WEAK $G$-IDENTITIES FOR THE PAIR $(M_2( \mathbb{C}),sl_2( \mathbb{C}))$

In this paper we study algebras acted on by a finite group $G$ and the corresponding $G$-identities. Let $M_2( \mathbb{C})$ be the $2\times 2$ matrix algebra over the field of complex numbers $\mathbb{C}$ and let $sl_2( \mathbb{C})$ be the Lie algebra of traceless matrices in $M_2( \mathbb{C})$. Assume that $G$ is a finite group acting as a group of automorphisms on $M_2( \mathbb{C})$. These groups were described in the Nineteenth century, they consist of the finite subgroups of $PGL_2( \mathbb{C})$, which are, up to conjugacy, the cyclic groups $\mathbb{Z}_n$, the dihedral groups $D_n$ (of order $2n$), the alternating groups $A_4$ and $A_5$, and the symmetric group $S_4$. The $G$-identities for $M_2( \mathbb{C})$ were described by Berele. The finite groups acting on $sl_2( \mathbb{C})$ are the same as those acting on $M_2( \mathbb{C})$. The $G$-identities for the Lie algebra of the traceless $sl_2( \mathbb{C})$ were obtained by Mortari and by the second author. We study the weak $G$-identities of the pair $(M_2( \mathbb{C}), sl_2( \mathbb{C}))$, when $G$ is a finite group. Since every automorphism of the pair is an automorphism for $M_2( \mathbb{C})$, 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 $(M_2( \mathbb{C}), sl_2( \mathbb{C}))$ when $G$ is a finite group acting as a group of automorphisms.