PreprintCC BY 4.0arXiv
On the non-existence of a left-symmetric structure on osp(1|2) in any characteristic
Abstract
Burde proved that the Lie algebra sl2(𝕂), over a field 𝕂 of characteristic p > 2, admits compatible left-symmetric algebra structures if and only if p = 3. Since the even part of the Lie superalgebra osp(1|2) is isomorphic to sl2(𝕂), we ask whether Burde's structures can be extended to compatible left-symmetric superalgebra structures on osp(1|2). We verify Burde's left-symmetric structures on sl2(𝕂), and then test their possible extensions to osp(1|2). We show that no such extension exists in characteristic 3. Together with the obstruction coming from the even part for characteristic $0$ and p > 3, this gives non-existence for a field 𝕂 of characteristic different from $2$.
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BibTeX
@misc{loza2026nonexistence,
title = {{On the non-existence of a left-symmetric structure on osp(1|2) in any characteristic}},
author = {Josemaria Loza},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12489},
eprint = {2610.12489},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12489},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the non-existence of a left-symmetric structure on osp(1|2) in any characteristic
AU - Josemaria Loza
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.12489
UR - https://arxiv.org/abs/2610.12489
AB - Burde proved that the Lie algebra sl_2(𝕂), over a field 𝕂 of characteristic p > 2, admits compatible left-symmetric algebra structures if and only if p = 3. Since the even part of the Lie superalgebra osp(1|2) is isomorphic to sl_2(𝕂), we ask whether Burde's structures can be extended to compatible left-symmetric superalgebra structures on osp(1|2). We verify Burde's left-symmetric structures on sl_2(𝕂), and then test their possible extensions to osp(1|2). We show that no such extension exists in characteristic 3. Together with the obstruction coming from the even part for characteristic $0$ and p > 3, this gives non-existence for a field 𝕂 of characteristic different from $2$.
ER -