The Automorphism For The Derived Lie Subalgebra Of Gl₂(C_q²)

Given a non-zero non-root of unity q. Denote by gl₂'(C_q²) the derived Lie sub-algebraof gl₂(C_q²). In this paper, we first recall a realization of gl₂(C_q²) given in the paper[8], and then determine all the automorphisms of gl₂'(C_q²) which preserve a certain Cartan subalgebra of gl₂'(C_q²) invariant. As a consequence, we also determineall of its graded automorphisms of gl₂'(C_q²). Finally, we study the structure of the subgroup consisting of such kind of automorphisms of gl₂'(C_q²).