## Dissertations

#### Title

A lie algebra related to the universal Askey-Wilson algebra

2016

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics

#### Subject Categories

Algebra | Mathematics

#### College

College of Science

#### Department/Unit

Mathematics and Statistics Department

Arlene A. Pascasio

#### Abstract/Summary

Let F denote an algebraically closed eld. Denote the three-element set by X = fA B Cg, and let F hX i denote the free unital associative F-algebra on X. Fix a nonzero q 2 F such that q4 6= 1. The universal Askey-Wilson algebra is the quotient space F hX i =I, where I is the two-sided ideal of F hX i generated by the nine elements UV {u100000} V U, where U is one of A B C, and V is one of (q + q{u100000}1)A + qBC {u100000} q{u100000}1CB q {u100000} q{u100000}1 (q + q{u100000}1)B + qCA {u100000} q{u100000}1AC q {u100000} q{u100000}1 (q + q{u100000}1)C + qAB {u100000} q{u100000}1BA q {u100000} q{u100000}1 : Turn F hX i into a Lie algebra with Lie bracket [X Y ] = XY {u100000} Y X for all X Y 2 F hX i. Let L denote the Lie subalgebra of F hX i generated by X, which is also the free Lie algebra on X. Let L denote the Lie subalgebra of generated by A B C. Since the given set of de ning relations of are not in L, it is natural to conjecture that L is freely generated by A B C. We give an answer in the negative by showing that the kernel of the canonical map F hX i ! has a nonzero intersection with L. Denote the span of all Hall basis ele- ments of L of length n by Ln, and denote the image of Pn i=1 Li under the canonical map L ! L by Ln. We show that the simplest nontrivial Lie algebra relations on L occur in L5. We exhibit a basis for L4, and we also exhibit a basis for L5 if q is not a sixth root of unity.

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English

Electronic

CDTG006752

#### Shelf Location

Archives, The Learning Commons, 12F Henry Sy Sr. Hall

#### Physical Description

1 computer optical disc; 4 3/4 in

Lie algebras

COinS