×

Bases of Feigin-Stoyanovsky’s type subspaces for \(C_\ell ^{(1)}\). (English) Zbl 1430.17074

A Feigin-Stoyanovsky’s type subspace is a subspace of a standard module for an affine Lie algebra. These subspaces are similar to the subspaces first defined in Stoyanovky-Feigin [G. Trupčević, Commun. Algebra 38, No. 10, 3913–3940 (2010; Zbl 1221.17026)]. Bases for these have been found for some affine algebras and some levels such as \(A_{\ell}^{(1)}\), \(B_{2}^{(1)}\), \(D_{4}^{(1)}\). These bases are described in terms of certain monomials starting from the Poincaré-Birkhoff-Witt theorem, satisfying certain difference conditions and initial conditions. The bases are described combinatorially by the use of paths in the set of colors \(\Gamma\). The present paper gives combinatorial bases in the case of level \(k\) modules of the affine Lie algebra \(C_{\ell}^{(1)}\).

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B69 Vertex operators; vertex operator algebras and related structures
05A19 Combinatorial identities, bijective combinatorics

Citations:

Zbl 1221.17026
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ardonne, E., Kedem, R., Stone, M.: Fermionic characters and arbitrary highest-weight integrable \[\widehat{\mathfrak{sl}}_{r+1}\] sl^r+1 -modules. Commun. Math. Phys. 264, 427-464 (2006) · Zbl 1233.17017 · doi:10.1007/s00220-005-1486-3
[2] Baranović, I.: Combinatorial bases of Feigin-Stoyanovsky’s type subspaces of level \[22\] standard modules for \[D_4^{(1)}\] D4(1). Commun. Algebr. 39, 1007-1051 (2011) · Zbl 1269.17011 · doi:10.1080/00927871003639329
[3] Bourbaki, N.: Groupes et Algebres de Lie. Hermann, Paris (1975) · Zbl 0329.17002
[4] Butorac, M.: Combinatorial bases of principal subspaces for the affine Lie algebra of type \[B_2^{(1)}\] B2(1). J. Pure Appl. Algebr. 218, 424-447 (2014) · Zbl 1281.17025 · doi:10.1016/j.jpaa.2013.06.013
[5] Calinescu, C.: Principal subspaces of higher-level standard \[\widehat{\mathfrak{sl}(3)}\] sl(3)^-modules. J. Pure Appl. Algebr. 210, 559-575 (2007) · Zbl 1184.17010 · doi:10.1016/j.jpaa.2006.10.018
[6] Calinescu, C., Lepowsky, J., Milas, A.: Vertex-algebraic structure of the principal subspaces of certain \[A_1^{(1)}\] A1(1) -modules, II: higher-level case. J. Pure Appl. Algebr. 212, 1928-1950 (2008) · Zbl 1184.17013 · doi:10.1016/j.jpaa.2008.01.003
[7] Calinescu, C., Lepowsky, J., Milas, A.: Vertex-algebraic structure of the principal subspaces of level one modules for the untwisted affine Lie algebras of types A, D, E. J. Algebr. 323, 167-192 (2010) · Zbl 1221.17032 · doi:10.1016/j.jalgebra.2009.09.029
[8] Capparelli, S., Lepowsky, J., Milas, A.: The Rogers-Ramanujan recursion and intertwining operators. Commun. Contemp. Math. 5, 947-966 (2003) · Zbl 1039.05005 · doi:10.1142/S0219199703001191
[9] Capparelli, S., Lepowsky, J., Milas, A.: The Rogers-Selberg recursions, the Gordon-Andrews identities and intertwining operators. Ramanujan J. 12, 379-397 (2006) · Zbl 1166.17009 · doi:10.1007/s11139-006-0150-7
[10] Dong, C., Lepowsky, J.: Generalized Vertex Algebras and Relative Vertex Operators. Progress in Mathematics, vol. 112. Birkhaüser, Boston (1993) · Zbl 0803.17009 · doi:10.1007/978-1-4612-0353-7
[11] Dong, C., Li, H., Mason, G.: Simple currents and extensions of vertex operator algebras. Commun. Math. Phys. 180, 671-707 (1996) · Zbl 0873.17027 · doi:10.1007/BF02099628
[12] Feigin, B., Jimbo, M., Loktev, S., Miwa, T., Mukhin, E.: Bosonic formulas for \[(k,\ell )\](k,ℓ)-admissible partitions. Ramanujan J. 7, 485-517 (2003); Addendum to ‘Bosonic formulas for \[(k,\ell )\](k,ℓ)-admissible partitions’. Ramanujan J. 7, 519-530 (2003) · Zbl 1039.05008
[13] Feigin, B., Jimbo, M., Miwa, T., Mukhin, E., Takeyama, Y.: Fermionic formulas for \[(k,3)\](k,3)-admissible configurations. Publ. RIMS 40, 125-162 (2004) · Zbl 1134.17311 · doi:10.2977/prims/1145475968
