Identities for the special linear Lie algebra with the Pauli and Cartan gradings

Abstract

Let $$\mathbb{K}$$ be a field of characteristic zero. We study the graded identities of the special linear Lie algebra with the Pauli and Cartan gradings. Given a prime number p we provide a finite basis for the graded identities of $$s{l_p}\left(\mathbb{K}\right)$$ with the Pauli grading by the group ℤp × ℤp and compute its graded codimensions. We also prove that $${{\mathop{\rm var}} ^{{\mathbb{Z}_p} \times {\mathbb{Z}_p}}}\left( {s{l_p}\left(\mathbb{K} \right)} \right)$$ is a minimal variety and satisfies the Specht property. As a by-product we determine a basis for the identities of certain graded Lie algebras with a grading in which every homogeneous subspace has dimension ≤ 1. For $$s{l_m}\left(\mathbb{K}\right)$$ with the Cartan grading a finite basis for the graded identities is determined, moreover a basis for the subspace of the multilinear polynomials in the relatively free algebra $$L\left\langle {{X_G}} \right\rangle /{T_G}\left( {s{l_m}\left( \right)} \right)$$ , as a vector space, is exhibited. As a consequence we compute the graded codimensions for m = 2 and provide bases for the graded identities and for the subspace of the multilinear polynomials in the relatively free algebra of certain Lie subalgebras of $${M_m}{\left(\mathbb{K}\right)^{\left( - \right)}}$$ with the Cartan grading.