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.