Quasi-particle fermionic formulas for (k, 3)-admissible configurations

Miroslav Jerković; Mirko Primc

Open Mathematics (2012)

  • Volume: 10, Issue: 2, page 703-721
  • ISSN: 2391-5455

Abstract

top
We construct new monomial quasi-particle bases of Feigin-Stoyanovsky type subspaces for the affine Lie algebra sl(3;ℂ)∧ from which the known fermionic-type formulas for (k, 3)-admissible configurations follow naturally. In the proof we use vertex operator algebra relations for standard modules and coefficients of intertwining operators.

How to cite

top

Miroslav Jerković, and Mirko Primc. "Quasi-particle fermionic formulas for (k, 3)-admissible configurations." Open Mathematics 10.2 (2012): 703-721. <http://eudml.org/doc/269365>.

@article{MiroslavJerković2012,
abstract = {We construct new monomial quasi-particle bases of Feigin-Stoyanovsky type subspaces for the affine Lie algebra sl(3;ℂ)∧ from which the known fermionic-type formulas for (k, 3)-admissible configurations follow naturally. In the proof we use vertex operator algebra relations for standard modules and coefficients of intertwining operators.},
author = {Miroslav Jerković, Mirko Primc},
journal = {Open Mathematics},
keywords = {Quasi-particle bases; Feigin-Stoyanovsky type subspaces; Affine Lie algebras; Fermionic-type formulas; Admissible configurations; quasi-particle bases; affine Lie algebras; fermionic-type formulas; admissible configurations},
language = {eng},
number = {2},
pages = {703-721},
title = {Quasi-particle fermionic formulas for (k, 3)-admissible configurations},
url = {http://eudml.org/doc/269365},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Miroslav Jerković
AU - Mirko Primc
TI - Quasi-particle fermionic formulas for (k, 3)-admissible configurations
JO - Open Mathematics
PY - 2012
VL - 10
IS - 2
SP - 703
EP - 721
AB - We construct new monomial quasi-particle bases of Feigin-Stoyanovsky type subspaces for the affine Lie algebra sl(3;ℂ)∧ from which the known fermionic-type formulas for (k, 3)-admissible configurations follow naturally. In the proof we use vertex operator algebra relations for standard modules and coefficients of intertwining operators.
LA - eng
KW - Quasi-particle bases; Feigin-Stoyanovsky type subspaces; Affine Lie algebras; Fermionic-type formulas; Admissible configurations; quasi-particle bases; affine Lie algebras; fermionic-type formulas; admissible configurations
UR - http://eudml.org/doc/269365
ER -

