Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6414592 | Journal of Algebra | 2014 | 19 Pages |
We prove two structure theorems for simple, locally finite dimensional Lie algebras over an algebraically closed field of characteristic p which give sufficient conditions for the algebras to be of the form [R(â),R(â)]/(Z(R)â©[R(â),R(â)]) or [K(R,â),K(R,â)] for a simple, locally finite dimensional associative algebra R with involution â. The first proves that a condition we introduce, known as locally nondegenerate, along with the existence of an ad-nilpotent element suffice. The second proves that an ad-integrable Lie algebra is of this type if the characteristic of the ground field is sufficiently large. Lastly we construct a simple, locally finite dimensional associative algebra R with involution â such that K(R,â)â [K(R,â),K(R,â)] to demonstrate the necessity of considering the commutator in the first two theorems.