Local derivation on the Schro¨dinger Lie algebra in (n+1)-dimensional space-time
DOI:
https://doi.org/10.31489/2026m1/55-66Keywords:
Lie algebras, semisimple Lie algebra, solvable Lie algebra, nilpotent Lie algebra, Schr¨odinger algebras, inner derivations, derivations, local derivationsAbstract
This paper investigates local derivations on the Schro¨dinger Lie algebra sn, the Lie algebra of the (n+1)dimensional space-time Schro¨dinger group. As a finite-dimensional Lie algebra that is neither semisimple nor solvable, the Schr¨odinger algebra plays an important role in mathematical physics, particularly as the symmetry algebra of the free Schr¨odinger equation. While local derivations are well understood for semisimple, solvable, and certain infinite-dimensional Lie algebras, much less is known for non-semisimple algebras. We prove that for all integers n≥ 3, every local derivation on sn is a derivation. Our approach uses the explicit structure of the Schro¨dinger algebra together with a detailed description of its derivation algebra. First, we reduce the problem to derivations that act trivially on the semisimple part, and then we perform a coefficient-wise analysis in a fixed basis. This shows that every local derivation is an ordinary derivation. Moreover, such derivations decompose in the usual way into inner derivations and the known outer derivations. Our result extends earlier low-dimensional cases and shows a uniform rigidity phenomenon for all higher-dimensional Schro¨dinger algebras.
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