Primary 17B67, 81R10
- DOI: https://doi.org/10.1090/S1088-4165-09-00349-5
- MathSciNet review: 2506263