The talk is devoted to description of local and 2-local derivations (respectively, automorphisms) on octonian algebras over fields with zero characteristics. We shall give a general form of local derivations on the real octonion algebra O(ℝ). This description implies that the space of all local derivations on O(ℝ) when equipped with Lie bracket is isomorphic to the Lie algebra 𝔰𝔬7(ℝ) of all real skew-symmetric 7 × 7-matrices. We also consider 2-local derivations on the octonion algebra O(F) over an algebraically closed field F and prove that every 2-local derivation on O(F) is a derivation. Further, we apply these results to problems for the simple 7-dimensional Malcev algebra. As a corollary we obtain that the real octonion algebra O(ℝ) and Malcev algebra M7(ℝ) are simple non-associative algebras which admit pure local derivations, that is, local derivations which are not derivation. Further, we shall give a general form of local automorphisms on the octonion algebra O(F) over a field F. This description implies that the group of all local automorphisms on O(F) is isomorphic to the group O7(F) of all orthogonal 7 × 7-matrices over F. We also consider 2-local automorphisms on the octonion algebra O(F) over an algebraically closed field F and prove that every 2-local automorphism on O(F) is an automorphism. As a corollary we obtain descriptions of local and 2-local automorphisms of seven dimensional simple Malcev algebra.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
