Extensions at Nonzero p-Characters for Cartan-Type Lie Algebras
Abstract
For restricted simple Cartan-type Lie algebras over an algebraically closed field of characteristic p≥5, the reduced enveloping algebra at every nonzero p-character is a matrix algebra over a terminal symmetric corner, and this corner determines the extensions. Central-socle traces recover its radical and its simple factors, and the first radical layer determines the degree-one extensions. For the Witt algebra W(1;1), at even height 2≤ h≤ p−3, the unique simple module has self-extension dimension min{p−h−2,h+2}. At height two in characteristic seven, we present the Yoneda algebra as an eight-dimensional symmetric algebra tensored with a polynomial algebra on three central degree-two classes. An equivariant Tate resolution and cocycle identities prove the presentation in all degrees.
Related
- Extensions of Restricted Simple Modules for Restricted Lie Algebras of Cartan Type
- Elementary divisors of the Cartan matrix for partial characters
- The Cartan determinant conjecture for representation-finite Artin algebras
- simple cartan-free modules of finite rank over the super-virasoro algebras
- Weights and characters for affine Hecke algebras
BibTeX
@misc{ma2026extensions,
title = {{Extensions at Nonzero p-Characters for Cartan-Type Lie Algebras}},
author = {Chao Ma},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02683},
eprint = {2611.02683},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02683},
}
RIS
TY - UNPB
TI - Extensions at Nonzero p-Characters for Cartan-Type Lie Algebras
AU - Chao Ma
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02683
UR - https://arxiv.org/abs/2611.02683
AB - For restricted simple Cartan-type Lie algebras over an algebraically closed field of characteristic p≥5, the reduced enveloping algebra at every nonzero p-character is a matrix algebra over a terminal symmetric corner, and this corner determines the extensions. Central-socle traces recover its radical and its simple factors, and the first radical layer determines the degree-one extensions. For the Witt algebra W(1;1), at even height 2≤ h≤ p−3, the unique simple module has self-extension dimension min{p−h−2,h+2}. At height two in characteristic seven, we present the Yoneda algebra as an eight-dimensional symmetric algebra tensored with a polynomial algebra on three central degree-two classes. An equivariant Tate resolution and cocycle identities prove the presentation in all degrees.
ER -