References

top
  1. [1] Ardonne E., Kedem R., Stone M., Fermionic characters and arbitrary highest-weight integrable bs 𝔰 l ^ r + 1 -modules, Comm. Math. Phys., 2006, 264(2), 427–464 http://dx.doi.org/10.1007/s00220-005-1486-3 Zbl1233.17017
  2. [2] Baranović I., Combinatorial bases of Feigin-Stoyanovsky’s type subspaces of level 2 standard modules for D 4(1), Comm. Algebra, 2011, 39(3), 1007–1051 http://dx.doi.org/10.1080/00927871003639329 Zbl1269.17011
  3. [3] Calinescu C., Principal subspaces of higher-level standard 𝔰 𝔩 ( 3 ) ^ -modules, J. Pure Appl. Algebra, 2007, 210(2), 559–575 http://dx.doi.org/10.1016/j.jpaa.2006.10.018 
  4. [4] Calinescu C., Intertwining vertex operators and certain representations of 𝔰 𝔩 ( n ) ^ , Commun. Contemp. Math., 2008, 10(1), 47–79 http://dx.doi.org/10.1142/S0219199708002703 
  5. [5] Calinescu C., Lepowsky J., Milas A., Vertex-algebraic structure of the principal subspaces of certain A 1(1)-modules I: Level one case, Internat. J. Math., 2008, 19(1), 71–92 http://dx.doi.org/10.1142/S0129167X08004571 Zbl1184.17012
  6. [6] Calinescu C., Lepowsky J., Milas A., Vertex-algebraic structure of the principal subspaces of certain A 1(1)-modules II: Higher-level case, J. Pure Appl. Algebra, 2008, 212(8), 1928–1950 http://dx.doi.org/10.1016/j.jpaa.2008.01.003 Zbl1184.17013
  7. [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. Algebra, 2010, 323(1), 167–192 http://dx.doi.org/10.1016/j.jalgebra.2009.09.029 Zbl1221.17032
  8. [8] Capparelli S., Lepowsky J., Milas A., The Rogers-Ramanujan recursion and intertwining operators, Commun. Contemp. Math., 2003, 5(6), 947–966 http://dx.doi.org/10.1142/S0219199703001191 Zbl1039.05005
  9. [9] Capparelli S., Lepowsky J., Milas A., The Rogers-Selberg recursions, the Gordon-Andrews identities and intertwining operators, Ramanujan J., 2006, 12(3), 379–397 http://dx.doi.org/10.1007/s11139-006-0150-7 Zbl1166.17009
  10. [10] Dong C., Lepowsky J., Generalized Vertex Algebras and Relative Vertex Operators, Progr. Math., 112, Birkhäuser, Boston, 1993 http://dx.doi.org/10.1007/978-1-4612-0353-7 Zbl0803.17009
  11. [11] Feigin B., Jimbo M., Loktev S., Miwa T., Mukhin E., Bosonic formulas for (k; l)-admissible partitions, Ramanujan J., 2003, 7(4), 485–517, 519–530 http://dx.doi.org/10.1023/B:RAMA.0000012430.68976.c0 Zbl1039.05008
  12. [12] Feigin B., Jimbo M., Miwa T., Mukhin E., Takeyama Y., Fermionic formulas for (k; 3)-admissible configurations, Publ. Res. Inst. Math. Sci., 2004, 40(1), 125–162 http://dx.doi.org/10.2977/prims/1145475968 Zbl1134.17311
  13. [13] Feigin B., Jimbo M., Miwa T., Mukhin E., Takeyama Y., Particle content of the (k; 3)-configurations, Publ. Res. Inst. Math. Sci., 2004, 40(1), 163–220 http://dx.doi.org/10.2977/prims/1145475969 Zbl1062.05010
  14. [14] Frenkel I.B., Kac V.G., Basic representations of affine Lie algebras and dual resonance models, Invent. Math., 1980, 62(1), 23–66 http://dx.doi.org/10.1007/BF01391662 Zbl0493.17010
  15. [15] Frenkel I., Lepowsky J., Meurman A., Vertex Operator Algebras and the Monster, Pure Appl. Math., 134, Academic Press, Boston, 1988 Zbl0674.17001
  16. [16] Georgiev G., Combinatorial constructions of modules for infinite-dimensional Lie algebras I. Principal subspace, J. Pure Appl. Algebra, 1996, 112(3), 247–286 http://dx.doi.org/10.1016/0022-4049(95)00143-3 
  17. [17] Jerković M., Recurrence relations for characters of affine Lie algebra A ℓ(1), J. Pure Appl. Algebra, 2009, 213(6), 913–926 http://dx.doi.org/10.1016/j.jpaa.2008.10.001 Zbl1183.17012
  18. [18] Jerković M., Recurrences and characters of Feigin-Stoyanovsky’s type subspaces, In: Vertex Operator Algebras and Related Areas, Contemp. Math., 497, American Mathematical Society, Providence, 2009, 113–123 
  19. [19] Jerković M., Character formulas for Feigin-Stoyanovsky’s type subspaces of standard sl(3, ℂ)∧-modules, preprint avaliable at http://arxiv.org/abs/1105.2927 
  20. [20] Kac V.G., Infinite-Dimensional Lie Algebras, 3rd ed., Cambridge University Press, Cambridge, 1990 http://dx.doi.org/10.1017/CBO9780511626234 Zbl0716.17022
  21. [21] Lepowsky J., Wilson R.L., The structure of standard modules I. Universal algebras and the Rogers-Ramanujan identities, Invent. Math., 1984, 77(2), 199–290; The structure of standard modules II. The case A 1(1), principal gradation, Invent. Math., 1985, 79(5), 417–442 http://dx.doi.org/10.1007/BF01388447 Zbl0577.17009
  22. [22] Primc M., Vertex operator construction of standard modules for A n(1), Pacific J. Math., 1994, 162(1), 143–187 Zbl0787.17024
  23. [23] Primc M., (k; r)-admissible configurations and intertwining operators, In: Lie Algebras, Vertex Operator Algebras and their Applications, Contemp. Math., 442, American Mathematical Society, Providence, 2007, 425–434 Zbl1152.17012
  24. [24] Segal G., Unitary representations of some infinite-dimensional groups, Commun. Math. Phys., 1981, 80(3), 301–342 http://dx.doi.org/10.1007/BF01208274 Zbl0495.22017
  25. [25] Stoyanovsky A.V., Feigin B.L., Functional models of the representations of current algebras, and semi-infinite Schubert cells, Funktsional. Anal. i Prilozhen., 1994, 28(1), 68–90 (in Russian) http://dx.doi.org/10.1007/BF01079010 
  26. [26] Trupčević G., Combinatorial bases of Feigin-Stoyanovsky’s type subspaces of higher-level standard 𝔰 l ( + 1 , ) -modules, J. Algebra, 2009, 322(10), 3744–3774 http://dx.doi.org/10.1016/j.jalgebra.2009.07.024 Zbl1216.17008
  27. [27] Trupčević G., Combinatorial bases of Feigin-Stoyanovsky’s type subspaces of level 1 standard modules for 𝔰 l ( + 1 , ) , Comm. Algebra, 2010, 38(10), 3913–3940 http://dx.doi.org/10.1080/00927870903366827 Zbl1221.17026

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.