## On homomorphisms of diagonal Lie algebras

- Diagonal Lie algebras are defined as direct limits of finite-dimensional Lie algebras under diagonal injective homomorphisms. An explicit description of the isomorphism classes of diagonal locally simple Lie algebras is given in the paper [A. A. Baranov, A. G. Zhilinskii, Diagonal direct limits of simple Lie algebras, Comm. Algebra, 27 (1998), 2749-2766]. The three finitary infinite-dimensional Lie algebras sl(1), so(1), and sp(1) are important special cases of diagonal locally simple Lie algebras. Many classical results have been extended to these three infinite-dimensional Lie algebras. In particular, in the paper [I. Dimitrov, I. Penkov, Locally semisimple and maximal subalgebras of the finitary Lie algebras gl(1), sl(1), so(1), and sp(1), J. Algebra 322 (2009), 2069-2081] all locally semisimple subalgebras of g $= sl(1), so(1), and sp(1) are described, and moreover all injective homomorphisms s ! g are described in terms of the action of s on the natural and the conatural g-modules. The present dissertation makes a substantial contribution to further extending these results to the class of diagonal locally simple Lie algebras.
In Chapter 3 all locally simple Lie subalgebras of any diagonal locally simple Lie algebra are described up to isomorphism. The main result of the dissertation, Theorem 3.1.11, provides a list of conditions under which there exists an injective homomorphism s ! g of a locally simple Lie algebra s into a diagonal locally simple Lie algebra g.
In Chapter 4, with Ivan Penkov, we study certain invariants of homomorphisms of diagonal locally simple Lie algebras. The ideas and partial results presented in this Chapter may lead to a description of such homomorphisms in the future.