Lie polynomials in q-deformed Heisenberg algebras

An extension of a q-deformed Heisenberg algebra and its Lie polynomials

“…Denote by L(q) the Lie subalgebra of H (q) generated by A, B. Following the notation in [3,Section 5], if q is nonzero and not a root of unity, we denote H (q) by H (q), and L(q) by L(q). Then by [2, Lemma 5.1] the elements in (2), except any power of A or B with exponent not equal to 1, can be expressed as elements of L(q).…”

“…Furthermore, the above elements form a basis for L(q) [3, Theorem 5.8]. It was also shown in [3] that L(q) is a Lie ideal of H (q). Thus, if we compute the Lie bracket of an element in (30) with an element in (2), whether the latter is in (30) or not, then the result is a linear combination of (30).…”

“…Thus, if we compute the Lie bracket of an element in (30) with an element in (2), whether the latter is in (30) or not, then the result is a linear combination of (30). A further result in [3] is that the resulting quotient Lie algebra H (q)/L(q) is one-step nilpotent. This implies that the Lie bracket of any two elements in (2) that are both not in (30) can be expressed uniquely as a linear combination of (30).…”

“…This implies that the Lie bracket of any two elements in (2) that are both not in (30) can be expressed uniquely as a linear combination of (30). The Lie polynomials for the case q = 0 are discussed in [3,Section 4], while the case q = 1 leads to a trivial low-dimensional Lie algebra as discussed in [3, Section 1].…”

“…The Lie subalgebras for q-deformed Heisenberg algebra H (q) when q is not a root of unity has been considered in [3] where especially the Lie subalgebra generated by generators A and B has been studied in more detail and in particular its bases and some properties have been described. This paper is devoted to Lie subalgebras for q-deformed Heisenberg algebras H (q) when q is a root of unity, which is a natural continuation of [3]. Specifically, for the Lie subalgebra generated by the generators A and B the basis is described in terms of A and B, and thus a characterization of elements of the Lie subalgebra generated by them is achieved.…”