Virasoro extensions of generalized Kac-Moody algebras

@misc{zveryk2026virasoro,
  title = {{Virasoro extensions of generalized Kac-Moody algebras}},
  author = {Vladyslav Zveryk},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15729},
  eprint = {2609.15729},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15729},
}

TY  - UNPB
TI  - Virasoro extensions of generalized Kac-Moody algebras
AU  - Vladyslav Zveryk
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.15729
UR  - https://arxiv.org/abs/2609.15729
AB  - Borcherds-Kac-Moody (BKM) Lie algebras are constructed by combining a number of sl_2-s and $3$-dimensional Heisenberg algebras in a certain way, generalizing the celebrated Kac-Moody algebras. Such algebras have a wide range of applications, including the examples of the monster and fake monster Lie algebras being BKM algebras. Among their main features are existence of an invariant bilinear form, denominator identity and Weyl-Kac character formulas for irreducible representations. We introduce a construction of closely related algebras that we call Virasoro-Borcherds-Kac-Moody (VBKM) algebras: if it happens that certain $3$-dimensional Heisenberg algebras inside a BKM algebra form the infinite-dimensional Heisenberg algebra, we replace it by the Virasoro algebra in the construction. We construct a family of Lie algebras over ℙ^1, where the generic fiber is a VBKM algebra and a special fiber is a BKM algebra. This allows us to visualize VBKM algebras as deformations of BKM algebras and prove structural properties about VBKM algebras similar to those of BKM algebras, despite the absence of an invariant bilinear form. Those include denominator identity and character formulas of irreducible representations. We give examples of realizations of such ℙ^1-families of VBKM algebras in lattice vertex operator algebras. In particular, we include the fake monster Lie algebra into such a family.
ER  -