Jordan-Lie Inner Ideals of Finite Dimensional Associative Algebras

A subspace B of a Lie algebra L is said to be an inner ideal if [B, [B,L]] ⊆ B. Suppose that L is a Lie subalgebra of an associative algebra A. Then an inner ideal B of L is said to be Jordan-Lie if B2 = 0. In this thesis, we study Jordan-Lie inner ideals of finite dimensional associative algebras (with involution) and their corresponding Lie algebras over an algebraically closed field F of characteristic not 2 or 3. Let A be a finite dimensional associative algebra over F. Recall that A becomes a Lie algebra A(-) under the Lie bracket defined by [x,y] = xy - yx for all x,y ∈ A. Put A(0) = A(-) and A(k) = [A(k-1),A(k-1)] for all k ≥ 1. Let L be the Lie algebra A(k) (k ≥ 0). In the first half of this thesis, we prove that every Jordan-Lie inner ideal of L admits Levi decomposition. We get full classification of Jordan-Lie inner ideals of L satisfying a certain minimality condition. In the second half of this thesis, we study Jordan-Lie inner ideals of Lie subalgebras of finite dimensional associative algebras with involution. Let A be a finite dimensional associative algebra over F with involution * and let K(1) be the derived Lie subalgebra of the Lie algebra K of the skew-symmetric elements of A with respect to *. We classify * -regular inner ideals of K and K(1) satisfying a certain minimality condition and show that every bar-minimal * -regular inner ideal of K or K(1) is of the form eKe* for some idempotent e in A with e*e = 0. Finally, we study Jordan-Lie inner ideals of K(1) in the case when A does not have “small” quotients and show that they admit *-invariant Levi decomposition.