WEAK G-IDENTITIES FOR THE PAIR (M₂( C),sl₂( C))

In this paper we study algebras acted on by a finite group G and the corresponding G-identities. Let M₂( C) be the 2× 2 matrix algebra over the field of complex numbers C and let sl₂( C) be the Lie algebra of traceless matrices in M₂( C). Assume that G is a finite group acting as a group of automorphisms on M₂( C). These groups were described in the Nineteenth century, they consist of the finite subgroups of PGL₂( C), which are, up to conjugacy, the cyclic groups Zₙ, the dihedral groups Dₙ (of order $2n$), the alternating groups A₄ and A₅, and the symmetric group S₄. The G-identities for M₂( C) were described by Berele. The finite groups acting on sl₂( C) are the same as those acting on M₂( C). The G-identities for the Lie algebra of the traceless sl₂( C) were obtained by Mortari and by the second author. We study the weak G-identities of the pair (M₂( C), sl₂( C)), when G is a finite group. Since every automorphism of the pair is an automorphism for M₂( C), it follows from this that G is one of the groups above. In this paper we obtain bases of the weak G-identities for the pair (M₂( C), sl₂( C)) when G is a finite group acting as a group of automorphisms.