Bivariate polynomial mappings associated with simple complex Lie algebras

There are three families of bivariate polynomial maps associated with the rank-2 simple complex Lie algebras A2,B2≅C2A2,B2≅C2 and G2G2. It is known that the bivariate polynomial map associated with A2A2 induces a permutation of Fq2 if and only if gcd⁡(k,qs−1)=1gcd⁡(k,qs−1)=1 for s=1,2,3s=1,2,3. In this paper, we give similar criteria for the other two families. As an application, a counterexample is given to a conjecture posed by Lidl and Wells about the generalized Schur's problem.