On the non-existence of a left-symmetric structure on osp(1|2) in any characteristic

Josemaria Loza^* (corresponding author)

* Corresponding author: Josemaria Loza.

College Mathematics, Vol 1, Issue 1 (Winter 2026), pp. 21-54.

Open access. © 2026 the author(s). Published under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

This article belongs to an issue that is still in progress. Bibliographic details may change before formal publication in December, 2026.

Abstract

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), 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.

How to cite

Josemaria Loza. “On the non-existence of a left-symmetric structure on osp(1|2) in any characteristic.” College Mathematics 1, no. 1 (2026): 21-54.