A lie algebra related to the universal Askey-Wilson algebra

• Main Author: Cantuba, Rafael Reno S.
• Format: text
• Language: English
• Published: Animo Repository 2016
• Subjects: Lie algebras; Algebra; Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/etd_doctoral/474
• Institution: De La Salle University

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.