TKK algebras and vertex operator representations

A class of irreducible integrable modules for the extended baby TKK algebra

“…They are natural generalizations of finite-dimensional simple Lie algebras and affine Kac-Moody algebras. There are many examples of EALAs, such as toroidal algebras and TKK algebras [Moody et al 1990;Mao and Tan 2007a;2007b;Eswara Rao 2004;Tan 1999]. In [Eswara Rao 2004], the author studied the irreducible integrable weight modules of toroidal algebras.…”

“…Its vertex operator representation and quantum analogue were studied in [Tan 1999;Gao and Jing 2010].…”

“…We recall this construction [Allison et al 1997;Tan 1999]: Let e 1 = (1, 0) and e 2 = (0, 1) be the unit elements in the lattice ‫ޚ‬ 2 . Let S i for 0 ≤ i ≤ 3 be the cosets of ‫ޚ2‬ 2 in ‫ޚ‬ 2 defined by (1-1) S 0 = ‫ޚ2‬ 2 , S 1 = e 1 + ‫ޚ2‬ 2 , S 2 = e 2 + ‫ޚ2‬ 2 , S 3 = e 1 + e 2 + ‫ޚ2‬ 2 .…”

“…We will use a, b to denote the element a ⊗ b + I in ((S) ⊗ (S))/I . In [Tan 1999], the baby TKK algebraᏳ((S)) is realized as the vector space …”

“…)ރ(‬ A vertex operator representation ofᏳ((S)) was given in [Tan 1999] on a mixed bosonic-fermionic Fock space.…”