Weak G-identities for the Pair \((M_2( \mathbb {C}),sl_2( \mathbb {C}))\)

Abstract

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.