Rota-Baxter operators and skew left brace structures over Heisenberg group

Abstract

Rota-Baxter operators on groups have been recently defined in [L. Guo, H. Lang and Y. Sheng, Integration and geometrization of Rota-Baxter Lie algebras, Adv. Math. 387 (2021) 107834], and they share a close connection with skew braces, as demonstrated in [V. Bardakov and V. Gubarev, Rota-Baxter groups, skew left braces, and the Yang-Baxter equation, J. Algebra 587 (2022) 328-351]. In this paper, we classify all Rota-Baxter operators of weight 1 on the Heisenberg Lie algebra of dimension 3 up to the Jordan canonical form of a 2 × 2 matrix block by automorphisms of the Heisenberg Lie algebra. Using the fact that the exponential map from the Heisenberg Lie algebra to the Heisenberg group is bijective, we induce these operators on the Heisenberg group. Finally, we enumerate all skew left brace structures on the Heisenberg group induced by these Rota-Baxter operators.