Bulletin of the
Korean Mathematical Society BKMS

We define subalgebras $V_q^{mZ \times nZ}$ of $V_q$ where $V_q$ are in the paper \cite{KPS}. We show that the Lie algebra $V_q^{mZ \times nZ}$ is simple and maximally abelian decomposing. We may define a Lie algebra is maximally abelian decomposing, if it has a maximally abelian decomposition of it. The $F$-algebra automorphism group of the Laurent extension of the quantum plane is found in the paper \cite{KPS}, so we find the Lie automorphism group of $V_q^{mZ\times nZ}$ in this paper.

Keywords: simple Lie algebra, maximally abelian decomposing, algebra automorphism, Lie automorphism, isomorphism

MSC numbers: Primary 17B40, 17B56