Advertisement

Algebras and Representation Theory

, Volume 22, Issue 6, pp 1457–1478 | Cite as

Presentations of Principal Subspaces of Higher Level Standard \({A}_{2}^{(2)}\)-Modules

  • Corina CalinescuEmail author
  • Michael Penn
  • Christopher Sadowski
Article
First Online:
  • 15 Downloads

Abstract

We study the principal subspaces of higher level standard \({A}_{2}^{(2)}\)-modules, extending earlier work in the level one case, by Calinescu, Lepowsky and Milas. We prove natural presentations of principal subspaces and also of certain related spaces. By using these presentations we obtain exact sequences, which yield recursions satisfied by the characters of the principal subspaces and related spaces. We conjecture a formula for a specialized character of the principal subspace, given by the Nahm sum of the inverse of the tadpole Cartan matrix.

Keywords

Twisted affine Lie algebras Twisted vertex-algebra modules Principal subspaces 

Mathematics Subject Classification (2010)

17B69 17B65 05A17 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

We thank the referee for providing us with constructive comments.

References

  1. 1.
    Borcherds, R.E.: Vertex, algebras, Kac-Moody, algebras, and the Monster. Proc. Natl. Acad. Sci. USA 83, 3068–3071 (1986)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Butorac, M.: Combinatorial bases of principal subspaces for the affine Lie algebra of type \({B}_{2}^{(1)}\). J. Pure Appl. Algebra 218, 424–447 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Butorac, M.: Quasi-particle bases of principal subspaces for the affine Lie algebras of types \({B}_{l}^{(1)}\) and \({C}_{l}^{(1)}\). Glas. Mat. Ser. III(51), 59–108 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Butorac, M.: Quasi-particle bases of principal subspaces of the affine Lie algebra of type \({G}_{2}^{(1)}\). Glas. Mat. Ser. III(52), 79–98 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Butorac, M., Sadowski, C.: Combinatorial bases of principal subspaces of modules for twisted affine Lie algebras of type \({A}_{2l-1}^{(2)}\), \({D}_{l}^{(2)}\), \({E}_{6}^{(2)}\) and \({D}_{4}^{(3)}\), preprintGoogle Scholar
  6. 6.
    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. 19, 71–92 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    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 212, 1928–1950 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    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 323, 167–192 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Calinescu, C., Lepowsky, J., Milas, A.: Vertex-algebraic structure of principal subspaces of standard \({A}_{2}^{(2)}\)-modules, I. Internat. J. Math. 25(1450063) (2014)Google Scholar
  10. 10.
    Capparelli, S., Lepowsky, J., Milas, A.: The Rogers-Ramanujan recursion and intertwining operators. Comm. Contemp. Math. 5, 947–966 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Capparelli, S., Lepowsky, J., Milas, A.: The Rogers-Selberg recursions, the Gordon-Andrews identities and intertwining operators. The Ramanujan Journal 12, 379–397 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Calinescu, C., Milas, A., Penn, M.: Vertex-algebraic structure of principal subspaces of basic \({A}_{2n}^{(2)}\)-modules. J. Pure Appl. Algebra 220, 1752–1784 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Feigin, B., Stoyanovsky, A.: Quasi-particles models for the representations of Lie algebras and geometry of flag manifold; arXiv:hep-th/9308079
  14. 14.
    Feigin, B., Stoyanovsky, A.: Functional models for representations of current algebras and semi-infinite Schubert cells (Russian). Funktsional Anal. i Prilozhen 28, 68–90 (1994). translation in: Funct. Anal. Appl. 28 (1994), 55–72MathSciNetGoogle Scholar
  15. 15.
    Frenkel, I., Huang, Y.-Z., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Memoirs American Math. Soc. 104 (1993)Google Scholar
