Twisted fermionic and bosonic representations for a class of BC-graded Lie algebras

Abstract

In this paper, we study the fermionic and bosonic representations for a class of BC-graded Lie algebras coordinatized by skew Laurent polynomial rings. This generalizes the fermionic and bosonic constructions for the affine Kac-Moody algebras of type A _N ⁽²⁾.

Keywords

Fermionic and bosonic representation / graded Lie algebra / unitary representation

References

[1] Allison B. N., Azam S., Berman S., Gao Y., Pianzola A. Extended Affine Lie Algebras and Their Root Systems, 1997, Providence: Am Math Soc.

[2] Allison B. N., Benkart G., Gao Y. Central extensions of Lie algebras graded by finite root systems. Math Ann, 2000, 316, 499-527

[3] Allison B. N., Benkart G., Gao Y. Lie Algebras Graded by the Root Systems BC _{r, r} ⩾ 2, 2002, Providence: Am Math Soc.

[4] Benkart G., Zelmanov E. Lie algebras graded by finite root systems and intersection matrix algebras. Invent Math, 1996, 126, 1-45

[5] Berman S., Moody R. V. Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy. Invent Math, 1992, 108, 323-347

[6] Chen H., Gao Y. BCN-graded Lie algebras arising from fermionic representations. J Alg, 2007, 308, 545-566

[7] Feingold A. J., Frenkel I. B. Classical affine Lie algebras. Adv Math, 1985, 56, 117-172

[8] Frenkel I. B. Spinor representation of affine Lie algebras. Proc Natl Acad Sci USA, 1980, 77, 6303-6306

[9] Frenkel I. B. Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory. J Funct Anal, 1981, 44, 259-327

[10] Gao Y. Vertex operators arising from the homogeneous realization for $\widehat{gl}_N$. Comm Math Phys, 2000, 211, 23-66

[11] Gao Y. Fermionic and bosonic representations of the extended affine Lie algebra $\widetilde{gl_N \left( {\mathbb{C}_q } \right)}$. Canada Math Bull, 2002, 45, 623-633

[12] Jing N., Misra K. C. Fermionic realization of toroidal Lie algebras of classical types. J Alg, 2010, 324, 183-194

[13] Jing N., Misra K. C., Xu C. Bosonic realization of toroidal Lie algebras of classical types. Proc Am Math Soc, 2009, 137, 3609-3618

[14] Kac V. G., Peterson D. H. Spin and wedge representations of infinite-dimensional Lie algebras and groups. Proc Natl Acad Sci USA, 1981, 78, 3308-3312

[15] Lau M. Bosonic and fermionic representations of Lie algebra central extensions. Adv Math, 2005, 194, 225-245

[16] Neher E. Lie algebras graded by 3-graded root systems. Am J Math, 1996, 118, 439-491