<p>In this work we investigate the structure of this type of Lie algebras over arbitrary fields F by constructing them from their Abelian ideal. To accomplish this, an algorithm is developed and as application a classification up to 7-dimensional Lie Algebras is given. We discuss a recent example of financial mathematics as well.</p>

In this work we investigate the structure of this type of Lie algebras over arbitrary fields F by constructing them from their Abelian ideal. To accomplish this, an algorithm is developed and as application a classification up to 7-dimensional Lie Algebras is given. We discuss a recent example of financial mathematics as well.

No description available.

The results exhibited in this thesis are related to Schrodinger operators in three dimensions and are subdivided in two parts based on two published papers, [15] and [14]. A variant of Turbiner's conjecture is proved in the first paper while a partial classification of quasi-exactly solvable Lie algebras of first order differential operators in dimension three is exhibited in the second paper. This classification is then used to construct new quasi-exactly solvable Schrodinger operators in three dimensions. / Turbiner's conjecture posits that, for a Lie algebraic Schrodinger operator in dimension two, the Schrodinger equation is separable if the underlying metric is locally flat. This conjecture is false in general. However, if the generating Lie algebra is imprimitive and if a certain compactness requirement holds, Rob Milson proved that in two dimensions, the Schrodinger equation separates in a Cartesian or polar coordinate system. In [15], the first paper included in this thesis, a similar theorem is proved in three variables. The imprimitivity and compactness hypotheses are still necessary and another condition, related to the underlying metric, must be imposed. In three dimensions, the separation is only partial and the separation will occur in either a spherical, cylindrical or Cartesian coordinate system. / In the second paper [14], a partial classification of quasi-exactly solvable Lie algebras of first order differential operators is performed in three dimensions. Such a classification was known in one and two dimensions but the three dimensional case was still open before the beginning of this research. These new quasi-exactly solvable Lie algebras are used to construct new quasi-exactly solvable Schrodinger operators with the property that part of their spectrum can be explicitly determined. This classification is based on a classification of Lie algebras of vector fields in three variables due to Lie and Amaldi.

No description available.

No description available.

Thesis (Ph.D.)--York University, 2006. Graduate Programme in Mathematics. / Typescript. Includes bibliographical references (leaves 95-97). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR19776

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität, Bonn, 1986. / Includes bibliographical references (p. 139-142).

Thesis (Ph.D.) - University of St Andrews, November 2007.

In this thesis we study subalgebras in free nilpotent and polynilpotent Lie algebras. Chapter 1 sets up the notation and includes definitions and elementary properties of free and certain reduced free Lie algebras that we use throughout this thesis. We also describe a Hall basis of a free Lie algebra as in [4] and a basis for a free polynilpotent Lie algebra which was developed in [24]. In Chapter 2 we first consider the class of nilpotency of subalbebras of free nilpotent Lie algebras starting with two-generator subalgebras. Then we study those subalgebras in a free nilpotent Lie algebra which, are themselves free nilpotent. We give necessary and sufficient conditions in the case of two-generator subalgebras. Chapter 3 extends the results obtained in Chapter 2 to the polynilpotent case. First we look at two-generator subalgebras of a free polynilpotent Lie algebra. Then we consider more general subalgebras. Finally we study those subalgebras which are themselves free polynilpotent and give necessary and sufficient conditions for two-generator subalgebras to be free polynilpotent. In Chapter 4 we first study certain properties of ideals in free, free nilpotent and free polynilpotent Lie algebras and establish the fact that in a free polynilpotent Lie algebra a nonzero ideal which is finitely-generated as a subalgebra must be equal to the whole algebra. Then we consider the quotient Lie algebra of a lower central term of a free Lie algebra by a term of the lower central series of an ideal. We then generalize the results to cover the free nilpotent and free polynilpotent cases. In the last section of Chapter 4 we consider ideals of free nilpotent (and later polynilpotent) Lie algebras as free nilpotent (polynilpotent) subalgebras and establish the fact that in most non-trivial cases such an ideal cannot be free nilpotent (polynilpotent). In the last chapter we consider the m+k-th term of the lower central series of a free Lie algebra as a subalgebra of the m-th term for m ⩽ k and generalize the results proved in [25]. We give reasons for the failure of these results in the case m > k.