Glasnik Matematicki, Vol. 53, No. 1 (2018), 115-121.

PRESENTATIONS OF FEIGIN-STOYANOVSKY'S TYPE SUBSPACES OF STANDARD MODULES FOR AFFINE LIE ALGEBRAS OF TYPE Cl(1)

Goran Trupčević

Faculty of Teacher Education, University of Zagreb, 10000 Zagreb, Croatia
e-mail: goran.trupcevic@ufzg.hr

Abstract.   Feigin-Stoyanovsky's type subspace W(Λ) of a standard -module L(Λ) is a 1-submodule of L(Λ) generated by the highest-weight vector vΛ, where 1 is a certain commutative subalgebra of . Based on the description of basis of W(Λ) for of type Cl(1), we give a presentation of this subspace in terms of generators and relations W(Λ)≃ U(1-)/J.

2010 Mathematics Subject Classification.   17B67, 17B69, 05A19.

Key words and phrases.   Affine Lie algebras, principal subspaces, generators and relations.


Full text (PDF) (access from subscribing institutions only)

DOI: 10.3336/gm.53.1.08


References:

  1. E. Ardonne, R. Kedem and M. Stone, Fermionic characters and arbitrary highest-weight integrable r+1-modules, Comm. Math. Phys. 264 (2006), 427-464.
    MathSciNet     CrossRef

  2. I. Baranović, Combinatorial bases of Feigin-Stoyanovsky's type subspaces of level 2 standard modules for D4(1), Comm. Algebra 39 (2011), 1007-1051.
    MathSciNet     CrossRef

  3. I. Baranović, M. Primc and G. Trupčević, Bases of Feigin-Stoyanovsky's type subspaces for Cl(1), Ramanujan J (2016). doi:10.1007/s11139-016-9840-y 1007-1051.

  4. M. Butorac, Combinatorial bases of principal subspaces for the affine Lie algebra of type B2(1), J. Pure Appl. Algebra 218 (2014), 424-447.
    MathSciNet     CrossRef

  5. M. Butorac, Quasi particle bases of principal subspaces of the affine Lie algebra of type G2(1), Glas. Mat. Ser. III 52(72) (2017), 79-98.
    MathSciNet     CrossRef

  6. C. Calinescu, Intertwining vertex operators and certain representations of , Commun. Contemp. Math. 10 (2008), 47-79.
    MathSciNet     CrossRef

  7. C. Calinescu, J. Lepowsky and A. Milas, Vertex-algebraic structure of the principal subspaces of certain A1(1)-modules, I. Level one case, Internat. J. Math. 19 (2008), 71-92.
    MathSciNet     CrossRef

  8. C. Calinescu, J. Lepowsky and A. Milas, Vertex-algebraic structure of the principal subspaces of certain A1(1)-modules, II. Higher-level case, J. Pure Appl. Algebra 212 (2008), 1928-1950.
    MathSciNet     CrossRef

  9. C. Calinescu, J. Lepowsky and A. Milas, Vertex-algebraic structure of the principal subspaces of level one modules for the untwisted affine Lie algebras of types A, D, E, J. Algebra 323 (2010), 167-192.
    MathSciNet     CrossRef

  10. S. Capparelli, J. Lepowsky and A. Milas, The Rogers-Ramanujan recursion and intertwining operators, Commun. Contemp. Math. 5 (2003), 947-966.
    MathSciNet     CrossRef

  11. S. Capparelli, J. Lepowsky and A. Milas, The Rogers-Selberg recursions, the Gordon-Andrews identities and intertwining operators, Ramanujan J. 12 (2006), 379-397.
    MathSciNet     CrossRef

  12. A. V. Stoyanovsky and B. L. Feigin, Functional models of the representations of current algebras, and semi-infinite Schubert cells, (Russian) Funktsional. Anal. i Prilozhen. 28 (1994), 68-90; translation in Funct. Anal. Appl. 28 (1994), 55-72.
    MathSciNet     CrossRef

  13. B. Feigin, M. Jimbo, S. Loktev, T. Miwa and E. Mukhin, Bosonic formulas for (k,l)-admissible partitions, Ramanujan J. 7 (2003), 485-517.; Addendum to `Bosonic formulas for (k,l)-admissible partitions', Ramanujan J. 7 (2003), 519-530.
    MathSciNet     CrossRef

  14. B. Feigin, M. Jimbo, T. Miwa, E. Mukhin and Y. Takeyama, Fermionic formulas for (k,3)-admissible configurations, Publ. Res. Inst. Math. Sci. 40 (2004), 163-220.
    MathSciNet     CrossRef

  15. G. Georgiev, Combinatorial constructions of modules for infinite-dimensional Lie algebras. I. Principal subspace, J. Pure Appl. Algebra 112 (1996), 247-286.
    MathSciNet     CrossRef

  16. J. Humphreys Introduction to Lie algebras and representation theory, Springer, New-York, 1978.
    MathSciNet    

  17. M. Jerković, Recurrence relations for characters of affine Lie algebra Al(1), J. Pure Appl. Algebra 213, 913-926.
    MathSciNet     CrossRef

  18. M. Jerković and M. Primc, Quasi-particle fermionic formulas for (k, 3)-admissible configurations, Cent. Eur. J. Math. 10 (2012), 703-721.
    MathSciNet     CrossRef

  19. V. G. Kac, Infinite-dimensional Lie algebras, Cambridge University Press, Cambridge, 1990.
    MathSciNet     CrossRef

  20. J. Lepowsky and H.-S. Li, Introduction to vertex operator algebras and their representations, Birkhäuser, Boston, 2004.
    MathSciNet     CrossRef

  21. J. Lepowsky and M. Primc, Structure of the standard modules for the affine Lie algebra A1(1), Contemporary Mathematics 46, American Mathematical Society, 1985.
    MathSciNet     CrossRef

  22. A. Meurman and M. Primc, Annihilating fields of standard modules of (2,ℂ)˜ and combinatorial identities, Mem. Amer. Math. Soc. 652, 1999.
    MathSciNet     CrossRef

  23. M. Penn, Lattice vertex superalgebras, I: Presentation of the principal subalgebra, Comm. Algebra 42 (2014), 933-961.
    MathSciNet     CrossRef

  24. M. Primc, Vertex operator construction of standard modules for An(1), Pacific J. Math. 162 (1994), 143-187.
    MathSciNet     CrossRef

  25. M. Primc, Basic representations for classical affine Lie algebras, J. Algebra 228 (2000), 1-50.
    MathSciNet     CrossRef

  26. M. Primc, Combinatorial bases of modules for affine Lie algebra B2(1), Cent. Eur. J. Math. 11 (2013), 197-225.
    MathSciNet     CrossRef

  27. C. Sadowski, Presentations of the principal subspaces of the higher level standard -modules, J. Pure Appl. Algebra 219 (2015), 2300-2345.
    MathSciNet     CrossRef

  28. A. V. Stoyanovsky, Lie algebra deformations and character formulas (Russian) Funktsional. Anal. i Prilozhen. 32 (1998), 84-86; translation in Funct. Anal. Appl. 32 (1998), 66-68.
    MathSciNet     CrossRef

  29. G. Trupčević, Bases of standard modules for affine Lie algebras of type Cl(1), Comm. Algebra 46 (2018), 3663-3673.
    MathSciNet     CrossRef

  30. G. Trupčević, Combinatorial bases of Feigin-Stoyanovsky's type subspaces of level 1 standard (l+1,ℂ)-modules, Comm. Algebra 38 (2010), 3913-3940.
    MathSciNet     CrossRef

Glasnik Matematicki Home Page