  16. 16.
    Frenkel, I., Lepowsky, J., Meurman, A.: Vertex Operator Calculus. In: Mathematical Aspects of String Theory, Proc. 1986 Conference, San Diego, ed. by S.-T. Yau, World Scientific, Singapore, pp 150–188 (1987)Google Scholar
  17. 17.
    Frenkel, I., Lepowsky, J., Meurman, A.: Vertex operator algebras and the monster. Pure and Applied Math., vol. 134 academic press (1988)Google Scholar
  18. 18.
    Jerkovic, M.: Recurrences and characters of Feigin-Stoyanovsky’s type subspaces. Vertex operator algebras and related areas. Contemp. Math. 497, 113–123 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kac, V.: Infinite Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  20. 20.
    Kožic, S.: Principal subspaces for quantum affine algebra \(u_{q}({A}_{n}^{(1)})\). J. Pure Appl. Algebra 218, 2119–2148 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lepowsky, J.: Calculus of twisted vertex operators. Proc. Nat. Acad. Sci. USA 82, 8295–8299 (1985)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lepowsky, J.: Perspectives on vertex operators and the Monster. in: Proceedings 1987 Symposium on the Mathematical Heritage of Hermann Weyl, Duke Univ. Proc. Symp. Pure Math., American Math. Soc. 48, 181–197 (1988)CrossRefGoogle Scholar
  23. 23.
    Lepowsky, J., Li, H.: Introduction to Vertex Operator Algebras and Their Representations Progress in Mathematics, vol. 227. Birkhäuser, Boston (2003)Google Scholar
  24. 24.
    Li, H.-S.: Local systems of twisted vertex operators, vertex superalgebras and twisted modules. Contemp. Math. 193, 203–236 (1996)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Milas, A., Penn, M.: Lattice vertex algebras and combinatorial bases: general case and \(\mathcal {W}\)-algebras. New York J. Math. 18, 621–650 (2012)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Penn, M., Lattice Vertex Superalgebras, I: Presentation of the principal subspace. Commun. Algebra 42(3), 933–961 (2014)CrossRefGoogle Scholar
  27. 27.
    Penn, M., Sadowski, C.: Vertex-algebraic structure of principal subspaces of basic \({D}_{4}^{(3)}\)-modules. The Ramanujan Journal 43(4), 571–617 (2017)CrossRefGoogle Scholar
  28. 28.
    Penn, M., Sadowski, C.: Vertex-algebraic structure of principal subspaces of basic modules for twisted affine Kac-Moody Lie algebras of type \({A}_{2n + 1}^{(2)}, {D}_{n}^{(2)}, {E}_{6}^{(2)}\). J. Algebra 496, 242–291 (2018)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Penn, M., Sadowski, C., Webb, G.: Twisted Modules of Principal Subalgebras of Lattice Vertex Algebras, 40 pgs, submittedGoogle Scholar
  30. 30.
    Primc, M.: (K, r)-admissible configurations and intertwining operators. Contemp. Math 422, 425–434 (2007)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Sadowski, C.: Presentations of the principal suspaces of higher level \(\widehat {\mathfrak {sl}(3)}\)-modules. J. Pure Appl. Algebra 219, 2300–2345 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Sadowski, C.: Principal subspaces of \(\widehat {\mathfrak {sl}(n)}\)-modules. Int. J. Math. 26(08), 1550063 (2015)CrossRefGoogle Scholar
  33. 33.
    Trupcevic, G.: Combinatorial bases of Feigin-Stoyanovsky’s type subspaces of level one standard modules for \(\tilde {\mathfrak {sl}}(l + 1, \mathbb {C})\). Comm. Algebra 38, 3913–3940 (2010)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Trupcevic, G.: Combinatorial bases of Feigin-Stoyanovsky’s type subspaces of higher-level standard \(\tilde {\mathfrak {sl}}(l + 1, \mathbb {C})\)-modules. J. Algebra 322, 3744–3774 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Mathematics, New York City College of Technology and the Graduate CenterCity University of New YorkNew YorkUSA
  2. 2.Department of MathematicsRandolph CollegeLynchburgUSA
  3. 3.Department of Mathematics and Computer ScienceUrsinus CollegeCollegevilleUSA

Personalised recommendations

Cite article

Buy options