Graded identities for Kac–Moody and Heisenberg algebras with the Cartan grading

Abstract

Kac–Moody algebras, g(A), are Lie algebras defined by generators and relations given by generalized Cartan matrices A. In this paper, we study the graded identities for Kac–Moody algebras when the matrix A is diagonal. More precisely, we provide a basis for the graded identities of g(A) equipped with its natural grading, the grading of Cartan type. These results are obtained over an arbitrary infinite field. We also compute the graded codimensions for these algebras and provide a basis for the vector space of the multihomogeneous polynomials of any given multidegree in the relatively free algebra. As the base field is infinite we have a vector space basis of the relatively free algebra. As a consequence of our results, we give an alternative proof of Theorem 17 in [15], and generalize it to characteristic two. Finally, we also describe a basis of the graded identities for the Heisenberg algebra with its natural grading, over any field.