An extension of a q-deformed Heisenberg algebra and a solution to the corresponding Lie polynomial characterization problem - Merciales, Mark Anthony C. (Animo Repository, 2018)
Let F be a field and fix a q ϵ F. The q-deformed Heisenberg algebra H(q) is the unital associative algebra over F with generators A, B and relation AB - qBA = I, where I is the multiplicative identity in H(q). We extend H(q) into an algebra R(q) defined by generators A, B and a relation which asserts that AB - qBA is central in R(q). We introduce a new generator γ:= AB - qBA and give an alternate presentation for R(q) by generators and relations; the generators are A, B, γ. We show that the vectors γhBmAn (h, m, n ϵ N) form a basis for the algebra R(q). We introduce a new generator C := [A,B] = AB - BA and give another presentation for R(q) by generators and relations; the generators are A, B, C, γ. We also show that given nonzero q 6= 1, the vectors γhCkBl, γhCkAt (h, k, l ϵ N; t ϵ Z+) form a basis for the algebra R(q). We determine the equivalent expression of AnBm for any natural number m and n, as linear combination of basis elements wherein scalar coefficients involved are related to q-special combinatorics. Also, we investigate properties of R(q) involving the Lie bracket. We denote by L the Lie subalgebra R(q) generated by A, B. We show that the vectors γhCnAm, γhBmCn, {n + 1}qγhCn+1 - nqγh+1Cn, A, B, C (m; n ϵ Z+; h ϵ N) form a basis for the Lie subalgebra L.