Description of 2-local derivations on some Lie rings of skew-adjoint matrices

TL;DR

This paper proves that all 2-local derivations on certain Lie rings of skew-adjoint matrices are actually inner or spatial derivations, extending the understanding of derivation structures in these algebraic systems.

Contribution

It establishes that 2-local derivations on Lie rings of skew-adjoint matrices are inner, and extends this result to infinite-dimensional cases of matrix-valued maps.

Findings

Every 2-local inner derivation on the Lie ring of skew-symmetric matrices is an inner derivation.

Every 2-local spatial derivation on infinite-dimensional skew-adjoint matrix-valued maps is a spatial derivation.

The techniques used can be applied to various Lie algebras of infinite-dimensional skew-adjoint matrices.

Abstract

In the present paper we prove that every 2-local inner derivation on the Lie ring of skew-symmetric matrices over a commutative ring is an inner derivation. We also apply our technique to various Lie algebras of infinite dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation.