عنوان

Jordan-Lie inner ideals of finite dimensional associative algebras

Number

TLets755336

Title Proper

Jordan-Lie inner ideals of finite dimensional associative algebras

General Material Designation

[Thesis]

First Statement of Responsibility

Shlaka, Hasan Mohammed Ali Saeed

Subsequent Statement of Responsibility

Baranov, Alexander ; Pirashvili, Teimuraz

Name of Publisher, Distributor, etc.

University of Leicester

Date of Publication, Distribution, etc.

2018

Dissertation or thesis details and type of degree

Thesis (Ph.D.)

Text preceding or following the note

2018

Text of Note

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.

Shlaka, Hasan Mohammed Ali Saeed

Baranov, Alexander ; Pirashvili, Teimuraz

University of Leicester

Electronic name

p

[Thesis]

276903

a

Y