**Graduate #66**

**Discipline:**Mathematics and Statistics

**Subcategory:**Mathematics and Statistics

**Juan M. Escobar Salsedo**

**- California State University, Los Angeles**

**Co-Author(s):**Crystal Salas, Sonoma State University, Rohnert Park CA ; Terris Becker, Sonoma State University. Rohnert Park C; Dr. Rustam Turdibaev, INHA University, Tashkent, Uzbekistan

A linear map ∆ is said to be a local automorphism if for all x there exists some automorphism φ(x), such that ∆(x) = φx(x). It has been proven for an associative algebra of square matrices Mn(R) that any local automorphism is either an automorphism or an anti-automorphism [1]. Furthermore, Ayupov et. al. have classified local automorphisms of simple Leibniz algebras proving that if ∆ is a local automorphism over a complex simple Leibniz algebra L = G+I, then ∆ is an automorphism if and only if its Lie part ∆G,G is an automorphism [2]. This result simplifies the classification of local automorphisms of simple Leibniz algebras down to the Lie case and gives us reason to classify local automorphisms of Lie algebras. In our work we consider the simple Lie algebra of traceless n x n matrices (sln). There is a well-known classification of automorphisms of simple Lie algebras which states that over an algebraically closed field of characteristic 0, the group of automorphisms of the Lie algebra sl2 is the set of mappings X → A−1XA and the group of automorphisms of the Lie algebra sln (n ≥ 3) is the set of mappings X → A−1XA and X → −AX⊤A−1 [3]. Using this fact, our motivation is to classify local automorphisms of the simple Lie algebra sln and see if it is possible to obtain results similar to those for associative algebras. For the Lie algebra of 2 x 2 matrices with trace zero, sl2, we are able to classify all local automorphisms that fix the basis element h = [{1, 0}, {0 −1}] and we obtain the following result: Theorem: The only local automorphisms of sl2 are all automorphisms and all anti-automorphism. We also construct a local automorphism for sln (n ≥ 3) that is neither an automorphism nor an anti-automorphism and prove the following collorary: Collorary: For n ≥ 3 we prove that AUT±(sln) is a proper subgroup of LAut(sln) of an infinite index. We do not have a full description of the group LAut(sln), but we find that obtaining a full description is a much bigger problem that may or may not be possible of obtaining. Thus, we find different results from those for associative algebras. If we do obtain a full description, we can find if this group is algebraic or not. REFERENCES 1. D. R. Larson, A. R. Sourour, Local derivations and local automorphisms of B(X), Proc. Sympos. Pure Math. 51 (1990) 187-194. 2. S. Ayupov, K. Kudaybergenov, B. Omirov, Local and 2-local Derivations and Automorphisms on Simple Leibniz Algebras, arXiv:1703.10506 [math.RA]. 3. N. Jacobson, Lie algebras, Interscience Publishers, Wiley, New York, 1962. Funding Acknowledgements The first three authors were supported by the National Science Foundation, project number 1658672.

EscobarSalsedoJuan-On Local Automorphisms of .pdf**Funder Acknowledgement(s):** The first three authors were supported by the National Science Foundation, project number 1658672.

**Faculty Advisor: **Rustam Turdibaev,
r.turdibaev@inha.uz

**Role:** We all individually proved Theorem 1.
I came up with an example of a local automorphism of SL3 that was neither an automorphism nor an anti-automorphism.
With the latter result, Dr. Turdibaev and I came up with a generalized example for all SLn, where n is greater than or equal to 3, of a local that is neither an automorphism nor an anti-automorphism.
Terris, Crystal and myself each showed that our example was indeed correct.