[14] Feingold, A.J., Fredenhagen, S.: A new perspective on the Frenkel-Zhu fusion rule theorem. J. Algebr. 320, 2079-2100 (2008) · Zbl 1213.17022 · doi:10.1016/j.jalgebra.2008.05.026
[15] Frenkel, I., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules, preprint (1989). Mem. Am. Math. Soc. 104 (1993) · Zbl 0789.17022
[16] Frenkel, I., Lepowsky, J., Meurman, A.: Vertex Operator Algebras and the Monster. Pure and Applied Mathematics, vol. 134. Academic Press, Boston (1988) · Zbl 0674.17001
[17] Georgiev, G.: Combinatorial constructions of modules for infinite-dimensional Lie algebras, I. Principal subspace. J. Pure Appl. Algebr. 112, 247-286 (1996) · Zbl 0871.17018 · doi:10.1016/0022-4049(95)00143-3
[18] Gannon, T.: The automorhisms of affine fusion rings. Adv. Math. 165, 165-193 (2002) · Zbl 1053.17017 · doi:10.1006/aima.2001.2006
[19] Humphreys, J.: Introduction to Lie Algebras and Representation Theory. Springer, New York (1994) · Zbl 0254.17004
[20] Kac, V.G.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990) · Zbl 0716.17022 · doi:10.1017/CBO9780511626234
[21] Lepowsky, J., Li, H.-S.: Introduction to Vertex Operator Algebras and Their Representations. Progress in Mathematics, vol. 227. Birkhäuser, Boston (2004) · Zbl 1055.17001 · doi:10.1007/978-0-8176-8186-9
[22] Lepowsky, J., Primc, M.: Structure of the standard modules for the affine Lie algebra \[A_1^{(1)}\] A1(1). Contemp. Math. 46, 1-84 (1985) · Zbl 0569.17007 · doi:10.1090/conm/046/01
[23] Meurman, A., Primc, M., Meurman, A., Primc, M.: Annihilating fields of standard modules of \[\mathfrak{sl}(2,\mathbb{C})^{\widetilde{} }\] sl(2,C)  and combinatorial identities. Mem. Am. Math. Soc. 652, 1-89 (1999) · Zbl 0918.17018
[24] Primc, M.: Vertex operator construction of standard modules for \[A_n^{(1)}\] An(1). Pac. J. Math. 162, 143-187 (1994) · Zbl 0787.17024 · doi:10.2140/pjm.1994.162.143
[25] Primc, M.: Basic representations sor classical affine Lie algebras. J. Algebr. 228, 1-50 (2000) · Zbl 0960.17002 · doi:10.1006/jabr.1999.7899
[26] Primc, M.: \[(k, r)\](k,r)-admissible configurations and intertwining operators. Contemp. Math. 442, 425-434 (2007) · Zbl 1152.17012 · doi:10.1090/conm/442/08540
[27] Primc, M.: Combinatorial bases of modules for affine Lie algebra \[B_2^{(1)}\] B2(1). Cent. Eur. J. Math. 11, 197-225 (2013) · Zbl 1293.17032
[28] Primc, M., Šikić, T.: Leading terms of relations for standard modules of affine Lie algebras \[C_n^{(1)}Cn(1)\]. arXiv:1506.05026 [math.QA] · Zbl 1462.17028
[29] Sadowski, C.: Presentations of the principal subspaces of the higher level \[\widehat{{{\mathfrak{s}l}} (3)}\] sl(3)^-modules. J. Pure Appl. Algebr. 219, 2300-2345 (2015) · Zbl 1333.17019 · doi:10.1016/j.jpaa.2014.09.002
[30] Stoyanovsky, A.V.: Deformations, Lie algebra, formulas, character (in Russian). Funktsional. Anal. i Prilozhen. 32, 84-86; translation in Funct. Anal. Appl. 32, 66-68 (1998) · Zbl 0938.17005
[31] Stoyanovsky, A.V., Feigin, B.L.: Functional models of the representations of current algebras, and semi-infinite Schubert cells (in Russian). Funktsional. Anal. i Prilozhen. 28, 68-90 (1994); translation in Funct. Anal. Appl. 28, 55-72 (1994); preprint Feigin, B. Stoyanovsky, A.: Quasi-particles models for the representations of Lie algebras and geometry of flag manifold, hep-th/9308079, RIMS 942 · Zbl 0905.17030
[32] Trupčević, G.: Combinatorial bases of Feigin-Stoyanovsky’s type subspaces of higher-level standard \[\tilde{\mathfrak{s}l}(\ell +1,)\] sl (ℓ+1,)-modules. J. Algebr. 322, 3744-3774 (2009) · Zbl 1216.17008 · doi:10.1016/j.jalgebra.2009.07.024
[33] Trupčević, G.: Combinatorial bases of Feigin-Stoyanovsky’s type subspaces of level standard \[\tilde{\mathfrak{s}l}(\ell +1,)\] sl (ℓ+1,)-modules. Commun. Algebr. 38, 3913-3940 (2010) · Zbl 1221.17026 · doi:10.1080/00927870903366827
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.