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PreprintCC BY 4.0arXiv

W-algebras with involution: polynomial identities and asymptotic growth

  • Ginevra Giordani
  • Antonio Ioppolo
  • Carla Rizzo
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Abstract

Let W be an algebra with involution over a field of characteristic zero. We develop a theory of W-polynomial identities with involution for finite-dimensional (W,*)-algebras A, using the multiplier algebra with involution of A. We prove that the (W,*)-exponent of A exists and coincides with the ordinary *-exponent. We then consider a four-dimensional subalgebra M of the algebra of 4×4 upper triangular matrices, endowed with the reflection involution, and study two non-equivalent (W,*)-algebra structures on it. For both structures, we determine the corresponding TW*-ideals of W-polynomial identities with involution and compute the (W,*)-codimension sequences explicitly; for one of them, we also determine the complete (W,*)-cocharacter sequence. Finally, we prove that these two W-algebras with involution generate distinct varieties of almost polynomial growth.

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Cite

BibTeXRIS
BibTeX
@misc{giordani2026walgebras,
  title = {{$W$-algebras with involution: polynomial identities and asymptotic growth}},
  author = {Ginevra Giordani and Antonio Ioppolo and Carla Rizzo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10406},
  eprint = {2610.10406},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10406},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - W-algebras with involution: polynomial identities and asymptotic growth
AU  - Ginevra Giordani
AU  - Antonio Ioppolo
AU  - Carla Rizzo
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10406
UR  - https://arxiv.org/abs/2610.10406
AB  - Let W be an algebra with involution over a field of characteristic zero. We develop a theory of W-polynomial identities with involution for finite-dimensional (W,*)-algebras A, using the multiplier algebra with involution of A. We prove that the (W,*)-exponent of A exists and coincides with the ordinary *-exponent. We then consider a four-dimensional subalgebra M of the algebra of 4×4 upper triangular matrices, endowed with the reflection involution, and study two non-equivalent (W,*)-algebra structures on it. For both structures, we determine the corresponding T_W^*-ideals of W-polynomial identities with involution and compute the (W,*)-codimension sequences explicitly; for one of them, we also determine the complete (W,*)-cocharacter sequence. Finally, we prove that these two W-algebras with involution generate distinct varieties of almost polynomial growth.
ER  -