Matrix bosonic realizations of a Lie colour algebra with three generators and five relations of Heisenberg Lie type (Q2655930)

The motivation of this paper is that representations of three-dimensional Lie algebras play an important role in the representation theory of general Lie algebras, and people expect the same to be true for three-dimensional Lie colour algebras. This paper studies the realizations of a Lie colour algebra with three generators \(A_1\), \(A_2\), \(A_3\) and five relations: \[ \begin{gathered} A_1A_2+A_2A_1= A_3\tag{1}\\ A_1A_3+A_3A_1= 0\tag{2}\\ A_2A_3-A_3A_2= 0\tag{3}\\ A_2^2= 0\tag{4}\\ A_3^2= 0\tag{5} \end{gathered} \] The author shows that with a natural choice for \(A_1\) as the first generator of the Heisenberg algebra corresponding to differentiation and \(A_2\), \(A_3\) are power series in Heisenberg generators, then (1)-(5) hold if and only if \(A_2=A_3=0\). Moreover, the author constructs nontrivial realizations of (1)-(5) using two \(2\times2\) matrices with entries chosen as formal power series in the noncommuting indeterminates \(A\) and \(B\) satisfying Heisenberg's canonical commutation relations: \[ AB-BA = I \] In the end, the author constructs the concrete operator representations by applying the above technique to the canonical representation of Heisenberg canonical commutation relations and to the simple quantum mechanical harmonic oscillator.