ABSTRACT

In this paper we study Z x Z-graded Lie algebras A = ∑ A i , j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187674/f8337d7b-6b74-4da5-ae84-c225ed353b0a/content/in259_1.tif"/> with dim Aij ≤ 1, where A0-1 1 A0,0, A0,1 span a Heisenberg algebra H, and [A-1,0, A1,0] = 0. We show that there are no such algebras satisfying the additional conditions (i) that A is generated by A±1)0 and A0±1, (ii) A0,0 is not in the center of A, and (iii) that H ∩ [ A , A ] ≠ A 0 , 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187674/f8337d7b-6b74-4da5-ae84-c225ed353b0a/content/in259_2.tif"/> . If Condition (iii) is removed, then there are many such Lie algebras, but they do not seem interesting because of their weaker structure.