Israel Journal of Mathematics | 2021
Identities for the special linear Lie algebra with the Pauli and Cartan gradings
Abstract
Let $$\\mathbb{K}$$ 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 $$s{l_p}\\left(\\mathbb{K}\\right)$$ s l p ( K ) with the Pauli grading by the group ℤ p × ℤ p and compute its graded codimensions. We also prove that $${{\\mathop{\\rm var}} ^{{\\mathbb{Z}_p} \\times {\\mathbb{Z}_p}}}\\left( {s{l_p}\\left(\\mathbb{K} \\right)} \\right)$$ var ℤ p × ℤ p ( s l p ( K ) ) 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 $$s{l_m}\\left(\\mathbb{K}\\right)$$ s l m ( K ) 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 $$L\\left\\langle {{X_G}} \\right\\rangle /{T_G}\\left( {s{l_m}\\left( \\right)} \\right)$$ L 〈 X G 〉 / T G ( s l m ( K ) ) , 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 $${M_m}{\\left(\\mathbb{K}\\right)^{\\left( - \\right)}}$$ M m ( K ) ( − ) with the Cartan grading.