Total views : 221

Quasi Affine Generalized Kac Moody Algebras QAGGD3(2): Dynkin diagrams and root multiplicities for a class of QAGGD3(2)

Affiliations

  • Quaid-e-millath Government College for Women (Autonomous), Chennai – 600 002, Tamil Nadu, India

Abstract


In this paper quasi affine generalized Kac Moody algebras(GKM) QAGGD3(2) are considered; the non isomorphic, connected Dynkin diagrams associated with this particular class of QAGGD3(2) are classified with their general form given; Specific class of these quasi affine generalized Kac Moody algebras QAGGD3(2) are then considered and some root multiplicities are computed; First , we begin with  a general family QAGGD3(2)  of symmetrizable Generalized Generalized Cartan Matrices (GGCM) of quasi affine type, with one imaginary simple root,  which are obtained from the affine GCM D3(2) . The multiplicities of roots of a class GKM algebras  QAGGD3(2), which are obtained as extensions of the affine family D3(2) are then determined;


Keywords

Dynkin Diagram, Generalized Kac Moody Algebras (GKM), Imaginary Roots, Quasi Affine, Root Multiplicity.

Full Text:

 |  (PDF views: 199)

References


  • Borcheds RE. Generalized Kac-Moody algebras. J Algebra. 1988; 115:501–12.
  • Sthanumoorthy N, Lilly PL, Uma Maheswari A. Root multiplicities of some classes of extended-hyperbolic Kac-Moody and extended - hyperbolic generalized KacMoody algebras. Contemporary Mathematics, AMS. 2004; 343:315–47.
  • Sthanumoorthy N, Lilly PL. On the root systems of generalized Kac-Moody algebras. J Madras University (WMY2000 special issue) Section B: Sciences. 2000; 52:81–103.
  • Sthanumoorthy N, Lilly PL. Special imaginary roots of generalized Kac-Moody algebras. Comm Algebra. 2002; 30:4771–87.
  • Sthanumoorthy N, Lilly PL. A note on purely imaginary roots of generalized Kac-Moody algebras. Comm Algebra. 2003; 31:5467–80.
  • Sthanumoorthy N, Lilly PL. On some classes of root systems of generalized Kac-Moody algebras. Contemp Mathematics, AMS. 2004; 343:289–313.
  • Sthanumoorthy N, Lilly PL. Complete classifications of Generalized Kac-Moody algebras possessing special imaginary roots and strictly imaginary property. to appear in Communications in algebra (USA). 2007; 35(8):2450–71.
  • Sthanumoorthy N, Lilly PL. Root Multiplicities of some generalized Kac-Moody algebras. Indian J Pure Appl Math. 2007 Apr; 38(2):55–78.
  • Kang SJ. Generalized Kac-Moody algebras and the modular function. J Math Ann. 1994; 298:373–84.
  • Kang SJ. Root multiplicities of graded Lie algebras, in: Lie algebras and their representations. In: Kang SJ, Kim MH, Lee IS editors. Contemp Math. 1996; 194:161–76.
  • Kang SJ, Kim MH. Dimension formula for graded Lie algebras and its applications. Trans. Amer. Math. Soc. 1999; 351: 4281–4336.
  • Kim K, Shin DU. The recursive dimension formula for graded Lie algebras and its applications. Comm Algebra. 1999; 27:2627–52.
  • Barwald O, Gebert R. On th imaginary simple roots of the Borcherds algebra gII9,1. Nuclear Phys B. 1998; 510:721–38.
  • Song X, Guo Y. Root Multiplicity of a Special Generalized Kac- Moody Algebra EB2. Mathematical Computation. 2014 Sep; 3(2):76–82.
  • Song X, Wang X, Guo Y. Root Structure of a Special generalized Kac-Moody algebras, Mathematical Computation. 2014 Sep; 3(3):83–8.
  • Uma Maheswari A. Root multiplicities for a class of Quasi affine generalized Kac Moody algebras QAGGA2(1) of rank 4. International Journal of Mathematical Archive. 2016; 7(2):5–13.
  • Jurisich E. An exposition of generalized Kac-Moody algebras, in: Lie algebras and their representations. In: Kang SJ, Kim MH, Lee IS editors. Contemp Math. 1996; 194:121–59.
  • Jurisich E. Generalized Kac-Moody Lie algebras, free Lie algebras, and the structure of the monster Lie algebra. J Pure and Applied Algebra. 1998; 126:233–66.
  • Kac VG. Infinite Dimensional Lie Algebras, Dedkind’s η-function, classical MÖbius function and the very strange formula. Adv Math. 1978; 30:85–136.
  • Uma Maheswari A. A Study on the Structure of Indefinite Quasi-Affine Kac-Moody Algebras QAC2(1). International Mathematical Forum. 2014; 9(32):1595–609.
  • Sthanumoorthy N, Uma Maheswari A. Purely imaginary roots of Kac-Moody algebras. Communication in Algebra. 1996; 24(2):677–93.
  • Sthanumoorthy N, Uma Maheswari A. Root multiplicities of extended hyperbolic Kac-Moody algebras. Communication in Algebra. 1996; 24(14):4495–512.
  • Sthanumoorthy N, Uma Maheswari A, Lilly PL. Extendedhyperbolic Kac-Moody algebras EHA2(2) structure and Root Multiplicities”. Communication in Algebra. 2004; 32(6):2457–76.
  • Sthanumoorthy N, Uma Maheswari A. Structure and Root Multiplicities for two classes of Extended Hyberbolic KacMoody Algebras EHA1(1) and EHA2(2) for all cases. Communications in Algebra. 2012; 40:632–65.
  • Berman S, Moody RV. Multiplicities in Lie algebras. Proc Amer Math Soc. 1979; 76:223–8.
  • Jurisich E, Lepowsky J, Wilson RL. Realization of the Monster Lie algebra. Selecta Mathematica New Serie. 1995; 1:129–61.
  • Kac VG. Infinite Dimensional Lie Algebras, 3rd ed. Cambridge: Cambridge University Press, 1990.
  • Naito S. The strong Bernstein-Gelfand-Gelfand resolution for generalized Kac-Moody algebras I: The existence of the resolution. Publ RIMS, Kyoto Univ. 1993; 29:709–30.
  • Liu LS. Kostant’s formula for Kac-Moody Lie algebras. J Algebr. 1992; 149:155–78.
  • Frenkel IB, Kac VG. Basic representation of affine Lie algebras and dual resonance models. Invent Math. 1980; 62:23–66.
  • Kac VG, Peterson DH. Affine Lie algebras and Hecke modular forms, Bull, Amer Math Soc. 1980; 3:1059–61.

Refbacks

  • There are currently no refbacks.


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.